Download WARM-UPS - Institut Pere Fontdevila

1999–2000 MATHCOUNTS School Handbook
WARM-UPS
Answers to the Warm-Ups include one-letter codes, in parentheses, indicating appropriate problem
solving strategies, as desribed in the Problem Solving section. It should be noted that the strategies
indicated may not be the only applicable strategies. A calculator icon indicates problems which may be
more easily solved with a calculator.
The following codes will be used in the answer keys:
(C)
(F)
(M)
(T)
(G)
(S)
(E)
(P)
Compute or Simplify
Use a Formula
Make a Model or Diagram
Make a Table, Chart or List
Guess, Check and Revise
Consider a Simpler Case
Eliminate
Look for Patterns
The answer key to each Warm-Up appears on the following page. A detailed solution to one of the ten
problems is also provided on the accompanying answer key, and, as appropriate, a mathematical
connection to a problem or an investigation and exploration activity has been noted.
MATHCOUNTS Symbols and Notation
Standard abbreviations have been used for units of measure. Complete words or symbols are also
acceptable. Square units or cube units may be expressed as units2 or units3.
Typesetting of the MATHCOUNTS handbook and competition materials provided by EducAide Software, Vallejo, California.
WARM-UP 1
1.
What is the maximum number of 33/c stamps that can be
purchased with 5 dollars?
1.
2.
A rep-date occurs when the number formed by the number of the
month and the number of the day is the same as the last two digits
of the year. For example, 9/8/98 is a rep-date since 98 = 98. In
which year of the 21st century will the first rep-date occur?
2.
3.
How many different four-digit numbers can be formed using each
of the digits in 1999 exactly once?
3.
4.
Manu and Janani are playing a coin toss game with a fair penny.
Manu gets a point if the penny lands on heads, and Janani gets a
point if the penny lands on tails. The score is Janani 9, Manu 7,
in a game to 10 points. What is the probability that Janani will
win the game?
4.
5.
A cube has a surface area of 900 cm2 . What is the number of
cubic centimeters in the volume of the cube? Express your answer
in simplest radical form.
5.
6.
What is the degree measure of the indicated angle?
6.
7.
What is the units digit of 248 ?
7.
8.
The perimeter of a rectangle is 48 units, and its length is twice
its width. What is the number of square units in the area of the
rectangle?
8.
9.
What is the ratio of the number of degrees in the complement of a
60-degree angle to the number of degrees in the supplement of a
60-degree angle? Express your answer as a common fraction.
9.
10. All clocks on a NASA space shuttle are set to Mission Elapsed
Time (MET). The MET clock is set at midnight and begins when
the shuttle is launched. Hence, one hour after liftoff, the shuttle’s
clock reads 1:00 (MET). If a shuttle launches at 8:09 a.m., at what
time that afternoon will the shuttle clock read 4:55 (MET)?
10.
c MATHCOUNTS 19992000
ANSWER KEY
1.
15
(C)
2.
7
8
(TEP)
(SP)
(C)
5.
8.
4.
7. 6
10. 1:04
WARM-UP 1
2011
√
750 6
128
(EP)
3.
4
(FT)
(FM)
(M)
6.
9.
45
(M)
(FM)
1
4
SOLUTION
Problem #7
FIND OUT
What would we like to find? The units digit of the 48th power of 2.
CHOOSE A
STRATEGY
A good first attempt would be to simply enter the expression into a scientific calculator.
Unfortunately, most calculators only show eight digits in the display, and the value of this
expression contains fifteen digits. So that won’t work. An alternative is to try smaller powers
of 2 to search for a pattern and see where that leads.
SOLVE IT
The pattern of the units digit in powers of 2 is predictable. Notice that
21
22
23
24
25
has
has
has
has
has
units
units
units
units
units
digit
digit
digit
digit
digit
2,
4,
8,
6,
2,
and so on. The pattern of units digits is 2, 4, 8, 6, 2, 4, 8, 6, . . ., and it repeats every fourth power.
Since we want the units digit of 248 , and because 48 is a multiple of 4, the units digit will be
the same as it is for 24 . The answer is 6.
LOOK BACK Our logic makes sense, so we can have some confidence in our answer. An interesting extension
is to look for the units digit of powers of other numbers. The pattern of units digits for powers
of any digit 19 repeats in a similar way.
MAKING CONNECTIONS. . . to Biology
Problem #5
The surface area to volume ratio is of extreme importance in the study of cell size. The average human body
contains about 65 trillion cells, so it’s not surprising that individual cells are very small. Human red blood cells,
for instance, are typically only 7 to 8 microns in diameter. (A micron is one-millionth of a meter.) The diameter
of most cells fall within the narrow range of 10100 microns.
Why are cells so small? One factor is the size of the cell membrane. Cells obtain nutrients and eliminate waste
through the cell membrane. As a cell increases in size, its need for nutrients and production of waste increases.
Therefore, larger cells require a membrane with a larger surface area for the rapid exchange of materials to the
environment. As the size of a cell increases, its surface area to volume ratio decreases, thus making it difficult for
a large cell to transport needed nutrients in and harmful wastes out. Evolution has kept cell size relatively small
to facilitate these processes.
INVESTIGATION & EXPLORATION
Problem #7
An obvious pattern emerges when the units digit of the powers of a number are analyzed. Similarly, patterns can
be used to solve puzzles. Consider the classic Tower of Hanoi puzzle.
Three pegs are on a board. Three disks are on one peg: a large disk is at the bottom, a medium disk is in the
middle, and a small disk is on top. The object of the puzzle is to move all three disks to a different peg, while
obeying two simple rules:
• You may only move one disk at a time.
• You may never place a larger disk on a smaller disk.
Try to solve this puzzle. If your solution is as efficient as possible, it should only take you 7 moves. Now, increase
the number of disks to 4, then to 5, and more. Record your results of the least number of moves it takes to solve
the puzzle depending on the number of disks. Do you see a pattern? Can you explain the pattern?
c MATHCOUNTS 19992000
WARM-UP 2
1.
A telephone pole is supported by a steel cable which extends
from the top of the pole to a point on the ground 3 meters from
its base. When Leah walks 2.5 meters from the base of the pole
toward the point where the cable is attached to the ground, her
head just touches the cable. Leah is 1.5 meters tall. How many
meters tall is the pole? (Problem submitted by Jane Lataille,
P.E.)
1.
2.
In linear measure, 7 palms equal 1 cubit, and 28 digits equal
1 cubit. What is the number of cubits in 8 palms, 6 digits?
Express your answer as a mixed number.
2.
3.
Each new triangle shown below has one more dot per side than
the previous triangle. What is the total number of dots on the
triangle with 358 dots per side?
3.
4.
What percent of the quadrilaterals in the diagram below are
parallelograms?
4.
5.
Mikela drove 500 miles on her three-wheeler. She rotated a spare
tire with the other tires so that all four tires got the same amount
of wear. How many miles of wear did each tire accumulate?
5.
6.
Start with a positive integer; add 4; multiply by 2; subtract 3;
multiply by 2; add 2; divide by 4; subtract 3. If the final result
is 6, what was the value of the original integer?
6.
7.
Find the least prime number greater than 2000.
7.
8.
What is the mean of all three-digit numbers that can be created
using each of the digits 1, 2 and 3 exactly once?
8.
9.
Find the least integer value of x for which 2|x| + 7 < 17.
9.
10. What is the positive difference between the greatest and least
prime factors of 2000?
10.
c MATHCOUNTS 19992000
ANSWER KEY
WARM-UP 2
1.
9
(M)
2.
5
1 14
(CF)
3.
1071
(FSP)
4.
50
(MT)
5.
375
(C)
6.
6
(CMG)
7.
2003
(TE)
8.
222
(TP)
9.
−4
(CG)
10. 3
(TE)
SOLUTION
Problem #7
FIND OUT
We are asked to find the first prime number greater than 2000.
CHOOSE A
STRATEGY
There are many divisibility rules that can be used. Let’s take the odd numbers
greater
√
than 2000 in order and check them for divisibility by all primes less than 2500 = 50. (Why?)
Also note that every even number is divisible by 2, and not prime, so we don’t need to check
them. The first number we come to that is prime will be the answer.
SOLVE IT
We know that 2001 isn’t prime by using our divisibility rule for 3 (the sum of the digits is 3, so
2001 is divisible by 3).
To test if 2003 is prime, use the divisibility rules for some of the smaller primes. It’s obviously
not divisible by 3 (the sum of its digits is 5), nor by 5 (its units digit is 3), nor by 9 (again, the
sum of its digits is 5), nor by 11 (the first and third digit have a sum of 2, while the second
and fourth digit have a sum of 3). For larger prime numbers (and even for these ones, if you
don’t know the divisibility rules), a calculator could be used to check. None of the primes less
than 50 evenly divide 2003, so it is prime.
LOOK BACK Because 2001 and 2002 are not prime, 2003 is the least prime number greater than 2000.
MAKING CONNECTIONS. . . to Measurement
Problem #2
The cubit, palm and digit were actually the ancient Egyptians’ three linear units of measure. The cubit was the
length of a man’s forearm from the tip of his finger to his elbow. The palm was one-seventh of a cubit, and
the digit was one-fourth of a palm. Today, the word digit has different meanings. How do we usually use it in
mathematics? How is it being used in this measurement system? Measure your cubit, palm and digit. How do
your measurements compare to the Egyptian values?
The English system of measurementthe system still used in the United Statesis a bit more standardized than
the Egyptian system, but it is similarly confusing to use. When talking about linear distance, the basic unit is
the foot. A foot is divided into 12 inches, and inches are continually divided in half to form halves, quarters,
eighths, sixteenths, thirty-seconds and even sixty-fourths. Feet are also combined to form yards (3 feet) and
miles (5280 feet). Unlike the metric system, which is based on powers of 10, the conversions in the English and
Egyptian system evolved through tradition and appear to be somewhat arbitrary.
INVESTIGATION & EXPLORATION
Problem #6
You can use algebra to show why the following trick works:
Choose three different digits.
Add 3 to the first digit.
Multiply by 10.
Add the second digit.
Add 3.
Multiply by 10.
Add the last digit.
Subtract 330.
The result is a three-digit number consisting of the original digits.
Let’s say the digits chosen were p, q and r. The process then gives 10(p + 3) = 10p + 30 after the first three steps,
10(10p + 30 + q + 3) = 100p + 10q + 330 after the next three steps, and 100p + 10q + 330 + r − 330 = 100p + 10q + r
as the final result. Notice that the result is a three-digit number consisting of the three digits chosen.
Create a number trick of your own. Exchange your trick with a partner. Can you tell why your partner’s trick
works, and can your partner tell why your trick works?
c MATHCOUNTS 19992000
WARM-UP 3
1.
