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Arithmetic w/ Percent
December 1994
1. A compact disc as purchased from a store which was having a 15% discount sale.
The cost, including 6% sales tax was $8.65. What was the regular selling price?
Express your answer in dollars and cents to the nearest cent.
2. An elementary school had an enrollment of 180 students last year. They used 103
boxes of crayons. This year there are 207 students. The school wants to increase
its per student use of boxes of crayons over last year’s by 15%, because they ran
short last year. How many boxes of crayons will the students need?
3. A store changed its price on an item in the following pattern for a four week
period: week 1, 10% markdown; week 2, 10% markdown; week 3, 15% markup;
week 4, 25% markdown. The store buys an ad in the newspaper which reads:
“Over the past four weeks we have reduced our price on this item by a total of
N%.” Find N to the nearest whole %.
Arithmetic w/ Percent
December 1995
1. What minimum score should Cheryl get on her next exam so that her grade
average will improve by 2 points? Her exam grades so far are 75, 86 and 82.
2. How many books each containing 250 pages can a student read in 30 days, if she
2
can read 1 pages per minute? She reads two-thirds of her leisure time and she
3
has 2 hours of leisure time each day.
3. A merchant paid $960 on a bill of goods on which he had been given successive
discounts of 25%, 20% and 20%. What was the original bill?
Arithmetic w/ Percent
December 1996
1. Find the sum of these two decimals as a fraction in lowest terms. .096 + .853
2. A used-car dealer reduced the price on one of his cats by 25%. He then took off
$500, but the customer still did not buy it. He offered the next customer another
25% below this, it still remained on the lot. He finally dropped the now very low
tag by 20% and sold the car to his nephew for $960. What was the original ticket
price on the car?
3. Find the only whole number from 0 to 1000 which, when divided by 7 gives a
remainder of 2, when divided by 8 gives a remainder of 3, and when divided by 9
gives a remainder of 4.
Arithmetic w/ Percent
December 1998
1. A CD sells for $24.95. The next week (week 2), the owner has a “20% off” sale.
The week after (week 3), the owner returns all sales to their original prices. What
% were the CD’s marked up from week 2 to week 3?
2. For the past three years, the inflation rate has been 4%, 3.17%, and 1.75%. For
the past three years, income has increased at rates of 1.5%, 1.2% and 1.1%,
respectively. If the inflation rate for this year is 1.2%, what income rate for this
year would be necessary to match the rate of inflation for the past 4 years?
Express answer to nearest 1000th of a percent.
3. The product of the GCF and the LCM of x and y is 3456. The sum of the GCF
and the LCM of x and y is 168. List all possible pairs of (x, y) that meet these
criteria.
Arithmetic w/ Percent
December 1999
1. Judy’s average bowling score is exactly 115. If she bowls one more game and
scores 125, her average will improve to exactly 117. Find her average if, instead
of bowling one more game and getting the 125, she bowls two more games and
scores 104 and 117. Give answer to nearest 10th.
2. Erik is 20% older than Martin. Tony is 20% older than Erik. How old is Martin
if Tony was 14 when Erik was 8?
3. John Martin invested $8000, simple interest for a year, part at 6% and the
remainder at 5%. The income from the 5% investment yielded $48 more than the
6% investment. How much more interest would he have received, if he had
switched the investments with the percents?
Arithmetic w/ Percent
December 2000
1. Find the single discount that is equivalent to successive discounts of 20%, 10%,
and 15%.
2. A family paid 52 cents/gallon for oil for 2 years ago. This year it looks like
they’ll be paying 91 cent/gallon. If the family paid $1,640 for oil during 1998, at
this rate of change, how much will they pay for oil in 2000?
3. An alloy of brass and aluminum is 72% brass. How much brass should be melted
with 60 grams of this alloy to make an alloy which is 80% brass?
Arithmetic w/ Percent
December 2001
1. John took a picture to a copy machine and had a copy made, not realizing that
someone had left the machine set for a reduction to 80 percent of the original
size. What percent setting will John have to use on the copy machine to obtain
from his reduced copy a picture with the original dimensions?
2. If A is 35% less than B and if C is 75% more than B, what percent of C is A?
Round your answer to the nearest hundredth of a percent.
3. A pencil distributor divided a certain number of pencils equally into 100 boxes
and had 50 pencils left over. He changed his mind and decided to divide the
pencils equally into 110 boxes and had five fewer pencils in each box than he
had before, and none were left over. How many pencils did he have?
Arithmetic w/ Percent
December 2002
2
1
1.
is what percent of ?
3
6
2. 212 base 8 is what percent of 10220 base 6 ? Write your answer as a percent in base 10.
3. DollMart is selling a product at price A. Price A was reduced by 20%. Thus
creating the new price B. Price B was then increased to price C. Price C
represents a 10% reduction from price A. By what percent was price B increased
to reach price C?
4.
Arithmetic w/ Percent (Calculator)
December 2003
1. Suppose that you are told that the non-zero number A is 2A percent of B. What
must the number B equal?
2. Elmer kept a log of the high temperatures for all five days of a school week. The
high Tuesday was 20% lower that the Monday high. Wednesday was 50% higher
than Tuesday. Thursday was 25% lower than Wednesday. Friday was 25%
higher than Thursday and 5 degrees warmer than Monday. What was the high on
Wednesday in degrees?