Express the reciprocal of 2.3 as a common fraction.
2.
For how many positive integers n will
3.
What is the median of the composite integers that are greater
than 20 and less than 35?
3.
4.
Evaluate ( 23 + 12 )−3 . Express your answer as a common fraction.
4.
5.
If May 1 falls on a Saturday, what is the sum of all the weekend
dates (Saturdays and Sundays) in May?
5.
6.
The radius of circle O is 12 inches, and AB and CD are tangent
to the circle at B and D, respectively. AB = 16 00 , and CD = 5 00 .
What is the sum of the number of inches in OC + OA?
6.
7.
The sum of three numbers is 81 and their ratio is 3 : 7 : 17. What
is the value of the smallest number?
7.
8.
What is the sum of the fifth prime number, the sixth composite
number, and the third perfect square?
8.
9.
The two arithmetic sequences, 1, 5, 9, 13, . . . and 1, 6, 11, 16, . . .,
have infinitely many terms in common. What is the sum of the
first three common terms?
9.
10. Rectangle ABCD lies in circle D
with AB = 6 cm and CE = 4 cm.
What is the number of centimeters
in the length of diagonal AC?
60
n
also be an integer?
1.
2.
10.
c MATHCOUNTS 19992000
ANSWER KEY
WARM-UP 3
(C)
2.
12
(TP)
3.
27
(CT)
4.
10
23
216
343
(C)
5.
155
(MT)
6.
33
(FM)
7.
9
(CG)
8.
32
(CT)
9.
63
(TP)
1.
10. 10
(M)
SOLUTION
Problem #2
FIND OUT
What do we want to know? The number of integer values that will result when 60 is divided by
a positive integer.
CHOOSE A
STRATEGY
The value of n must be positive, so we need to check values greater than 0. Using a bit more
logic, any value of n greater than 60 will yield a fraction with value less than 1 but greater
than 0; hence, we can exclude any value greater than 60. Consequently, let’s check all values
between 0 and 60.
SOLVE IT
Employing the strategy identified, and using a calculator, the integers 1, 2, 3, 4, 5, 6, 10, 12,
15, 20, 30 and 60 all yield an integer value when divided into 60. The answer, then, is 12.
LOOK BACK In reviewing the list above, it probably seems obvious that the numbers which yield an integer
value also happen to be the factors of 60. That shouldn’t be too surprising, because by
definition they are one and the samefor each integer factor of 60, there is a corresponding
integer co-factor (that is, an integer by which the factor can be multiplied to give 60).
Therefore, our answer of 12 must be correct.
MAKING CONNECTIONS. . . to the Calendar
Problem #5
There’s a poem that is supposed to help schoolchildren remember how many days in each month:
Thirty days hath September,
April, June, and November;
All the rest have thirty-one
Excepting February alone:
Which hath but twenty-eight, in fine,
Till leap year gives it twenty-nine.
Why is the number of days in February so different? The reason February has 29 days once every four years is
fairly easy to explain. The amount of time it takes the Earth to orbit the Suna yearis slightly longer than
365 days; it’s actually about 365.2422 days. Adding a day every four years roughly puts the calendar right. But
why does February have only 28 instead of 30 or 31 days like the other months?
According to a Basque legend, a shepherd in the hills of Euskal Herria was thankful because he had not lost many
sheep one season. The shepherd thanked the elements: March Weather, you killed none of my sheep this year,
and for that I thank you. But March Weather was proud of his fierce reputation, and he was angry that he
might lose that reputation; so, he stole two days from February so that he might have more time to be fierce, and
since then, February has had only 28 days.
That’s just a legend, however. The truth is that February originally had 30 days in leap years, and it had 29 days
in other years. August, named after the Roman emperor Augustus, originally had 30 days, too. In an egotistical
act by Augustus, he increased the number of days in August to 31, so that it would have as many days as July,
the month named for his predecessor, Julius Caesar. The extra day was taken from February.
INVESTIGATION & EXPLORATION
Problem #8
With a calculator, find the sum of the first seven odd positive integers; that is, find the sum
1 + 3 + 5 + 7 + 9 + 11 + 13. To that, add 15; then, add 17, 19, 21, 23, and so on. What pattern develops?
While working with square numbers, you may notice some other interesting facts. For instance, no square number
has a units digit of 2, 3, 7 or 8. You may also notice that, for every square number n2 , either n2 − 1, n2 or n2 + 1
is divisible by 3. Can you find any other integers which always divide either n2 − 1, n2 or n2 + 1, regardless of the
integer n?
c MATHCOUNTS 19992000
WARM-UP 4
1.
A brick mantel over a fireplace consists of rectangles as shown.
What is the total number of rectangles in the pattern?
1.
2.
Jared has nine coins in his pocket. They all look alike, but
one coin is counterfeit and weighs less than the others. What
is the least number of weighings on a balance scale needed to
guarantee that the counterfeit coin is found?
2.
3.
Compute: 4 + 5 − 7 × 9 ÷ 3. (Problem submitted by mathlete
Marc Costanzo.)
3.
4.
The first term of an arithmetic sequence is 15, and the seventh
term is 57. What is the third term of the sequence?
4.
5.
The perimeter of a square lot is lined with trees, and there are
three yards between the centers of adjacent trees. There are eight
trees on a side, and a tree is at each corner. What is the number
of yards in the perimeter of the lot?
5.
6.
What is the least whole number that is divisible by 7, but leaves a
remainder of 1 when divided by any integer 2 through 6?
6.
7.
Two small circles with radii 2 cm
and 3 cm are externally tangent. A
third circle is circumscribed about
the first two as shown. What is
the ratio of the area of the smallest
circle to the area of the shaded
region? Express your answer as a
common fraction.
7.
8.
What is the median of all values defined by the expression 2x − 1,
where x is a prime number between 0 and 20?
8.
9.
What is the sum of the integer solutions to |x + 2| < 5?
9.
10. At Agnesi Middle School, Mr. Eye, Mr. Love and Mr. Problems
teach science, mathematics, and historybut not necessarily in
that order. The history teacher, who was an only child, has the
least experience. Mr. Problems, who married Mr. Eye’s sister, has
more experience than the science teacher. Who teaches science?
10.
c MATHCOUNTS 19992000
ANSWER KEY
WARM-UP 4
1.
165
(TP)
2.
2
(MEP)
3.
−12
(C)
4.
7.
29
(FTP)
(FM)
5.
8.
84
1087
(FM)
(CF)
6.
9.
301
−18
(TP)
(MG)
1
3
10. Mr. Eye
(E)
SOLUTION
Problem #6
FIND OUT
We are asked to find the least number that is divisible by 7 and when divided by each of 26
leaves a remainder of 1.
CHOOSE A
STRATEGY
Any number that is 1 greater than a multiple of the LCM of 26 will leave a remainder of 1
when divided by 26. Consequently, the first step should be to find numbers that are 1 greater
than the LCM, and then find the least of those which is divisible by 7.
SOLVE IT
Find the prime factorization of the integers 26 to find their least common multiple:
2
3
4 = 22
5
6 = 2· 3
22 · 3· 5
The least common multiple is
= 60. Thus, the arithmetic sequence 1, 61, 121, 181, . . .
consists of all the numbers that leave a remainder of 1 when divided by 26. The least number
in this sequence which is also a multiple of 7 is 301.
LOOK BACK When 301 is divided by each of the numbers 26, the remainder is 1, and 301 is divisible by 7.
By finding the least common multiple of 26, we know that we have found the least number for
our answer.
MAKING CONNECTIONS. . . to Logic
Problem #10
Sir Francis Bacon once said, Men imagine that their minds have the command of language, but it often happens
that language bears rule over their minds. And the photographer Minor White said, If we had no words,
perhaps we could understand each other better; the burden is ours, however. How words are used is very
important to the meaning implied. The study of logical reasoning dissects language and interprets an argument’s
validity and soundness. For instance, by pulling two premises from the text of this problem, a syllogisma
deductive argument that draws a conclusion because a common concept appears in both premisescan be
formed.
Mr. Eye had a sister.
The history teacher was an only child.
Therefore, Mr. Eye was not the history teacher.
Syllogisms, in general, obey the form A → B; B → C; therefore, A → C. In mathematics, such an argument is said
to obey the law of transitivity. In the case above, the argument actually takes a slightly different form: A → B;
C → −B; therefore, A → −C. Despite a different appearance, this argument is equally valid.
INVESTIGATION & EXPLORATION
Problem #1
The brick mantel shown is a 2 × 10 arrangement of rectangles. Consider a 1 × 3 arrangement of rectangles. How
many total rectangles are there? Consider arrangements of 1 × 4, 1 × 5 and 1 × 6. How many rectangles are there
in each of these arrangements? What is the pattern for the number of rectangles that will occur in a 1 × n
arrangement?
How many rectangles are in a 2 × 3, 2 × 4, 2 × 5 or 2 × 6 arrangement?
How many rectangles are in a 3 × 3, 3 × 4, 3 × 5 or 3 × 6 arrangement?
In general, how many rectangles will occur in an m × n arrangement of rectangles?
c MATHCOUNTS 19992000
WARM-UP 5
1.
The surface area of a cube is 294 square centimeters. What is
the ratio of the number of square centimeters in the surface area
to the number of cubic centimeters in the volume of the cube?
Express your answer as a common fraction.
1.
2.
July 4, 1903, was a Thursday. On what day of the week was
July 4, 1904?
2.
3.
Each of the squares shown is
inscribed in a larger square so
that the vertices of the inscribed
square bisect the sides of the larger
square. What fraction of the area
of the largest square is shaded?
Express your answer as a common
fraction.
3.
4.
Alia’s digital clock read 7:15 a.m. when she left for school. When
she returned home 7 hours and 15 minutes later, the clock read
5:55 a.m. because the power had gone off during the day. If her
clock automatically reset to 12:00 a.m. when power was restored,
at what time that morning did the power return?
4.
5.
A car holds exactly six people, but only two of those six people
can drive the car. What is the number of ways that the six people
can be seated in the car on a drive?
5.
6.
Terrell usually lifts two 20-pound weights 12 times. If he uses two
15-pound weights instead, how many times must Terrell lift them
in order to lift the same total weight?
6.
7.
Find the integer n such that n × 34 × 75 = 216 .
7.
8.
The point (4, 3) is reflected over the x-axis and then over the
y-axis. What is the sum of the coordinates of the new point?
8.
9.
The sides of a regular pentagon
are extended to form congruent
isosceles triangles as shown. What
is m6 A?
9.