3. The interest rate in a certain bank is compounded annually. On Jan. 2, 1990 it
was 6%. On Jan. 2, 1995 it changed to 5%. On Jan. 2, 2000 it changed to 1 1/2
%. If a person invested $4000 on January 2, 1990 and did not take it out until Jan
2, 2004, what amount of money should he be able to receive on that date?
Arithmetic w/ Percent (Calculator)
December 2004
1. If 8 is 4% of N, and 10 is 20% of M, what percent of M is N?
2. Find the LCM of 21, 25, and 27.
3. How many pints of an antifreeze solution, which is 72% alcohol, should be mixed
with 21 pints of an antifreeze solution 96% alcohol to make an antifreeze solution
that will be 81% alcohol?
Arithmetic w/ Percent (You may use Calculators)
December 2005
1. How many girls would have to leave a room containing 98 girls and 2 boys, in
order to reduce the percentage of girls in the room to 92%?
2. Half a strip of cloth is red. A fifth is blue. Half of the remaining strip is green
and the remaining 6 feet of the strip is yellow. How long is the strip?
3. In Mr. Thompson’s math class of thirty students, all took an exam on statistics. If
the average passing grade was 84%, the average failing grade was 60%, and the
overall average was 80%, how many students passed the exam?
Arithmetic w/ Percent (You may use Calculators)
December 2006
1. On March 1, at Jefferson Elementary School:
● 40% of the students rode the bus
● 60 students walked to school
● 35% of the students rode with their parents to school
● No one was absent
● No other form of transportation to school was used on that day
How many students attended Jefferson Elementary on March 1?
2. A Going out of Business sale is held. Products will be reduced in price by 10%
at the beginning of each day, based on the prices of the previous day. The store
is open every day of the week. The first day of the sale is May 12th. Find the
first date when the percent of the prices of items of May 11th will be more than
50% off?
3. Unlucky Luke lost 75% of his money gambling. He then had $500 stolen by a
pickpocket. He now has 18.75% of what he had first. How much did he lose
gambling?
Arithmetic w/ Percent (You may use Calculators)
December 2007
1. X is 75% of Y and 50% of (Y + 10). What percent is (Y – 5) of (X + 10)?
2. Anca buys shares on the commodities market at 20 % margin, meaning she foots
20% of the purchase price and borrows the other 80%. The shares increase 40%
in value and Anca sells, using part of the proceeds to satisfy her interest-free
loan. By what percent does Anca profit on her investment?
3. It takes Sam 3 hours and 18 minutes to mow the lawn, moving at an average
speed of ¾ miles per hour and cutting a swath of grass on average 20 inches
wide. How many acres are in the lawn? Hint: there are 640 acres in a square
mile.
Probability
December 1991
1. What is the probability that a 6-letter word chosen from the letters in the word
“factoring” ends with “a” and begins with “f”?
2. There are 12 people in a room. The number of permutations of N of these people
is 120 times the number of combinations of N of these people. Find N.
3. Find all real value(s) of x such that the sum of the last four terms of (x – 2)5 is
128.
Probability
December 1992
6
A B
3 1 3 7
1. The sum of the first three terms in the expansion of  +  is C . Find the
2
4 6
sum of A, B and C.
2. Urn I contains 5 red and 4 black balls. Urn II contains 4 red and 5 black balls. 2
balls are selected at random. If both are red, what is the probability that both came
from the same urn?
3. In how many ways can four envelopes be placed in three mailboxes, if each
mailbox can obtain any number of envelopes? (All envelopes are the same)
Probability
December 1993
1. When students get sodas from a machine at a certain school, 3 out of every 5
students choose a Pepsi. What is the probability that, of the next 5 students who
get a soda, 3 get a Pepsi and 2 do not? (Express your answer as a fraction in
simplest form.)
2. The Acme Electronics Company ordered 1000 computer chips. Of those, 600
were purchased from supplier A, 300 from supplier B, and 100 from supplier C. It
is known that 10% of supplier A’s chips, 20% of supplier B’s chips, and 25% of
supplier C’s chips are defective. If a chip selected at random is good, what is the
probability that it came from supplier B? (Give answer as a fraction in simplest
form.)
3. The digits 0, 1, 2, 3, 5, 7 are used to make all possible 3-digit numbers with
distinct digits. If a number is selected at random from this set, what is the
probability that it is even and greater than 300?
Probability
December 1994
1. Three coins are tossed. What is the probability that the result is more than one
head, if at least one head shows?
2. Each member of a nine person committee has a probability of
certain meeting. What is the probability that at least
1
of attending a
2
2
of the committee attends
3
the meeting? Find answer as an exact fraction.
3. There are blue, yellow and silver tokens in a container. A token is drawn at
2
random. The probability of drawing a blue or silver token is . The probability of
3
3
drawing a blue or yellow token is There are 42 of the yellow and silver tokens
4
combined. Find the number of tokens of each color.
Probability
December 1995
1. Let set A contain 43, 68, 73, 78, 80, 88, 92, 70, 75, 65, and 52. Find the difference
of the means of sets P and Q, where P is the set of numbers of set A which are
greater than the mean of set A, and Q is the set of numbers of set A which are less
than the mean of set A.
2. Harold is a regular customer at a Ron’s Soda Shop. The menu contains three
items: soda, hamburgers and fries. Ron has noticed that Harold’s probability of
ordering a hamburger is .85 and that of ordering fries is .7. What is the probability
that Harold orders more than a soda the next time he enters? Give answer to
nearest thousandth. Assume that Harold always orders soda.
3. A local lottery selects 6 numbers at random from the set of whole numbers from 1
to 12. What is the probability that the second smallest number selected is 5?
Express as a fraction in simplest form.
Probability
December 1996
1. Rob has a nickel, 2 dimes and 2 quarters in his pocket. If he selects 3 coins at
random, what is the probability that he gets 40 cents? Express answer in simplest
fractional form.
2. The data below is a sample of weights of girls at Bronton High School, which has
240 girls enrolled. Determine the sample standard deviation for the data. Round to
nearest hundredth.
128
133
142
106
121
130
112
125
156
124
108
129
130
141
123
116
119
128
145
109
141
102
113
125
116
3. Tickets are numbered from 100 to 400. If a ticket is chosen at random, what is the
probability that the number is a multiple of 3 or 5 but not both? Give exact
answer.
Probability
December 1999
1. On 40% of the days Barbara wears blue shoes; on 30% of the days Samuel wears
blue shoes; and on 20% of the days Mary wears blue shoes. They select their
shoes independently of each other. Find the probability that, on a given day,
Barbara and Samuel will wear blue shoes and Mary will not. Express your
answer as a decimal.
2. A box contains 3 red, 5 green, 4 yellow, ad 6 white balls. 6 balls are selected at
random. What is the probability that 1 red, 3 green and 2 white balls are selected?
Give answer to nearest 1000th or as a fraction reduced to lowest terms.
3. Three fair dice are tossed at random. What is the probability that the three
numbers that turn up can be arranged to form an arithmetic progression with a
common difference greater than or equal to 1. Give exact answer as a fraction.
Probability
December 2000
1. Mark has 3 quarters, 2 dimes and a nickel. He gives Paul 2 coins chosen at
random. Find the probability that Mark gives Paul more than 30 cents. Express
as a fraction in simplest form.
2. All possible 3-digit numbers with distinct digits are made from the numbers 1
through 7. What is the probability, if one is chosen at random, that it is less than
400? Express your answer as a fraction in simplest form.
3. 4 urns contain the marbles shown. If an urn is chosen at random and a marble is
chosen at random, what is the probability that the marble is green? Express your
answer as a fraction in simplest form.
Probability
December 2001
1. When two dice are tossed, what is the probability that the sum is 10? Assume
these are regular 6-sided dice. Give your answer as an exact fraction in simplest
form.
2. Twenty-one balls are in an urn. Four are blue, eight are red, and the rest are
yellow. If two balls are removed from the urn at random without replacement,
what is the probability that both balls are the same color? Give answer as a
fraction simplified to lowest terms.
3. In a certain country of 500,000 families, each family is expected to continue to
have children until it has a girl then stop. Assume that the probability of having a
boy is equal to that of having a girl, which is 50%. Assume that fertility is not a
problem and no multiple births occur. How many families are expected to have
five children?
Probability
December 2002
1. Find the difference between the probability of randomly selecting a heart from a
deck of 52 cards, and randomly selecting a face card form a deck of 52 cards.
Express the answer as a fraction in lowest terms. (Face cards include jacks,
queens, and kings only).
2. A fair coin is tossed six times. The outcomes are recorded. What is the
probability that the number of heads recorded was not the same as the number of
tails recorded? Express your answer as a fraction in lowest terms.
3. A monkey is given a box, which contains 3 of each kind of the following figures:
3 squares, 3 triangles, and 3 circles. The monkey picks all nine figures out of the
box, one at a time. What is the probability that the monkey picks them out by
kinds? That is, all of one kind, then all of another kind and then finally all of the
third kind. The kinds can be selected in any order.
Probability
December 2003
1. In room 106 there are 15 freshmen, composed of 5 girls and 10 boys. In room
108 there are 19 freshmen, composed of 12 girls and 7 boys. If a boy is chosen at
random from this group of freshmen, what is the probability that he came from
room 108?
2. On average, Mark flunks 25% of his Spanish quizzes. Find the probability Mark
will flunk at least one of his next three Spanish quizzes. Express your answer as a
fraction in simplest form.
3. Mudville High is playing Sludgetown in a best of 5 series. The first team to win 3
games wins the series. The probability Sludgetown will win any one of the games
is 3/5. However, Mudville wins the first game. Calculate the probability that
Mudville will win the series from this point?
Probability
December 2004
1. The Family Dollar Store has 5 goblin, 4 witch, and 6 Batman costumes left. If
each of the 15 costumes has an equal chance of selling, what is the probability
that the next costume will be a goblin or a Batman costume? Express in simplest
form.
2. From an ordinary deck of 52 playing cards, John is dealt three cards. What is the
probability that the 3 cards that John has are all from the same suit? Express in
simplest form.
3. From a set of 3 red, 4 white, and 5 blue balls, four are selected at random. Find
the probability that of those chosen, none are red or none are white. Express in
simplest form.