10. Tim and Kurt are playing a game in which players are awarded
either 3 points or 7 points for a correct answer. What is the
greatest score that cannot be attained?
10.
c MATHCOUNTS 19992000
ANSWER KEY
WARM-UP 5
1.
6
7
(FM)
2.
Saturday
(P)
3.
1
32
(MP)
4.
7.
8:35
63
(C)
(CMS)
5.
8.
240
−7
(FTP)
(M)
6.
9.
16
36
(CM)
(MP)
10. 11
(EP)
SOLUTION
Problem #5
FIND OUT
We are asked to find the number of ways six people can be arranged in a car, knowing that
only two of the six people are able to drive.
CHOOSE A
STRATEGY
The Fundamental Counting Principle says to multiply the number of ways each event can
happen by the number of ways every other event can happen to determine the total number
of arrangements. Determine the number of ways a driver can be chosen, then determine the
number of ways others can be placed in the remaining seats. When all of this information is
gathered, multiply to find the answer.
SOLVE IT
Only two of the six people are able to drive the car, so there are two ways to choose a driver.
The rest is then easy, but remember that each time a seat is filled, there are fewer people to fill
the remaining seats. In the front middle seat, there are now five people from whom to choose,
because one of the six is the driver. Then, there are two people who have been seated, so there
are four people who could fill the front right seat. Similarly, any of three people could fill the
rear left, two people could fill the rear middle, and the last person must sit in the right rear
seat. Hence, there are 2· 5· 4· 3· 2· 1 = 240 ways for these six people to fill the car.
LOOK BACK It would be hard to identify all 240 possible arrangements. But consider a simpler example. If
there were only three seats, and only one person could drive, there would be 1· 2· 1 = 2 ways to
fill the seats. It is fairly easy to see that the method of the Fundamental Counting Principle
works in this simpler case, so we can have some certainty that our answer is correct.
MAKING CONNECTIONS. . . to Girolamo Cardano
Problem #2
There are mathematicians who can tell on which day of the week a certain date will fall. This old parlor trick,
which uses a formula to turn the year, month and date into a day, has fascinated people for many generations.
But Girolamo Cardano was no ordinary mathematicianhe took this trick one step further, and he was able to
predict the day on which he would die!
Cardano (15011576) was an Italian mathematician, physician and astrologer in the sixteenth century. He was
the first mathematician to describe negative numbers and to comprehend the existence of negative roots. He was
also the first to recognize imaginary numbers. Cardano advanced the study of algebra and pioneered the study of
probability.
Although completely brilliant, Cardano was also a compulsive hypochondriac, continually complaining that he
could die at any minute. Finally, he became positive that he would die on September 20, 1576, four days before
his 75th birthday. However, nothing happened during the day of September 20, so to prove his prediction correct,
he drank a glass of poison that evening.
INVESTIGATION & EXPLORATION
Problem #10
The answer to problem #10 can be found with a fairly simple formula: if the two values possible in a game are p
and q, the greatest impossible score is pq − p − q. In this case, the greatest score that cannot be attained is
3(7) − 3 − 7 = 11 points.
From 0 through 11 points, what scores cannot be attained? What interesting pattern emerges? Try values other
than 3 and 7 to see if a similar pattern of unattainable scores results.
c MATHCOUNTS 19992000
WARM-UP 6
1.
Angel wants to sell 50 identical pencils in groups of 2 or 3. In how
many ways can the pencils be grouped?
1.
2.
Eight cubes form the figure shown.
If the side length of each cube is
3 cm, how many square centimeters
are in the surface area of the
figure?
2.
3.
Evaluate
4.
For what value of x does 32x −5x+2 = 32x
answer as a common fraction.
5.
The slant height of a cone is 13 cm, and the height from the vertex
to the center of the base is 12 cm. What is the number of cubic
centimeters in the volume of the cone? Express your answer in
terms of π.
5.
6.
Given five segments of length 2, 3, 5, 8 and 13, what is the number
of distinct triangles that can be formed using any three of the
segments?
6.
7.
How many squares of any size are
in this figure?
7.
8.
Two numbers are chosen at random, with replacement, from the
set {1, 2, 3, 4}. The two numbers are used as the numerator and
denominator of a fraction. What is the probability that the
fraction represents a whole number? Express your answer as a
common fraction.
8.
9.
What is the least possible positive integer with exactly five distinct
positive factors?
9.
p
3
(7!)(7!)(8!).
3.
2
2 +7x−4
? Express your
10. A digital, 12-hour clock shows hours and minutes. During what
fraction of the day will the clock show the digit 1 in its display?
Express your answer as a common fraction. (Problem submitted
by alumnus Michael Iachini.)
4.
10.
c MATHCOUNTS 19992000
ANSWER KEY
WARM-UP 6
1.
9
(TP)
2.
288
(CM)
3.
10,080
(CP)
4.
1
2
(M)
5.
100π
(FM)
6.
0
(TEP)
8.
1
2
(TP)
9.
16
(EP)
7.
30
(TP)
10.
1
2
(TP)
SOLUTION
Problem #9
FIND OUT
What do we want to know? The least positive integer with five distinct positive factors.
CHOOSE A
STRATEGY
The number of factors an integer has can be found by looking at the prime factorization.
If a number factors to 2a · 3b · 5c · · ·, the number of factors can be found with the formula
(a + 1)(b + 1)(c + 1) · · ·. For instance, the number 12 is prime factored as 22 · 31 , so it has
(2 + 1)(1 + 1) = 6 factors. Let’s use this knowledge to find the number for which we are looking.
SOLVE IT
In this problem, the number to be found must have 5 factors. Hence, the number must have
the form p4 , for some prime number p. Because the smallest possible positive integer for the
answer is required, choose the smallest prime for p. The smallest prime is 2, so the answer must
be 24 = 16.
LOOK BACK Clearly, the integer 16 has 5 factors, namely 1, 2, 4, 8 and 16. Further, each of the integers
less than 16 can be checked to see that none of them have exactly 5 factors. Notice, also,
that the number to be found must be a perfect square, because only perfect squares have an
odd number of factors. (Why?) This could have greatly reduced the search. But since this
information wasn’t used to find the answer, use it for verificationbecause the answer we
identified is a perfect square, we can have confidence in our answer.
MAKING CONNECTIONS. . . to the Fibonacci Sequence
Problem #6
The lengths of the segments used in this problem are from the Fibonacci Sequence. Each successive length is the
sum of the previous two lengths, and any attempt at building a triangle with three consecutive numbers from this
sequence comes up short. Any attempted triangle collapses.
Leonardo de Pisa (11751250) wrote Liber Abaci, a book that influenced the adoption of Hindu Arabic numerals
in Europe. In this book, a theoretical problem about rabbits was introduced, and the problem was based on the
Fibonacci sequence. The sequence was not given the name Fibonacci until the 19th century, at which time
mathematicians became intrigued with the properties of the sequence and its many connections to probability,
the golden ratio, and nature.
There are many interesting tricks with the Fibonacci sequence and its relatives. Each is based on mathematics
and can be proven algebraically. Try this one! Pick any two numbers (e.g., 5 and 7) to begin a Fibonacci-like
sequence; then, generate numbers in the sequence by adding the previous two terms. In this case, the sequence
becomes 5, 7, 12, 19, 31, 50, 81, 131, . . .. Now draw a line between any two numbers. The sum of the numbers
before the line will always be the same number as the difference between the second number after the line and
the second number in the sequence. For example, if a line is drawn between 50 and 81, the sum of all numbers
before the line is 124. The difference between the second number after the line, 131, and the second number in
the sequence, 7, is also 124. Cool, huh!?! Try the trick with other numbers. Why does this always work?
c MATHCOUNTS 19992000
WARM-UP 7
1.
How many different four-digit numbers can be obtained by using
any four of the digits 2, 3, 4, 4 and 4?
2.
What is the sum of all values of x for which
3.
Circles A, B and C are tangent as shown. The area of circle A
is 16π square centimeters, the area of circle C is 16π square
centimeters, and the area of circle B is π square centimeters.
What is the number of square units in the area of 4 ABC?
3.
4.
For what value of n is the four-digit number 712n, with units
digit n, divisible by 18?
4.
5.
Some bats were in a cave. Two bats could see out of their right
eye, three could see out of their left eye, four could not see out of
their left eye, and five could not see out of their right eye. What
is the minimum possible number of bats in the cave? (Problem
submitted by alumnus Dinesh Patel.)
5.
6.
How many different paths are
possible in moving from A to B
given that you must move down to
the right or down to the left?
6.
7.
>From a bag of coins, 13 were given to Mary, 15 to Norm, 16 to
Anna, and 14 to Bjorn. The six left were given to Troy. How many
coins were originally in the bag?
7.
8.
Evaluate: (2 + 3)−1 × (2−1 + 3−1 ).
8.
9.
The chickens and pigs in Farmer McCoy’s barn have a total of
50 heads and 170 legs. How many pigs are in the barn?
9.
10. A slug climbs ten inches in ten minutes. It then rests two minutes.
It continues climbing at a constant rate and rests for two minutes
after climbing ten minutes. How many minutes will it take the
slug to reach the top of a twenty-foot tower? (Problem submitted
by mathlete Lance Worth.)
10.
p
(x + 3)2 = 7?
1.
2.
c MATHCOUNTS 19992000
ANSWER KEY
1.
4.
20
8
7. 120
10. 286
WARM-UP 7
(TP)
(EP)
2.
5.
−6
7
(F)
(TE)
3.
6.
12
20
(FM)
(TP)
(CG)
(P)
8.
1
6
(C)
9.
35
(MG)
SOLUTION
Problem #3
FIND OUT
What are we asked to find? The area of 4 ABC.
CHOOSE A
STRATEGY
To determine the area of a triangle, the base and height of the triangle must be found. The
information provided about the areas of the circles can be used to determine the radius of
each circle, and adding the radii will determine the lengths of the sides of the triangle. The
Pythagorean theorem can then be used to calculate the height of the triangle, and from that
the area can be calculated.
SOLVE IT
Circle A has area 16π, and the formula for the area of a circle is πr2 . Hence, the radius of
circle A is 4 cm. Likewise, the radius of circle C is also 4 cm, and the radius of circle B is 1 cm.
4 ABC is isosceles with congruent sides of length 5 cm and base of length 8 cm. The height
from
√ vertex B forms two right triangles with hypotenuse 5 cm and leg 4 cm. The height, then,
is 52 − 42 = 3 cm.
The area of 4 ABC, then, is A = 12 bh = 12 (8)(3) = 12 cm2 .
LOOK BACK Does the answer make sense? Yes. The area of circle A is 16π cm2 , or approximately 50 cm2 .