Probability (No Calculator)
December 2005
1. A faculty council contains 8 members, two of which are men. If two members are
selected at random from the faculty council to fill vacancies on the technology
committee, what is the probability that both men will be selected? Leave your
answer as a fraction in lowest terms.
2. A five-digit number uses each of the digits 1, 2, 3, 4, and 5 exactly once. What is
the probability that the number is a multiple of 4? Express your answer as a
fraction in lowest terms.
3. If two marbles are removed at random from a bag containing only black and white
marbles (without replacement), the chances that they are both white is 1/3. If
three marbles are removed randomly (without replacement), the chances all three
are white is 1/6. What is the least number of white marbles in the bag?
Probability (No Calculator)
December 2006
1. 3 black socks, 4 white socks and 5 green socks are in a drawer. What is the
probability that if a person draws two socks at random, both will be the same
color? Express answer in simplest form.
2. Sam’s probability of hitting a target is ¾. Chris’s probability of hitting the same
target is 2/3. Chris and Sam shoot alternately at the target, Chris shooting first. If
each shoots two times, find the probability that the target is hit exactly 3 times.
Express in simplest form.
3. There are 8 white and 5 brown mice in a lab. 5 are chosen at random to be used
in an experiment. Find the probability that more white mice than brown mice are
used. Express answer in simplest form.
Probability (No Calculator)
December 2007
1. Byron has 2 nickels, 3 dimes and a quarter in his pocket. He draws out two
coins. What is the probability that he has at least 20 cents in his hand?
2. In an ultimate game, the first to score 15 goals wins. If one team leads another
team 13 to 11, and if each team has an equal likelihood of scoring each goal, find
the probability the team with 13 goals will win the game. Express answer in
simplest form.
3. A basket contains N marbles, including R red marbles and B blue marbles, so that
R + B= N. The probability that two marbles randomly selected without
replacement from the basket will be the same color is exactly ½. Find R − B in
terms of N.
Exponents and Radicals
December 1988
(
6
+ 1− 2
1. Find the value of −1
3 + 2 −1
)
2
− 25
−
1
2
+ 4 64
2. Order these numbers from the smallest to the largest 3 22 , 414 ,910 , 810
3. Find the value(s) of x and y so that 25 ⋅ 2 x +1 = 16 y and 2 ⋅ 5 x −1 = 5 y
Exponents and Radicals
December 1989
5
+ 3
3
1. Simplify:
2
+1
3
2. Find all real numbers x so that 4 x + 4 x +1 = 160
3. Find all real numbers t such that t + 2 = 4 + t 8 − t
Exponents and Radicals
December 1991
5x0 − 1
1. Simplify:
÷
2
125 3
2. Simplify:
2 63 +
(5 x )0 + 3
(9
2
+ 12 2
)
1
2
2
8+3 7
3. Find the solution set within the real numbers for x in the equation:
x + 7 + 3 x − 2 = (4 x + 9) / 3 x − 2
Exponents and Radicals
December 1992
1. Determine the value of A if 15 6 ⋅ 30 6 ⋅ 18 6 = A12
A
2. If
5
64 ⋅ 4 32 ⋅ 3 16 = 2 B , where A and B are relatively prime, find
3. Find all value(s) of x, so that 9 x −1 − 12 ⋅ 3 x− 2 + 3 = 0 .
A
.
B
Exponents and Radicals
December 1993
1. Simplify:
8−
7
6−
5
4−
3
2
2. For what value of x does 54 n −1 ⋅ 10 n +1 equal15 n +1 ⋅ 6 2 n −1 ⋅ x
3. Find all value(s) of w such that 9 w − 28 ⋅ 3 w −1 = −3
Exponents and Radicals
December 1994
( ) (2 x )
1. Solve for x: 2 3
2
3
= 2 2 + 2(2 )
2. Find all value(s) of x which satisfy the equation: x − 8 x − 1 + 14 = 0
1
3. If 5 n = 2025, what does 5 2
n −1
equal?
Exponents and Radicals
December 1995
1. Simplify:
5000 −
7
2
1
8
−
2. Find all solutions for x for which 4 ⋅ 3 x
( y − 2 )− 2 − ( y − 2 ) 2
1
3. Solve for y, if
2
−x
− 4 3 > −4(7 )
1
3− y
=
1
5
Exponents and Radicals
December 1996
( ) (a )
1. If a b a c a 2
b
3c 2
= a p , find p in terms of a, b and c in simplest form.
2. a 3 2 b c 2 = 21,600 where a, b and c are prime integers. Determine the value of
(a − b )3 (c − a )3
3. Express in simplest form:
n
54 ⋅ 5 n+ 2
25 ⋅ 3 2 n+ 3 ⋅ 2 3n +1
Exponents and Radicals
December 1999
1. Find x if 343 x = 7 7
7
2. Express
5
5 5 + 53 + 5 2 + 25
1+ 5
in simplest form.
3. Arrange A, B, C, D, E in order smallest to largest if,
A=
3
5 ⋅ 6 , B = 6 3 5 , C = 53 6 , D = 3 5 6 , E = 3 6 5
Exponents and Radicals
December 2000
5 2000 ⋅ 25 2001
1. Find P if 5 P =
1251999
 1 
2. Solve for x:  
 27 
x2 −x
1
= 
9
9
3. Solve for x, if x = 1 + 1 + 1 + ... Give exact answer.
Exponents and Radicals
December 2001
1. If 5 2 ⋅ 210 was multiplied out to produce a whole number, how many digits are in
the whole number?
2. Solve
x − 3 = 2 − x , where x is a real number.
(
3. Find all real values of x for which x 2 − 5 x + 5
)
x 2 − 9 x + 20
=1
Exponents and Radicals
December 2002
1. Find n in simplest form, if n = 3 81 ⋅ 6 81
2. Find the smallest possible sum of n + x + y, if each is a natural number and
3
a 2 b ⋅ 4 ab 2 = n a x b y
3. Find all x, such that x − 36 = 5 x
Exponents and Radicals
December 2003
9! 8!
Note: x!= 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ ... ⋅ x , where x is a natural number.
7!
1. Find the value of
2. If 2 4 − 2 3 + 2 2 − 21 + 2 0 − 2 −1 + 2 −2 − 2 −3 + 2 −4 = 11 − x , find x.
1
3. Let P =
1
−
3− 8 2 2 − 7
simplest possible form.
Exponents and Radicals
December 2004
(
1. If a ⋅ a 3
) = (a
6
2
+
1
7− 6
−
1
6− 5
+
1
. Express P in
5 −2
)
k
⋅ a 4 , find k.
3
2. Express the following as a single radical in simplest form:
3. Find all real values of x such that:
94 4
6
27
5 x + 4 − 3x − 11 = 2 x − 9
Exponents and Radicals (No Calculators)
December 2005
1. Simplify the following. Leave your answer in simplest radical form. 6
2. Solve the following for x:
2x − 1 = x − 2
(
)
3. Find all integral solutions for x such that: x 2 − x − 1
Exponents and Radicals (No Calculators)
December 2006
1. Solve the following for x:
x 2 − 3x + 12 = 4
2. Find all the rational values of x so that
64 5 x
2
−4 x−4
= 42x
3
+3 x 2 −2 x
3. What is the value of 17 − 12 2 + 17 + 12 2
x+2
=1
1
9
−
3
3
Exponents and Radicals (No Calculators)
December 2007
 144 
1. Find 

 225 
−
1
2
. Express in simplest form.
2. Find the value of
integers.
3. Solve for x if,
5 − 2 6 in the form
3x + 2 − 2 x = 1 − x .
A − B , where A and B are both
Lines, Angles, and Polygons
December 1988
1. Three lines meet at a point forming six angles, three of which have measures of
(3x + 6)ο , (7 x − 10 )ο and (9 x − 6)ο . If none of these angles have equal measure,
find the numerical ratio of the smallest measure to the largest measure. Express
your ratio in lowest terms.
2. In triangle ABC, AB ≅ AC , m∠A = 40 ο , and point O is within the triangle so that
∠OBC ≅ ∠OCA. Find m∠BOC in degrees.
3. Given : MN // PO
PO bisects ∠NOR
MN = x + 5
4x
MP =
+ 12
5
np = 2 x − 12
Find the perimeter of ∆MNP.
N
O
M
P
R
Lines, Angles, and Polygons
February 1989
1. Find the number of sides of a polygon given that each interior angle is 7 ½ times
the exterior angle at the same vertex.
2. Find m∠QAM + m∠MBN + m∠NCO + m∠ODE + m∠PED + m∠PFQ.
3. A regular octagon ABCDEFGH has sides of 8. Two line segments are drawn
from point F, one to point A and the other to point C. Find the number of square
units in the area of the quadrilateral ABCF.
Lines, Angles, and Polygons
December 1989
1. One side of a triangle is 6 cm longer than another side. A ray bisecting the angle
formed by these two sides divides the third side of the triangle into segments of 5
cm and 8 cm. Find the perimeter of the triangle in centimeters.
2. A pentagon has a 90 ο angle and a 100 ο angle. The other three angles are in the
ratio of 8 : 12 : 15. Find the measure of the other three angles.
3.
Given : AB // CD
Find : m∠1 + m∠ 2 + m∠3
Lines, Angles, and Polygons
January 1990
1. A regular hexagon is inscribed in a circle. Find the ratio of the length of a side of
the hexagon to the length of the shorter arc intercepted by the side. State your
answer in terms of π .
2. Find the number of sides of the regular polygon inscribed in a circle if each vertex
angle is three times as large as each central angle determined by a side of the
polygon.
3. Given two equiangular polygons P1 and P2 with different number of sides: each
angle of P1 is x degrees and each angle of P2 is kx degrees where k is an integer
greater than 1. Find all the possible number of sides for P1 and P2.
Lines, Angles, and Polygons
December 1991
1. Two consecutive angles of a quadrilateral are 62 and 83. Find the angle P made
by the bisectors of the other two angles (Refer to figure below).
2. The shorter diagonal of a rhombus is ¾ of the length of the other diagonal. If the
perimeter of the rhombus is 60 cm., find the perimeter of the quadrilateral that
joins the midpoints of the sides of the rhombus.