By visual comparison, it seems reasonable that the area of 4 ABC is roughly one-fourth the
area of circle A.
MAKING CONNECTIONS. . . to Pythagorean Theorem
Problem #3
Although the theorem about the lengths of the sides of right triangles was named the Pythagorean theorem
because it was associated with the Pythagorean school, variations on the proof of the theorem have been found
throughout the centuries, in different cultures and on various continents. Even United States President James
Garfield developed a proof based on two ways of determining the area of a trapezoid.
In the figure shown, the area of the trapezoid can be found in two different ways, and these expressions can be set
equal. The first way uses the typical formula, which multiplies the average of the bases by the height. The second
method finds the area by adding the areas of the three right triangles which comprise the trapezoid.
1
ab ab cc
(a + b)(a + b) =
+
+
2
2
2
2
a2 + 2ab + b2 = ab + ab + c2
a2 + b2 = c2
INVESTIGATION & EXPLORATION
Problem #4
All prime numbers greater than or equal to 7 share a divisibility rule. We can illustrate the rule by testing
68,198 for divisibility by 13. Starting with 0, list the first ten multiples of 13: 0, 13, 26, 39, 52, 65, 78, 91, 104, 117.
Then add to or subtract from 68,198 the multiple of 13 that will result in a 0 as the units digit. In this
case, add 52 to give 68,198 + 52 = 68,250. Truncate the units digit from the answer, which gives 6825.
Then, repeat this process until you get either 0 or a number from 1 to 12. Continuing this example,
6825 + 65 = 6890 → 689 − 39 = 650 → 65 − 65 = 0. Because the final result is 0, the original number is divisible
by 13; however, had the result been a number from 1 to 12, the original number would not be divisible by 13.
This technique may be applied to testing for divisibility for any prime number greater than or equal to 7. Create
a few multiples of 7, or 17, or 23, and investigate this technique. Can you explain why it works?
c MATHCOUNTS 19992000
WARM-UP 8
1.
For what value of n is the five-digit number 7n,933 divisible by 33?
1.
2.
Ben performed the following incorrect operations on a number.
First he added −5 instead of subtracting −5. Then he multiplied
his result by 41 instead of dividing by 41 . Finally, he squared the
last result instead of taking the square root. Ben’s final result was
225
16 . If Ben had performed the correct operations, what would the
result have been?
2.
3.
If each of the variables represents a different digit, what is the
value of a + b + c + d?
3.
abc
+ dca
1000
4.
A four-digit number is created by using each of the digits 4, 5, 8
and 9 exactly once. What is the probability that the number will
be a multiple of 4? Express your answer as a common fraction.
4.
5.
Each fair spinner below is divided into four congruent regions. Joe
used spinner A, and Sally used spinner B. They added the results.
What is the probability that the sum was even? Express your
answer as a common fraction.
5.
6.
Mrs. Read can knit one pair of children’s mittens with a ball of
yarn six inches in diameter. How many pairs of identical mittens
can she knit with a ball of yarn twelve inches in diameter? Assume
that the balls of yarn are rolled consistently.
6.
7.
Simplify:
7.
5
3!
+
5
4!
52
5!
8.
The complement of an angle is 5 ◦ more than four times the angle.
What is the number of degrees in the measure of the angle?
8.
9.
What is the total number of square
units in the shaded regions of the
3 × 4 grid of unit squares? Express
your answer as a common fraction.
9.
10. What is the value of the following expression? Express your
answer as a common fraction.
q
11(0.14 + 0.41 + 0.15 + 0.51)
10.
c MATHCOUNTS 19992000
ANSWER KEY
WARM-UP 8
1.
5
(EP)
2.
10
(FS)
3.
18
(EP)
4.
1
6
(TP)
5.
1
2
(TP)
6.
8
(FM)
7.
5
(C)
8.
17
(FM)
9.
3
4
(M)
10.
11
3
(CP)
SOLUTION
Problem #3
FIND OUT
What values do we need to identify? The values indicated by the variables a, b, c and d in the
addition shown.
CHOOSE A
STRATEGY
Relying on logic will solve this problem. Begin by realizing that the sum of the units digits
must have a sum of 0. Hence, a + c must equal 10. From there, other values can be tested.
SOLVE IT
The first possibility is to let c = 1 and a = 9. But we may notice immediately that this presents
a problem. The addition then becomes
9b1
+ d19
1000
and that can only have a solution if b = 8 and d = 0. However, to have a hundreds digit of 0
makes no sense. So, try a different combination.
By letting c = 2 and a = 8, the numbers fall into place to give 872 + 128 = 1000. The sum of
the digits is a + b + c + d = 8 + 7 + 2 + 1 = 18.
LOOK BACK There are several possible combinations of digits that will work in this problem. However,
the sum of the digits is 18 in each case. For more fun with such problems, check
out any of the Alphametic Puzzle Solvers on the Internet. For instance, the site
http://www.teleport.com/~tcollins/alphamet/alpha solve.html will let you enter
two addend words and a sum word and then return all possible numeric solutions.
MAKING CONNECTIONS. . . to Knitting
Problem #6
Hazel Read has been known as the mitten lady in Littleton, MA, for over 40 years. Everyone looks forward to
her hand-knitted mittens at the town Holiday Bazaar each year, and it is a sure thing that all the mittens will
sell. She has, indeed, become a legend in her own time. A rumor once circulated that each year she knits as many
pairs of mittens as there are years in her age, and that she had done so from age 70 until she turned 90. Hazel
says she can’t take credit for that many mittens, but at 96, she continues to keep the hands of many Littleton
residents warm.
But suppose Hazel Read really did live up to the legend? If a ball of yarn 6 inches in diameter makes a pair of
mittens, what would be the diameter of a single ball of yarn needed to knit all the mittens from age 70 through
age 90? A diameter expressed to the nearest foot seems more appropriate than to the nearest inch, considering
how much yarn would be needed.
When Hazel heard how big the ball of yarn would be, she said, It makes me tired just thinking about it. I’ve
only done 30 pairs so far this year.
INVESTIGATION & EXPLORATION
Problem #10
Divide a one-digit number by 9. Divide a two-digit number by 99. Divide a three-digit number by 999. What do
you think will happen when a four-digit number is divided by 9999? The pattern that results is fairly obvious.
When divided by other numbers, however, the pattern of repeating digits in the decimal representation may not
be so clear, though nonetheless interesting.
What are the repeating digits when 1 is divided by 7? . . .when 2 is divided by 7? . . .when any integer 16 is
divided by 7? How are the patterns of repeating digits related? How are the patterns different?
When an integer is divided by 13, how long is the string of repeating digits? How long is the string when an
integer is divided by 7, or by 11, or by 17?
c MATHCOUNTS 19992000
WARM-UP 9
1.
Brianna was having a party for 95 guests. Hot dogs are sold
in packages of eight; buns are sold in packages of ten. If she
purchased the minimum number of packages of each to guarantee
at least one hot dog and one bun for each guest, how many more
hot dogs than buns did she buy?
1.
2.
At 7:40 p.m., Bob passed mile marker 134. At 8:20 p.m., he passed
mile marker 176. What is the number of miles per hour in his
average speed?
2.
3.
What percent of the volume of a 10 00 × 10 00 × 10 00 box can be
filled with 4 00 × 4 00 × 4 00 wooden cubes? Express your answer as a
decimal to the nearest tenth.
3.
4.
How many quadrilaterals of any
size are in the diagram?
4.
5.
One number is chosen from the first three prime numbers, and a
second number is chosen from the first three positive composite
numbers. What is the probability that their sum is greater than
or equal to 9? Express your answer as a common fraction.
5.
6.
In a sequence, each term is obtained by calculating the sum of the
preceding two terms. The eighth term is 81, and the sixth term
is 31. What is the fourth term?
6.
7.
An advertisement read, Take an additional 10% off any item
which is already discounted 30%. A clerk with MATHCOUNTS
training said, Those two combined discounts are the same as one
n% discount. What is the value of n?
7.
8.
The vertices of square EFGH lie
on the edges of square ABCD.
AE
1
EB = 2 . What is the ratio of the
area of square EFGH to the area
of square ABCD?
8.
9.
Bertrand’s Postulate states that there is at least one prime
number between any counting number and its double. How many
prime numbers are there between 25 and 50?
9.
10. Two numbers, a and b, are randomly selected without replacement
from the set {2, 3, 4, 5, 6}. What is the probability that the fraction
a
b is less than 1 and can be expressed as a terminating decimal?
Express your answer as a common fraction.
10.
c MATHCOUNTS 19992000
ANSWER KEY
1.
4.
4
36
(CM)
(TSP)
2.
5.
7.
37
(C)
8.
10.
3
10
(TEP)
63
2
3
5
9
WARM-UP 9
(FP)
(TP)
3.
6.
51.2
12
(FM)
(TP)
(FM)
9.
6
(T)
SOLUTION
Problem #6
FIND OUT
What are we asked to find? The value of the fourth term in a sequence whose eighth term is 81,
whose six term is 31, and where each term is found by adding the previous two terms.
CHOOSE A
STRATEGY
We are given the sixth and eighth terms, so it is fairly easy to find the seventh term. We can
then work backwards to find the fourth term.
SOLVE IT
Because each term is found by adding the two previous terms, the sum of the sixth and seventh
terms equals the eighth term. That is, if we call the seventh term s, then 31 + s = 81. Quite
obviously, s = 50. Hence, the sequence has the form . . . , , , 31, 50, 81, . . ..
The fifth term can be found similarly. It is merely the difference between the seventh and sixth
terms, or 50 − 31 = 19. Likewise, the fourth term is the difference between the sixth and fifth
terms, or 31 − 19 = 12.
LOOK BACK By beginning with fourth term 12 and fifth term 19, we can reconstruct the sequence:
. . . , 12, 19, 31, 50, 81, . . .. In this sequence, the sixth term is 31 and the eighth term is 81, which
is what the original problem stated. Hence, our answer must be correct.
MAKING CONNECTIONS. . . to Patterns of Prime Numbers
Problem #9
For years, mathematicians have searched for a pattern to the prime numbers, yet the sequence of primes appears
to be highly irregular. One theorem in number theory states that an approximate value of the nth prime number,
for very large values of n, is n(1 + 12 + 31 + 14 + · · · + n1 ).
Within prime numbers, however, there are some fairly interesting patterns. One of the naughtiest prime
numbers is 2859433 − 1. This number has 258,716 digits, and 25,799 of them are 0’s. Roughly 9.97% of the
digits are naught (0). And the prime number 8 × 1011336 − 1 has 11,337 digits; of them, 11,336 are the digit 9,
representing 99.99% of the digits.