3. In the diagram.
AB = AC = CD = DE
1
m∠E = m∠CDE
10
Find m∠BAF
Lines, Angles, and Polygons
February 1992
1. Find the length of the shortest diagonal of a regular hexagon with a perimeter of
110 ft. Express your answer to the nearest tenth of a foot.
2. A regular octagon is formed by clipping the corners off a square that is 4 ft. by 4
ft. Find the exact perimeter of the octagon.
3. If the number of sides of a regular polygon is increased by 4, the resulting regular
polygon will have an interior angle that is 3 degrees more than the measure of an
interior angle of the original polygon. How many sides does the original polygon
have?
Lines, Angles, and Polygons
February 1993
1. Ted drew a large circle and labeled 39 distinct points on it. He then proceeded to
connect pairs of these points in as many ways as he possibly could. He found that
there were 39N ways. Find N.
2. The pentagon is circumscribed about the circle shown. The diameter of the circle
is 20. P, Q, R, S, and T are points of tangency. AS = 3, BS = 16, CQ = 6, DQ = 5,
and PE = 4. Find the area of the pentagon.
3. If the number of sides of a regular polygon is increased by three, the number of
degrees in each angle is increased by 20. Find the number of degrees in each
angle of the new polygon.
Lines, Angles, and Polygons
December 1992
1. Given :
DE = EF , AD = AF
m∠C = 37 ο, m∠ A = 34 ο
m∠G = 47 ο
Find : m∠ABH
2. Given : Quadrilateral ABCD with
right angles at A and C.
BF and DE are ⊥ to AC.
AF = 3, BF = 5, FC = 7.
Find the length of DE.
3. Given :
BC = 15, CF = 10
FE = 6, AE = 26
BE = CD
BD bisects ∠ABC
Find the length of AD
Lines, Angles, and Polygons
December 1993
1. Let ABCDEFGHIJ be a regular decagon. Find the sum of the angles,
alphabetically, K through T.
2. A regular convex polygon has 20 diagonals. If the length of longest diagonal is x
units, find the length of the shortest diagonal in terms of x.
3.
Given : CD = DE = AE
AC = AB
CE = CB
Find the measure of ∠ECB
Lines, Angles, and Polygons
December 1994
1. A regular pentagon has all of its diagonals drawn. Find the number of degrees in
the “point” of the star formed by the diagonals.
2.
Find the number of degrees in each angle of a regular polygon that has 54
diagonals.
3. Triangle ABC is isosceles. AB = AC = 10 cm. BC = 5 cm. BX is the median and
BY is the altitude to side AC . Find the centimeter length of XY in simplest form.
Lines, Angles, and Polygons
December 1995
1. Given quadrilateral ABCD with m∠D = 130 ο and m∠B = 72 ο . The bisector of
angle A intersects the line through C parallel to segment AB at point F. If
quadrilateral ADCF is a trapezoid, find the measure of angle BCD.
2. The sum of the exterior angles of polygon A is equal to the sum of the interior
angles of a polygon with ten less sides. How many sides does the polygon A
have?
3.
A regular hexagon has an area of 3 864 sq. cm. Three squares are attached to the
exterior of a hexagon so that a side of each square is a side of the hexagon. Three
equilateral triangles are attached in a similar fashion to the other three sides of the
hexagon. Find the perimeter in centimeters of the concave polygon formed in this
manner.
Lines, Angles, and Polygons
December 1996
1. What is the maximum number of segments that can be drawn to connect 6
different points?
2. Given : Regular pentagon QRSTV
and the rest of the figure as drawn.
TS = 20, QL is the altitude to MN .
If ML = p and the perimeter of
Pentagon ABCDE is x, find x in
terms of p.
3. In triangle ABC, AC = CD = AE
and m∠B = 45 ο . If m is the measure
of angle ACD, and n is the measure
of angle CAE, find the value of m + n.
Lines, Angles, and Polygons
December 1999
1. X is the midpoint of hypotenuse AC in right triangle ABC. Y is the midpoint of
side BC and BX = AB. Find m∠BXY .
2. Lizzie has determined the angle measures shown on the figure. Find the measure
b 10
of c in degrees, if = .
c 9
3. A regular hexagon with side length 4 is drawn with all nine diagonals. All
intersection points between line segments in the drawing are labeled with letters.
Find the shortest distance between two lettered points. Give exact answer.
Lines, Angles, and Polygons
December 2000
1. The measures of the angles of a pentagon are in the ratio of 7:8:9:10:11. Find the
sum of the smallest and largest angles of the polygon.
2. A polygon has ten times as many diagonals as it has sides. How many sides does
it have? The polygon has more than three sides.
3. The diagonals of regular hexagon ABCDE form another regular hexagon
MNOPQR. The diagonals of this hexagon form another regular hexagon
STUVWX. If the perimeter of ABCDE is 72, find the perimeter of STUVWX.
Lines, Angles, and Polygons
December 2001
1. On hypotenuse AB of right triangle ABC, D is the point for which CB = BD. If
the measure of angle ABC = 40 ο , find the measure of angle ACD.
2. In the diagram, BD = 6 km, AB = 3 km, and DE = 5 km. What is the number of
kilometers in AE?
3. Quadrilateral MATH contains right angles at vertices A and H. If
m∠AMH = 120 ο , MA = 10, and MH = 40, find TH. Give exact answer.