How many prime numbers are there? Actually, Euclid proved that there are infinite prime numbers. The
reasoning is fairly simple, and it is based on a reductio ad absurdum argumentan argument that establishes a
contradiction to prove that the premise is false. Assume that there is a greatest prime number, and call it P .
Then, compute the product of all prime numbers up to and including P , which is 2 × 3 × 5 × 7 × · · · × P . To that
value, add 1. This result, however, is not divisible by any of the prime numbers up to P , so it must be prime.
But the premise stated that P was the largest prime number. This contradiction implies that there must be no
largest prime number, so there are infinitely many prime numbers.
INVESTIGATION & EXPLORATION
Problem #6
Calculate the first twenty or so terms of the given sequence. What is the ratio of the first and second terms of the
sequence? What is the ratio of the fifth and sixth terms? Of the nineteenth and twentieth? As the terms of the
sequence get increasingly larger, the ratio of consecutive terms approaches a stable value. What is the value of
that ratio?
√
Compare the value of that ratio, for large terms in the sequence, with the value 1+2 5 . How do they compare?
What relationship do you think they have? (Hint: What is the solution to the equation x2 − x + 1 = 0?)
c MATHCOUNTS 19992000
WARM-UP 10
1.
Two girls and three boys sat in a five-seat row at the movie
theater. What is the probability that the two people at each end
of the row were both boys or both girls? Express your answer as a
common fraction.
1.
2.
A quarter is placed on the table. What is the number of quarters
that can be placed around the original quarter so that each quarter
is tangent to the original quarter and to two other quarters?
2.
3.
A square is divided into three congruent rectangles. Then, it is
divided diagonally as shown. If the area of the shaded trapezoid is
24 square centimeters, how many centimeters are in the perimeter
of the original square?
3.
4.
What is the least whole number value of x such that
f (x) = x2 + x + 11 is not prime?
4.
5.
If a + b = 8, b + c = −3, and a + c = −5, what is the value of the
product abc?
5.
6.
Two number cubes, each with the digits 16 on the six faces, are
rolled. What is the probability that the product of the numbers
on the top faces will be greater than 12? Express your answer as a
common fraction.
6.
7.
In May, the price of a pair of jeans was 250% of its wholesale cost.
In June, the price was reduced by 25%. After an additional 50%
discount in July, the jeans cost $22.50. What was the number of
dollars in the wholesale cost of the jeans?
7.
8.
What is the number of square centimeters in the shaded area?
8.
9.
What is the remainder when the sum of the first 100 positive
integers is divided by 9?
9.
10. A rectangular pool measuring 6 feet by 12 feet is surrounded by a
walkway. The width of the walkway is the same on all four sides of
the pool. If the total area of the walkway and pool is 520 square
feet, what is the number of feet in the width of the walkway?
10.
c MATHCOUNTS 19992000
ANSWER KEY
1.
2
5
2.
(MT)
WARM-UP 10
6
(M)
3.
48
(MP)
(TE)
(SP)
4.
10
(TG)
5.
−120
(MG)
6.
13
36
7.
24
(C)
8.
30
(FM)
9.
1
10. 7
(M)
SOLUTION
Problem #6
FIND OUT
What do we wish to know? The probability that the product of the numbers rolled on
two number cubes will be greater than 12.
CHOOSE A
STRATEGY
It will be easiest to keep track of the possible outcomes with a chart.
SOLVE IT
There are 36 possible outcomes, because each cube has 6 faces, and 6 × 6 = 36. These outcomes
are represented in a 6 × 6 chart below:
1
2
3
4
5
6
1
1
2
3
4
5
6
2
2
4
6
8 10 12
3
3
6
9 12 15 18
4
4
8 12 16 20 24
5
5 10 15 20 25 30
6
6 12 18 24 30 36
There are 2 + 3 + 4 + 4 = 13 products greater than 12. Since there are 36 possible outcomes,
the probability is 13
36 .
LOOK BACK The possible products range in value from 1 to 36. Because 12 is
that the probability is close to 13 .
MAKING CONNECTIONS. . . to Engineering
1
3
of 36, it seems reasonable
Problem #4
If the function y = x2 − 10x + 24 were graphed in a coordinate plane, its shape would be a parabola. Examples of
parabolas abound. For instance, the shape of a satellite dish, the headlight in an automobile, and even the path
traveled by a baseball are all parabolas. One man-made structure that appears to be parabolic is the Gateway
Arch of the Jefferson National Expansion Memorial in St. Louis, MO.
The history surrounding the Arch dates back to Thomas Jefferson. Jefferson authorized the Louisiana Purchase
in 1803, and this acquisition doubled the area of the United States. This marked the beginning of the pioneers’
exploration of the West. Many settlers used the strategic position of St. Louis as their starting point. That’s the
reason St. Louis was nicknamed Gateway to the West.
The Arch was designed by Eero Saarinen, who won a design competition in 1948. At 630 feet, the Arch
is the tallest monument in the United States. Although its shape can be approximated by the equation
1
y = 315
(−2x2 + 1260x), it is not a true parabola. It is actually an inverted catenary curve. (A catenary curve is
the shape assumed by a chain when its ends are supported.) The 630-foot span between the legs is equal to the
height, and the cross-section of each leg is an equilateral triangle. The design of the Arch allows it to withstand
winds up to 150 mph.
INVESTIGATION & EXPLORATION
Problem #2
Place three quarters on a flat surface so that each is tangent to the other two. What shape is formed when the
centers of these quarters are connected? (An equilateral triangle.) Now place a quarter on a flat surface, and then
place quarters around it as described in this problem. When the centers of these outer quarters are connected,
what shape is formed? (A hexagon.)
What relationship is there between the equilateral triangle formed by three quarters and the hexagon formed by
seven quarters?
c MATHCOUNTS 19992000
WARM-UP 11
1.
Each side of the square shown in Stage 0 measures 1 centimeter.
When the pattern is continued, what is the number of centimeters
in the perimeter of the figure formed in Stage 50?
1.
2.
All sixth-grade students are standing in line from shortest to
tallest. Three-fourths of them are less than 5 feet tall; two-thirds
are less than 4 12 feet tall; and twelve are not yet 4 feet tall. There
are twice as many between 4 and 4 12 feet as there are between 4 12
and 5 feet tall. How many students are standing in line?
2.
3.
The units digit of a six-digit number is removed, leaving a five-digit
number. The removed units digit is then placed at the far left of
the five-digit number, making a new six-digit number. If the new
number is 31 of the original number, what is the sum of the digits
of the original number?
3.
4.
How many congruent 4-foot tall cylindrical pipes with an inside
diameter of 2 inches are needed to hold the same amount of
water as one pipe of the same height with an inside diameter of
12 inches?
q
6!·4!·2!·0!
Evaluate:
5·3 .
4.
6.
In the diagram, ABCD is a square.
The area of rectangle NFMD is half
the area of ABCD, and ND = 12 CN.
If the area of ABCD is 36 square
centimeters, what is the number
of centimeters in the perimeter of
rectangle ABEM?
6.
7.
What is the number of positive factors of 648?
7.
8.
Corey is reading a 300-page book. After one hour, he had
finished 8% of the book. Assuming that he reads at a constant
rate, how many more hours will it take him to read the rest of the
book? Express your answer as a decimal to the nearest tenth.
8.
9.
How many different positive four-digit integers contain each of the
digits 0, 1, 2 and 3 exactly once?
9.
5.
10. What is the least value of x for which
x+2
4
=
?
3
x+1
5.
10.
c MATHCOUNTS 19992000
ANSWER KEY
1.
4.
7.
10.
404
36
20
−5
(TP)
(FM)
(TP)
(MG)
2.
5.
8.
WARM-UP 11
24
48
11.5
3.
6.
9.
(MG)
(CP)
(FP)
27
42
18
(EP)
(FMS)
(FT)
SOLUTION
Problem #1
FIND OUT
What are we asked to find? The perimeter of a shape in Stage 50 when a pattern for the first
several stages is known.
CHOOSE A
STRATEGY
By using a table to solve this problem, a pattern might be discovered.
SOLVE IT
The chart below shows the change in perimeter from stage to stage.
Stage (n)
0
1
2
3
4
...
n
Perimeter (p)
4
12
20
28
36
...
8n + 4
In each stage, the number of units in the perimeter increases by 8. Because there is a constant
increase, the perimeter at each stage can be described as a linear function, that is, p = an + b,
where p is the perimeter at stage n, a is the rate at which the perimeter changes, and b is the
value of the function at stage 0. (You may also note that the numbers shown obey the rule
p = 4(2n + 1), where 2n + 1 is the nth odd number. Plugging in the values already found, the
function becomes p = 8n + 4. At stage 50, the perimeter is p = 8(50) + 4 = 404 centimeters.
LOOK BACK The formula identified gives 8(0) + 4 = 4 for the 0th stage, 8(1) + 4 = 12 for the 1st stage and
8(2) + 4 = 20 for the 3rd stage. Because these numbers agree with the pictures given in the
problem, we can be certain that the formula is correct, and we can have confidence in our
answer.
MAKING CONNECTIONS. . . to Queues
Problem #2
People standing in line at the grocery store; cars waiting at a toll booth; callers on hold with a technical
assistance desk; copiers at a repair shop that need to be repairedall of these situations are examples of queues.
The word queue is a term borrowed from the British meaning to form a line. In mathematics, queuing theory
studies the phenomena of customers awaiting the delivery of a service.
There are three parts to any queue: the arrivals, the waiting line, and the service facility. At a bank, for instance,
the arrivals are the people waiting to make withdrawals, transfers or deposits; the waiting line is the line in
which they stand; and the service facility is a teller (or an ATM). Randomness is inherent to queues. Arrivals
to a queue occur at irregular intervals, and the amount of time to service each customer may vary. The study
of queues, therefore, is fairly complex and relies on sophisticated probability models. (A counterexample which
doesn’t involve randomness is a factory, where items on an assembly line arrive in a predictable manner and the
amount of time for each stage of production is known. The problem, then, is simply one of scheduling.)
Paul Davis, of the Worcester Polytechnic Institute, in an article for Math Awareness Week, explained queuing
theory as follows: Complexity is aggravated by uncertainty. For example, decisions about dynamic control of
traffic in telephone and computer networks are made more difficult by the uncertain patterns of demand. In a
simpler form, a bank faces a similar dilemma in deciding how many tellers to hire: how should resources be
allocated to maintain adequate service (shorter lines) when only the random characteristics of customer arrival
times are known? Queuing theory provides guidance for these kinds of decisions.
INVESTIGATION & EXPLORATION
Problem #4
How many 1 00 × 1 00 squares are needed to completely fill a 6 00 × 6 00 square? How many 2 00 × 2 00 squares are needed?