Lines, Angles, and Polygons
December 2002
1. A regular polygon has 20 diagonals. Find the degree measure of one of its angles.
2. In the figure, AC = CD = BD and m∠BDC = 5m∠ACD. Find m∠ACB.
3. ABCD is a trapezoid.
1
AB = BC = AD = DC , MQ ⊥ DC and AC , BD, and MQ meet at F . If BD = 12 units,
2
find the measuremen t of MQ.
Lines, Angles, and Polygons
December 2003
1. Given: AB // CD
PX bisects ∠BPQ
QX bisects ∠PQD
m∠RQS = 18ο
Find : m∠QPX
2. Four towns are shown at different locations on a map. It is 4 miles from town A
to town B, four miles from town A to town C, 4 miles from town B to town C, 4
miles from town B to town D, and 4 miles from town C to town D. Find the exact
distance from town A to town D.
3. The degree measure of each angle of a regular octagon is the same as the number
of diagonals in a regular n-gon. Find the sum of all the angles of the n-gon.
Lines, Angles, and Polygons
December 2004
1. Triangle ABC is equilateral. m∠PDQ = 90 ο . Find the sum of x and y.
2. Squares are attached to the nonconsecutive sides of the regular hexagon as shown
in the figure. The figure begins to form 3 more regular polygons at vertices P, A,
B, Q; R, C, D, S; and T, E, F, U. If AB = 6, find the sum of the perimeters of all
the regular polygons including the 3 to be formed.
3. Use the figure and information from number 2. Find the perimeter of the hexagon
PQRSTU. Express in simplest form.
Lines, Angles, and Polygons (No Calculators)
December 2005
1. The side lengths of a triangle are 4 cm, 6 cm, and 9 cm. One of the side lengths
of a similar triangle is 36 cm. What is the maximum number of centimeters
possible in the perimeter of the second triangle?
2. A regular convex polygon has 90 diagonals. Find the measure of each of its
angles.
3. What is the length of the radius of a circle inscribed in a rhombus with diagonals
of length 10 feet and 24 feet? Give your answer as a rational number in simplest
form.
Lines, Angles, and Polygons (No Calculators)
December 2006
1. A regular n-gon has 27 diagonals. How many vertices does the n-gon have?
2. Convex quadrilateral ABCD exists such that AC ∩ BD at x.
AC ⊥ BD, AC = 10 , and BX = DX = AX = 3 units. Find the unit measure of
the perimeter of ABCD. Express your answer in simplest radical form.
3. Define a “kite” to be a convex quadrilateral with one diagonal being a
perpendicular bisector of the other. The smallest and largest angles of a kite are
60° and 120°. The shortest side has length of 6 6 . Find the sum of the lengths
of the diagonals of all the different kites that fit these specifications.
Lines, Angles, and Polygons (No Calculators)
December 2007
1. A solid figure has 8 faces. Two of the faces are congruent regular hexagons in
horizontal planes, and the other six are congruent rectangles in vertical planes.
How many diagonals can be drawn between the vertices of the solid that pass
through the interior of it?
2. A, B, and C are vertices of a regular polygon with N sides, part of which is shown
in the drawing. Find the sum of angles θ , φ , α , β , and γ .
α = m∠ABC
3. Quadrilateral ABCD has the property that all four vertices are equidistant from a
point in the plane containing ABCD. If the measure of angle A is 70º, find the
measure of angle C.
Complex Numbers
December 1988
1. Divide 6 + 3i by 2 + 4i. Give your answer in a + bi form.
2. For what value(s) of k , k ≠ −i will (k + i ) be a real number?
4
3. Find a polynomial f(x) of degree five with real number coefficients that has zeros
of 1, 5i, and 1- i.
Complex Numbers
December 1989
1. Find the reciprocal of 3 – 4i. State your answer in a + bi form with a and b as real
numbers.
2. Find all the roots of 9 x 4 − 21x 3 − x 2 + 19 x − 10 = 0.
3. Find the value of
i ⋅ i 4 ⋅ i 7 ⋅ i 10 ⋅ i 13 ⋅ ... ⋅ i 58 ⋅ i 61 ⋅ i 64
in a + bi form.
i + i 4 + i 7 + i 10 + i 13 + ... + i 58 + i 61 + i 64
Complex Numbers
December 1991
1. Evaluate x 3 − 2 x − 4, if x = −1 + i
2. Simplify and express your answer in a + bi form:
i 18 − 3i 11
5i 100 + i 53
3. For what positive real value(s) of k will (2 + ki)3 be a real number?
Complex Numbers
December 1992
1. Find the complex conjugate of:
1
Express answer in simplest a + bi form.
4 + 3i
2. Find all value(s) of x for which 2 x 2 − ix + 1 = x 2 + 2ix + 3.
1
3
1
3
+
i and y = − −
i then what does x 7 + y 7 equal, in simplest
2 2
2 2
a + bi form?