3 00 × 3 00 squares?
How many 1 00 × 1 00 × 1 00 cubes are needed to completely fill a 6 00 × 6 00 × 6 00 cubes? How many 2 00 × 2 00 × 2 00 cubes
are needed? 3 00 × 3 00 × 3 00 cubes?
c MATHCOUNTS 19992000
WARM-UP 12
1.
It takes Amelia five hours to mow the yard and it takes her
brother Tom 7.5 hours to mow the same yard. If they have two
lawn mowers, how many hours will it take for them to mow the
yard together?
2.
Evaluate
3.
A box contains a dozen diamonds, a dozen emeralds and two dozen
sapphires. What is the least number of gems you must choose
from this box to guarantee that you have three of a kind?
3.
4.
The difference between two numbers is 9, and the sum of the
squares of each number is 153. What is the value of the product
of the two numbers?
4.
5.
Use numbers, written as words, to make the following sentence
true: This sentence contains
e’s,
t’s and
s’s. What is
the sum of the number of e’s, t’s and s’s in the previous sentence?
5.
6.
What is the fifth term in a geometric sequence if the first term
1
is 625 and the eighth term is 125
?
6.
7.
What is the least possible positive difference between two positive
integers whose squares differ by 400?
7.
8.
Two tangent congruent circles
are circumscribed by a larger
circle. The diameter of the larger
circle is 24 cm. How many square
centimeters are in the area of
the shaded region? Express your
answer in terms of π.
8.
9.
What is the sum of the three distinct prime factors of 47,432?
9.
√
√
3
4
272 − 162 .
1.
2.
10. For what value of n is the following equation true? Express your
answer as a mixed number.
3
=1
4 + 3+n 1
10.
7
c MATHCOUNTS 19992000
ANSWER KEY
WARM-UP 12
1.
3
(MS)
2.
5
(C)
3.
7
(SP)
4.
7.
36
2
(MG)
(MG)
5.
8.
19
72π
(GE)
(FM)
6.
9.
1
20
(FT)
(T)
10. −3 71
(MG)
SOLUTION
Problem #1
FIND OUT
We are asked to solve a problem using a combined rate of two people working together.
CHOOSE A
STRATEGY
Combined work problems are fairly typical in first-year algebra classes, but it’s not necessary
to resort to symbolic manipulation to solve this problem. By using proportional reasoning, we
can determine how much each person does in one hour. Then, we can figure out how long it
would take both of them working together.
SOLVE IT
1
2
It takes Tom 7.5 hours to mow the lawn, so in one hour he can complete 7.5
= 15
of it.
1
Likewise, it takes Amelia 5 hours to mow the lawn, so she can mow 5 of it in one hour. In one
2
2
3
5
hour working together, then, they can mow 15
+ 15 = 15
+ 15
= 15
= 13 of the yard. Since they
can mow 13 of the yard in one hour, they can mow the entire yard in three hours.
LOOK BACK Does our answer make sense? It makes sense that it takes less time for them working together
than it would take either of them working individually. Further, adding the pieces they each
2
) + 3( 15 ) = 52 + 53 = 55 = 1, which means that the job was completed.
worked gives 3( 15
MAKING CONNECTIONS. . . to Gemology
Problem #3
All diamonds are at least 990,000,000 years old. Many are 3,200,000,000 years (3.2 billion years) old!
Both diamonds and graphite are carbon-based minerals, but according to the MOHS hardness scale, diamonds
are the hardest and graphite is the softest. This may be surprising, but the explanation is simple: the atoms in
diamonds are linked together into a three-dimensional network, whereas the atoms in graphite are linked into
sheets with very little to hold the sheets together.
Diamonds form under extremely high pressures deep below the earth, whereas graphite is formed nearer to the
Earth’s surface. In the portion of the Earth’s mantle where diamonds form, approximately 100 to 200 km below
the surface, the temperature is between 900 ◦ C and 1300 ◦ C. To ensure that diamonds are not converted to
graphite when being moved from the mantle to the surface, they must be transported very quickly. Kimberlite
lava acts as a conveyor belt for diamonds, and diamonds are moved by the lava during volcanic eruptions below
the Earth’s surface. The kimberlite lavas carrying diamonds likely erupt at between 10 and 30 km/hr. Within
the last few kilometers, however, the eruption velocity probably increases to several hundred kilometers per hour!
INVESTIGATION & EXPLORATION
Problem #5
When solving this problem, begin with an arbitrary guess; for instance, five e’s, four t’s and six s’s. Those words
in the blanks yield 5 e’s, 4 t’s and 8 s’s. Then, putting in the words for 5, 4 and 8 give the result of 5 e’s, 5 t’s
and 9 e’s. This process can be written in an abbreviated form by representing the number words in the blanks
with ordered triples as follows:
(5, 4, 6) → (5, 4, 8) → (5, 5, 9) → (7, 4, 7) → . . .
Beginning with the triple (5, 4, 6), unfortunately leads to an infinite loop; that is, a solution will never be found!
For what original ordered triple can a solution be found in 1 step? . . .in 10 steps? . . .in 12 steps? What other
ordered triples will never lead to a solution?
c MATHCOUNTS 19992000
WARM-UP 13
1.
A billiard ball is hit at a 45-degree angle from a corner of a 4 foot
by 7 foot billiards table. How many times will the ball rebound off
an edge of the table before landing in a corner?
1.
2.
A magic square is an array of numbers in which the sum of the
numbers in each row, in each column, and along the two main
diagonals are equal. The numbers in the magic square shown are
not written in base 10. For what base will this be a magic square?
2.
3.
Rico can walk 3 miles in the same amount of time that Donna
can walk 2 miles. Rico walks a rate 2 miles per hour faster than
Donna. At that rate, what is the number of miles that Rico walks
in 2 hours and 10 minutes?
3.
4.
For the quadrilateral shown, how
many different whole numbers
could be the length of the diagonal
represented by the dashed line?
4.
5.
√
The length of the diagonal of a square is 2 6 cm. What is the
number of square centimeters in the area of the square?
5.
6.
The sequence 1, 2, 4, 7, . . . is generated by adding 1 to the first
term to get the second, adding 2 to the second term to get the
third, adding 3 to the third term to get the fourth, and so on.
What is the value of the 100th term in the sequence?
6.
7.
Two students set their digital watches to 10:00. One watch runs
one minute per hour too slow, and the other watch runs 2 minutes
per hour too fast. What time will the slow watch show when it is
exactly one hour behind the fast watch?
7.
8.
What is the positive difference between the sum of the first
100 positive odd integers and the sum of the first 100 positive even
integers?
8.
9.
What is the number of positive integral factors of 18,900?
9.
10. What is the remainder when 1313 + 5 is divided by 6?
10.
c MATHCOUNTS 19992000
ANSWER KEY
1.
4.
7.
9
13
5:40
(MP)
(MG)
(TP)
10. 0
2.
5.
8.
WARM-UP 13
5
12
100
(GP)
(M)
(CP)
3.
6.
9.
13
4951
72
(MG)
(TP)
(TP)
(P)
SOLUTION
Problem #3
FIND OUT
We want to know the number of miles Rico can walk in 2 hours and 10 minutes.
CHOOSE A
STRATEGY
Because some relationships between distances, rates and times are known, the distance-rate-time
formula (d = rt) will help guide a solution to this problem.
SOLVE IT
If we let x represent the rate at which Donna walks, the rate at which Rico walks is x + 2.
Further, we know that Rico walks 3 miles in the same amount of time that Donna walks
2 miles. Said another way, Rico’s speed is 23 Donna’s speed. Translated to an equation,
3
x+2
=
.
2
x
This equation can be solved using algebra, although guess-and-check more simply yields that
x = 4. Hence, Rico’s speed is x + 2 = 6 mph. In 2 hours and 10 minutes, Rico will walk
2 61 × 6 = 13 miles.
LOOK BACK Do the speeds found match with what the problem said? Yes. In 30 minutes, Rico can walk
3 miles while Donna walks 2 miles.
MAKING CONNECTIONS. . . to Magic Squares
Problem #2
According to Meridith Houlton in her article Magic Cubes (see http://www.inetworld.net/~houlton/),
construction of magic squares is an amusement of great antiquity; we hear of magic squares in India and China
before the Christian era, while they appear to have been introduced to Europe by Moscopulis. . . According to
Major P.A. MacMahon, in 1892, One method of construction is the DeLaBoubere method: Start with 1 in the
top row, middle cell. Move in a right hand upward diagonal. If the number in the diagonal position is outside of
the square, carry the number to its relative position. If the next cell is occupied, place the number beneath its
predecessor. Pretend there are two imaginary squares along each side of the square. Place the numbers which fall
out of the square in the relative position that is in the imaginary square. A 3 × 3 magic square formed with the
DeLaBoubere method is below.
8
1
6
3
5
7
4
9
2
Benjamin Franklin created a panmagic square, a magic square with a common sum of 260. Any half row or half
column total 130, and the four corners plus the middle total 260. The bent diagonals of his square also total 260.
There are many other magic squares to investigate, including alphamagic squares, antimagic squares, gnomin
squares, magic circles and magic cubes. Information about magic squares is prevalent on the Internet, and the
site http://www.geocities.com/CapeCanaveral/Lab/3469/examples.html is a good place to start.
INVESTIGATION & EXPLORATION
Problem #1
The number of rebounds a ball makes on a billiard table, when hit at a 45 ◦ angle from a corner, follows a very
regular pattern. Make a chart that shows various tables’ width, length, and the number of rebounds that occur
before the ball lands in a corner pocket. For starters, consider tables of size 4 × 9, 3 × 7 and 2 × 5. Then, try
some of your own. What patterns emerge? How might these patterns be explained?
c MATHCOUNTS 19992000
WARM-UP 14
1.
Of a group of boys and girls at Central Middle School’s after-school
party, 15 girls left early to play in a volleyball game. The ratio of
boys to girls then remaining was 2 to 1. Later, 45 boys left for a
football game. The ratio of girls to boys was then 5 to 1. How
many students attended the party?
1.
2.
For what value of x does 216 × 66 = 6x + 6x + 6x + 6x + 6x + 6x ?
2.
3.
Helen must read five books for her literature course. She may read
any one of three biographies, any two of four mysteries, and any
two of five science fiction books on her list. How many different
sets of five books can she choose?
3.
4.
What is the 80th term in the following pattern, where the
first term is 0, and terms are found by adding consecutive odd
numbers to previous terms?
4.
0, 3, 8, 15, 24, 35, 48, 63, . . .
(Problem submitted by mathlete James Cronican.)
5.
What is the mean of the elements in the sequence
−1, 3, −5, 7, −9, 11, . . . , −201?
5.
6.
How many digits are in the number 2521 × 248 ?
6.
7.
What is the positive difference between the 75th and 50th term of
the sequence in which the nth term is found by adding the first
n positive integers?
7.
8.
The symbols ? and ∗ represent different operations, either +, −,
×, or ÷, and x is a positive integer. Find x if 17 ? x = 54 ∗ x.
8.
9.
What is the greatest possible value of x + y such that x2 + y 2 = 90
and xy = 27?
9.
10. Micah places coins in the order of penny, nickel, dime, penny,
nickel, dime, and so on, so that each row contains one more coin
than the previous row, as shown. What is the number of cents in
the value of all coins in the 13th row?
10.
c MATHCOUNTS 19992000
ANSWER KEY
WARM-UP 14
1.
4.
90
6399
(MG)
(TP)
2.
5.
16
−1
(FS)
(P)
3.
6.
180
44
(FM)
(S)
7.
1575
(P)
8.
3
(GE)
9.
12
(MG)
10. 65
(P)
SOLUTION
Problem #5
FIND OUT
What are we asked to find? The mean of the elements in the alternating sequence
−1, 3, −5, 7, . . . , −201.
CHOOSE A
STRATEGY
Studying the method used by mathematician Karl Friedrich Gauss (17771855) will be helpful
in solving this problem. In primary school, Gauss was asked to find the sum of the numbers
from 1 to 100. Since calculators were not available during that period of time, Karl had to rely
on mathematical reasoning. The technique that he used is shown below:
1
+
2
+
+ . . . + 98 + 99 + 100 = S
3
100 + 99 + 98 + . . . +
3
+
2
+
= S
1
101 + 101 + 101 + . . . + 101 + 101 + 101 = 2S
There are 100 pairs which sum to 101, so this simplifies to S =
technique can be used to solve this problem.
SOLVE IT
100(101)
2
= 5050. A similar
= 101 terms because it consists of
The sequence of numbers described in the problem has 201+1
2
only odd numbers. To find the mean of the sequence, add the numbers and divide by the total
number of terms.
−1 + 3 + −5 + . . . + −197 + 199 + −201 = S
−201 + 199 + −197 + . . . +
−5
+
3
+
−1
= S
−202 + 202 + −202 + . . . + −202 + 202 + −202 = 2S
Obviously, −202 + 202 = 0. Fifty such pairs occur in the result; at the very end, a −202 is
left unpaired. Consequently, −202 = 2S, so the sum is −101. Since there are 101 terms in the
sequence, the mean is −101
101 = −1.
LOOK BACK Adding consecutive terms in the sequence gives alternating sums of −n and n, so the mean
alternates between −1 and 1. The sign of the mean in the same as the sign of the nth term.
The answer identified therefore seems reasonable.
INVESTIGATION & EXPLORATION
Problem #10
The pattern of coins in this problem bears a slight resemblance to Pascal’s triangle. Note, for instance, that the
sequence of numbers along the left side of the triangle repeat in the order 1, 5, 1. What interesting patterns can
you identify? The first several rows of Micah’s pattern are shown below.
1
5
1
1
5
1
1
5
1
5
5
5
10
1
10
1
5
1
1
5
1
10
10
10
1
10
5
10
5
10
5
5
10
5
10
10
10
1
10
1
5
1
10
5
10
c MATHCOUNTS 19992000
WARM-UP 15
1.
A bag contains one marble, either green or yellow. A yellow
marble is added to the bag, and one marble is randomly chosen.
The chosen marble is yellow. What is the probability that the
marble left in the bag is yellow? Express your answer as a common
fraction.
1.
2.
Which number can be subtracted from the numerator and added to
12
to create a fraction that is equivalent to 12 ?
the denominator of 15
2.
3.
Twenty-eight circular pepperoni
slices, each 1 00 in diameter, are
placed on a circular pizza. The
slices neither overlap nor hang off
the edge. The diameter of the
pizza is 14 00 . How many square
inches of pizza are not covered by
pepperoni slices? Express your
answer in terms of π. (Problem
submitted by mathlete Sashank
Veligati.)
3.
4.
What is the least positive integer divisible by each of the first
five composite numbers?
4.
5.
What is the greatest prime factor of 55100 + 55101 + 55102 ?
5.
6.
What is the maximum number of unit cubes that can be used to
create a structure with the front, top, and right-side views shown?
6.
7.
Five coins look the same, but one is a counterfeit coin with a
different weight than each of the four genuine coins. Using a
balance scale, what is the least number of weighings needed to
ensure that, in every case, the counterfeit coin is found and is
shown to be heavier or lighter?
7.
8.
What is the coefficient of a2 b in the expansion of (a + b)3 ?
8.
9.
A fish tank weighs 80 pounds when 40% full of water, and it
weighs 140 pounds when completely full. How many pounds does
the tank weigh when empty?
9.
10. Given that 12a + 10b = 1020, what is the value of
a
5
+ 6b ?
10.
c MATHCOUNTS 19992000
ANSWER KEY
WARM-UP 15
1.
2
3
(M)
2.
3
(G)
3.
42π
(FM)
4.
360
(TE)
5.
79
(S)
6.
18
(ME)
(MEP)
(MG)
8.
3
(FP)
9.
40
(FG)
7. 3
10. 17
SOLUTION
Problem #7
FIND OUT
We are asked to find the minimum number of weighings needed to guarantee that the
counterfeit coin is found and is shown to be heavier or lighter.
CHOOSE A
STRATEGY
To maximize the use of the balance scale, begin by weighing two coins against two coins, with
one coin off to the side. This results in two cases, and both cases need to be considered.
SOLVE IT
Case I. The coins balance. The fifth coin, then, must be the counterfeit coin. Compare it
with any of the genuine coins to determine if it is heavier or lighter. This case requires only
two weighings.
Case II. The coins do not balance. The fifth coin, then, is genuine, and one of the four coins
being weighed is counterfeit. The next step is to compare two coins that were on the same side
of the balance. If they were from the heavier side and they balance, then the lighter coin is one
of the other two. If they were from the heavier side and they do not balance, then the heavier
of them is the counterfeit coin. Similarly, if they were from the lighter side and they balance,
then the heavier coin is one of the other two; and if they were from the lighter side and they do
not balance, the lighter of them is the counterfeit coin. This case requires three weighings.
LOOK BACK If luck prevails, the weight of the counterfeit coin can be determined in just two weighings.
However, without a little luck, it will take a minimum of three weighings. And since the
problem asks for the minimum number needed to guarantee that the counterfeit coin is
identified, the answer is three weighings.
MAKING CONNECTIONS. . . to Pascal’s Triangle
Problem #8
Pascal’s Triangle is a familiar sight to many. In some cases, it can be used to solve probability problems. It
also provides a pattern for binomial expansion. Each row in Pascal’s Triangle represents the coefficients of the
binomial expansion of (a + b) raised to a particular power.
(a + b)0 = 1
1
1
1
1
1
2
3
4
(a + b)1 = a + b
1
3
6
(a + b)2 = a2 + 2ab + b2
1
(a + b)3 = a3 + 3a2 b + 3ab2 + b3
1
4
1
INVESTIGATION & EXPLORATION
(a + b)4 = a4 + 4a3 b + 6a2 b2 + 4ab3 + b4
Problem #1
One of the most effective ways to understand probability is to run an experiment and collect data. Give a friend a
yellow and a green marble. Have the friend randomly select one of the marbles and put it in the bag. Then, add
a yellow marble to the bag. Select a first marble. If the first marble selected is yellow, then remove the second
marble, and record the result. Continue this experiment several times. Do your results confirm that roughly 2 of
every 3 times the second marble is yellow?
But be careful! If the first marble chosen is not yellow, then you must remove the other marble from the bag and
start again without recording the results. This is an experiment in conditional probability, and the condition in
this problem is that the first marble selected must be yellow.
On the other hand, you could also run an experiment where you don’t care about the color of the first selected
marble. Again, record the color of the second marble. How do the results vary? Can you explain why the results
differ?
c MATHCOUNTS 19992000
WARM-UP 16
1.
Thirteen points have been placed in a plane so that no three points
are collinear. What is the number of different lines determined by
these points?
1.
2.
Find the number of cubic centimeters in the volume of the cylinder
formed by rotating a square with side length 14 centimeters about
its vertical line of symmetry. Express your answer in terms of π.
2.
3.
If each * represents a digit, and
none of the digits in the divisor
appear again in the problem, what
is the value of the quotient?
4.
Set A contains all the two-digit integers that equal the product of
their tens and units digits divided by the quotient of their tens
and units digits. What is the square root of the product of all
integers in set A?
4.
5.
The isosceles trapezoid shown
has area 24 cm2 . How many
centimeters are in the length of the
shorter base?
5.
6.
The perimeter of a regular hexagon is 48 inches. What is the
number of square inches in the positive difference between the
areas of the circumscribed and the inscribed circles of the hexagon?
Express your answer in terms of π.
6.
7.
The values −5, −3, and 4 randomly replace a, b and c in the
equation ax + b = c, and the equation is solved for x. What is the
probability that x is negative? Express your answer as a common
fraction.
7.
8.
How many four-digit odd integers greater than 6000 can be formed
from the digits 0, 1, 3, 5, 6 and 8, if no digit may be used more
than once?
8.
9.
Rachel subtracted two positive numbers and the difference was 12.
Britt multiplied the same two numbers and the product was 540.
What is the sum of the numbers?
9.
∗∗
135 ) ∗∗∗∗
∗∗∗
∗∗∗
∗∗∗
10. The 26 letters of the alphabet are assigned prime number values in
consecutive order beginning with 2. The product of the values of
the letters of a common math term is 595,034. What is the term?
3.
10.
c MATHCOUNTS 19992000
ANSWER KEY
WARM-UP 16
1.
78
(FMP)
2.
686π
(FM)
3.
22
(E)
4.
7.
30
1
2
(TP)
(T)
5.
8.
2
72
(FM)
(FT)
6.
9.
16π
48
(FM)
(MG)
10. Angle
(SE)
SOLUTION
Problem #1
FIND OUT
We are asked to determine the number of different lines determined by thirteen noncollinear
points in a plane.
CHOOSE A
STRATEGY
Systematically count the number of lines formed by connecting each point to every other point.
SOLVE IT
Choose a point with which to start. This first point can be connected to each of the twelve
other points. Then, a second point can be chosen. This second point can also be connected to
each of the other twelve points, but the connection made to the first point has already been
counted; hence, only eleven new lines have been formed. Similarly, a third point can form ten
new lines, a fourth point can form nine new lines, and so on. The answer, then, is the sum of
the positive integers 112, which is 12 + 11 + 10 + . . . + 1 = 78.