3. If x = −
Complex Numbers
December 1994
1. If z = 6, w = 4 - i and v = 2 – i, then find v (z – w) in simplest form.
2. Express (3-3i)10 as a complex number in simplest a + bi form, without exponents.
3. Determine the square root(s) of –5 + 12i.
Complex Numbers
March 1994
1. If z1 = 2 − 3i and z 2 = −4 − i, find 3z 1 2z 2 . Express your answer in a +bi form.
2. One of the roots of the equation z 3 + 3 z 2 − iz 2 − 3iz + 2 z + 6 = 0 is − i. What are
the others?
3. Find the 6 roots of the equation z 6 = −i .
Complex Numbers
December 1995
4. Simplify:
2 + 3i
Express in simplest a + bi form.
i (4 + i )
5. Solve the following for x and express in simplest a + bi form:
2 + 3ix − (4 + 5i ) = (4i − 3)x − 5i
6. 1 + 2i is a root of 7x3 – 29x2 + 65x – 75 = 0. Find all other roots.
Complex Numbers
December 1996
1. If A = 3 + 5i, B = 7 – 2i, C = 8 + 4i, then determine in a + bi form the value of
A – BC.
(
2. Evaluate − 1 − 3i
)
7
3. Find the two square roots of 5 + 12i.
Complex Numbers
December 1998
1. Solve for x, where x is a complex number in the form of a + bi and a and b are
real numbers, if 3 x + 2i = (4 + 6i )i
2. Express the sum of the reciprocals of z1 and z 2 in simplest a + bi form, where a
and b are real, if z1 = 2 + 3i and z 2 = 3 − 2i .
x1
π
π 
5π 
5π


3. If x1 = 6 cos
+ i sin
in a + bi form,
 and x 2 = 3 cos + i sin , find
12 
12
12 
12
x2


where a and b are real.
Complex Numbers
December 1999
1. Find all solutions for x, in a + bi form, if x 2 − 2ix − 2 x + 2i = 0.
2. There are only two numbers that meet the following specifications:
z3 is purely imaginary
z6 is real
z is purely imaginary
|z| = 2
Find the product of these two numbers.
3. Let r1 and r2 be two complex roots of a quadratic equation with integral
coefficients. If r1 + r2 = 4 and r1 r2 = 9, find
r1
r2
Complex Numbers
December 2000
1. Express the following fraction as a complex number in simplest a + bi
(2 + 3i )(2 − 3i )
form.
5 + 12i
2. Find the ordered pair (x, y) for which (15 + 9i )x + (4i − 7 ) y = 39 + 48i
3. Find all ordered pairs (x, y) which are solutions of the system of equations:
xy = −6
x− y =2
Complex Numbers
December 2001
1. Express (2 + i )(3 + i )(2 + i )(3 + i ) as a complex number in a + bi form. Then find
the sum of a + b.
2. One root of the equation 4 x 2 + kx + 13 = 0 is
2 + 3i
, find k.
2
3. If 1 + i is a root of x 4 − 2 x 3 + 7 x 2 − 10 x + 10 = 0 , find the other root(s).
Complex Numbers
December 2002
1. If x = 3 − 2i and y = 2 − 3i , find the complex number
x
in a + bi form, where a
y
and b are real numbers. Express in simplest form.
2. If n = (3 + 3i ) , find the complex number n in simplest a + bi form.
4
3. The equation x 4 − 10 x 3 + 42 x 2 − 82 x + 65 = 0 has 4 complex roots. If one of
these is 2 + i, what are the other three?
Complex Numbers
December 2003
1. Find the product of 4 + 12i and 3 – 9i.
2. If the product of the two solutions of a quadratic polynomial is 4 and the sum of
the solutions is -3, what is the difference of the two solutions? Express your
answer without a negative sign.
3. Define a number to be a squareback if its square equals its complex conjugate.
That is, z = a + bi, where a and b are real numbers, is squareback if z 2 = z .
Determine how many squareback numbers exist in the complex plane.
Complex Numbers
December 2004
1. Simplify completely: (3 + 2i)(3 – 2i) – (2 – 3i)(2 + 3i)
2. Express
4 + 3i
as a complex number in simplest a + bi form.
2−i
3. P is a complex number in the form a + bi, where a and b are integers. If
P(P + 1) = -19 + 77i, find P.
Complex Numbers (No Calculators)
December 2005
− 2 − 3i
1. Express
as a complex number in a + bi or (a + bi ) c form.
4 + 3i
2. Find the polynomial equation P( x ) = 0 , with real coefficients and leading
coefficient 1 (one) which has exactly three solutions, two of which are 2 and –3i.
Express your answer in standard form.
3. Given that -i is a root of the equation iz 3 − 2 z 2 − 1 = 0 , find the other roots.
Complex Numbers (No Calculators)
December 2006
1. If (2 + 3i )D = 3 − i , find D in the form of a + bi, where a and b are rational
numbers in simplest form.
2. x = 2 − 3i and y = 1 − 2i . Find xy in simplest radical form.
3. Expand (1 + i ) and express in simplest form.
18
Complex Numbers (No Calculators)
December 2007
1. Express in simplest form: 2i − 4i 2 i 3 − 1 .
(
)(
)
2. Find the value of y – x, if (8 + 9i )x + (12i − 11) y = 96i − 23 .
3. Find the least order polynomial equation in x with real coefficients and a real
constant, and 1 as a leading coefficient that has both 1 – i and -1 + i as roots.