LOOK BACK Another way to consider this solution is that each of the thirteen points can be connected with
twelve others, giving 13 × 12 = 156 lines. However, this counts each line twice, so divide by 2.
That yields 156 ÷ 2 = 78 lines, which agrees with the answer above.
MAKING CONNECTIONS. . . to Will Shortz
Problem #10
People who love puzzles love Will Shortz. He has been the editor of the New York Times Crossword Puzzle
since 1993. He’s had more than 150 puzzles appear in Games Magazine, and he directs the annual American
Crossword Puzzle Tournament. In 1974, he graduated from a major program at Indiana University that he
designed, and now he’s the only person in the world with a degree in enigmatology.
Will continually makes word (and sometimes number) puzzles that baffle and confuse. But, boy, are they ever fun
to solve! One of Will’s puzzles was similar to this problem:
Assign the letters of the alphabet, in order, the numbers 126. Now, take the numbers for each letter in a word,
and find the product of those numbers. (For instance, cat would have a product of 3 × 1 × 20 = 60.) Find the
unique English word for which the product of the letters is 3,000,000.
INVESTIGATION & EXPLORATION
Problem #1
In the SOLVE IT section of the solution above, it is stated that 1 + 2 + 3 + . . . + 12 = 78. Further, in the LOOK
BACK section, the check shows that 13(12)
= 78, also. Consequently, it seems obvious that
2
1 + 2 + 3 + . . . + 12 =
13(12)
.
2
There is a formulaknown as the Gaussian formula, because it was discovered by the mathematician Karl
Friedrich Gaussthat will give the value of 1 + 2 + 3 + . . . + n, the sum of the first n positive integers. Based
on the information shown above, as well as some further exploring on your own, can you discover the general
formula?
c MATHCOUNTS 19992000
WARM-UP 17
1.
Louise can travel only south or east along the roads shown from
her home to school. What is the number of different routes she
can take?
1.
2.
What is the number of units in the distance between (2, 5)
and (−6, −1)?
2.
3.
The first term of an arithmetic sequence is 8. The sum of the
first ten terms of the sequence is four times the sum of the first
five terms of the sequence. What is the common difference?
3.
4.
The pages of a book are numbered consecutively beginning with
page 1. Given that 768 digits are used to number the pages of the
book, how many pages does the book contain?
4.
5.
Two congruent cylinders each have radius 8 inches and height
3 inches. The radius of one cylinder and the height of the other
are both increased by the same number of inches. The resulting
volumes are equal. How many inches is the increase? Express your
answer as a common fraction.
5.
6.
Two congruent equilateral
triangles intersect so the region of
intersection is a regular hexagon as
shown. The area of each unshaded
equilateral triangle is 4 m2 . How
many square centimeters are in the
area of the shaded hexagon?
6.
7.
x
Given that f (x) = x−1 + 1+x
−1 , what is f (f (−2))? Express your
answer as a common fraction.
7.
8.
What is the number of square inches
in the surface area of a cube
√
with space diagonal of length 3 3 inches?
8.
9.
The arithmetic mean of four numbers is 70. When a fifth number
is added, the mean decreases to 60. What is the fifth number?
9.
−1
10. What is the units digit of the sum 386 + 768 ?
10.
c MATHCOUNTS 19992000
ANSWER KEY
WARM-UP 17
1.
4.
38
292
(FP)
(TP)
2.
5.
10
16
3
(FM)
(FM)
3.
6.
16
24
(F)
(MP)
7.
− 83
(C)
8.
54
(FM)
9.
20
(MG)
10. 0
(SP)
SOLUTION
Problem #3
FIND OUT
What do we wish to find? The common difference of an arithmetic sequence with first term 8
and which has some relationship among partial sums of the sequence.
CHOOSE A
STRATEGY
By representing the value of each term symbolically with the common difference d, we can
create expressions for the sum of the first ten terms and for four times the sum of the first
five terms.
SOLVE IT
The first ten terms can be expressed as 8, 8 + d, 8 + 2d, . . . , 8 + 9d. The sum of these ten terms
is (8 + 8 + 8 + . . . + 8) + (d + 2d + 3d + . . . + 9d) = 80 + 45d. The sum of the first five terms,
similarly, is 5(8) + (1 + 2 + 3 + 4)d = 40 + 10d. Because the sum of the first ten terms is equal
to four times the sum of the first five,
80 + 45d = 4(40 + 10d)
5d = 80
d = 16.
LOOK BACK With common difference 16, the sequence is 8, 24, 40, 56, . . . , 152. The sum of the first five terms
is 200. The sum of the first ten terms is 800, which is four times the sum of the first five, so the
answer checks.
INVESTIGATION & EXPLORATION
Problem #1
To solve this problem, you could count all possible routes. However, another approach would involve describing
any particular route by giving the direction of each block walked. For instance, to get to the intersection one block
south and one block east of Louise’s house, she could travel south then east, or east then south. Using shorthand,
these two routes could be described as SE or ES. Further, for Louise to travel from home to school requires that
she walk four blocks south and four blocks east. One possible route could be described as SSSSEEEE, and
another is SSEESSEE. She needs to walk a total of eight blocks to get from home to school,
four of those
and
8·7·6·5
blocks must be traveled south. The number of ways to get from home to school, then, is 84 = 4·3·2·1
= 70 routes.
Unfortunately, this method doesn’t quite work. For instance, one possible route using this method is SEEESSSE;
but notice that the fifth move is south, and that’s impossible given the map shown. If Louise tries to walk
south after traveling one block south and three blocks east, there’s no road! Consequently, some of the 70 routes
identified are not possible. Besides just counting the impossible routes, how could you use the method described
in the paragraph above to quickly count the routes that are not possible? How many routes, then, are possible?
c MATHCOUNTS 19992000
WARM-UP 18
1.
A magic cube has the first 27 positive integers placed in a 3 × 3 × 3
arrangement so that the sum of any three collinear numbers in a
column, row or diagonal is the same. What is the sum of any
three collinear numbers in the cube?
1.
2.
What is the units digit of 1757 − 1953 ? (Problem submitted by
mathlete Britt Kreiner.)
2.
3.
How many ordered triples (a, b, c) of real numbers have the
property that each number is the product of the other two?
3.
4.
Two chords intersect as shown.
What is the number of units in the
value of x?
4.
5.
The pressure P of wind exerted on a sail varies directly with the
area A of the sail and the square of the velocity V of the wind;
that is, P = kAV 2 , where k is a constant. The pressure exerted on
one square foot of sail is 1 pound when the velocity of the wind is
16 miles per hour. What is the number of miles per hour in the
velocity of the wind when the pressure on 9 square feet of sail is
49 pounds? Express your answer to the nearest whole number.
5.
6.
The point (5, 3) is reflected about the line x = 2. The image point
is then reflected about the line y = 2. The resulting point is (a, b).
Compute a + b.
6.
7.
A square is inscribed in a semicircle, and a second square is
inscribed in the whole circle, as shown below. What is the ratio
of the area of the smaller square to the area of the larger square?
Express your answer as a common fraction.
7.
8.
The three-digit base-six number, nmn6 , is equal to the product
of 234 and 345 . What digit is represented by n?
8.
9.
The volume of a cylinder is 60 cubic centimeters. What is the
number of cubic centimeters in the volume of the sphere it
circumscribes?
9.
10. For all integer values of n ≥ 2, k will divide n3 − n. What is the
greatest possible integer value of k?
10.
c MATHCOUNTS 19992000
ANSWER KEY
WARM-UP 18
1.
4.
42
6
(MP)
(FM)
2.
5.
8
37
(SP)
(F)
3.
6.
5
0
(TEP)
(M)
7.
2
5
(FM)
8.
5
(CFS)
9.
40
(FM)
10. 6
(MTP)
SOLUTION
Problem #3
FIND OUT
What are we asked to find? The number of possible ordered triples with the property that each
number in the triple is the product of the other two numbers.
CHOOSE A
STRATEGY
The statement can be translated into algebraic equations and then solved as a system.
SOLVE IT
If any of the numbers is zero, then all of the numbers are zero. This yields just one ordered
triple, (0, 0, 0).
If none of the numbers equal zero, we have ab = c, ac = b and bc = a. Consequently,
(ab)(ac)(bc) = abc, or a2 b2 c2 = abc, which gives abc = 1. Substituting a for bc gives a2 = 1, and
it follows that a = ±1.
If a = 1, then either b = 1, c = 1 or b = −1, c = −1. That gives two additional ordered triples,
(1, 1, 1) and (1, −1, −1).
Likewise, if a = −1, then either b = 1, c = −1 or b = −1, c = 1, giving two more ordered triples,
(−1, 1, −1) and (−1, −1, 1).
In total, there are five ordered triples.
LOOK BACK Does our answer make sense? Yes. Each of the triples identified satisfies the conditions of the
problem. Also, we considered the case if any value was 0 as well as solving the problem for
non-zero values, so we can be certain that no triples have been missed.
MAKING CONNECTIONS. . . to Computer Technology
Problem #8
Although base four or base five representations of numbers are not very common, it is common to represent
numbers in binary (base two). The Beginner’s Guide to Binary (http://www.brant.net/andrew/beg2.htm)
states that, In today’s electronic world, binary is the basic building block of technology. In any electronic circuit
there are only two possible states. The circuit can either be ON or OFF. This is the basis of the binary system.
Computers use strings of 1’s and 0’s to represent letters and numbers. These 1’s and 0’s tell a computer if a
particular location is on or off. The computer’s central processing unit (CPU) sends a signal that is converted
from a string of digits (which humans can’t understand) into a letter, a sound or a computation.
INVESTIGATION & EXPLORATION
Problem #8
To visualize number bases, use grid paper to create pieces for a variety of bases. Regardless of the base, call the
pieces units (base0 ), strips (base1 ), mats (base2 ) and stripmats (base3 ). In base 5, for example, there are units
of 1 square, strips of 5 squares, mats of 25 squares, and stripmats of 5 mats (125 squares), as shown below.
Using these pieces, the number 4235 is represented by 4 mats of 25 pieces each, 2 strips of 5 pieces each, and
3 units, giving 4(25) + 2(5) + 3 = 100 + 10 + 3 = 11310 . Understanding these conversions, numbers in base 5 can
be added, subtracted, multiplied and divided. For instance, after accumulating 5 of any piece, those pieces can be
exchanged for one of the next larger pieces. Hence the reason that the only digits used in base 5 are 0, 1, 2, 3
and 4, and this exchange of smaller pieces for a larger one is analogous to carrying when numbers are added.
c MATHCOUNTS 19992000