Download Problem Solving

Problem Solving(part 1)
Caution, This Induction May Induce Vomiting
2 3 4
3 4 5
1. Observe that 1  2  2  3 
, 1 2  2  3  3  4 
, and
3
3
456
.
1 2  2  3  3  4  4  5 
3
Use inductive reasoning to make a conjecture about the value
1  2  2  3  3  4   n  n  1 .
Use your conjecture to determine the value of 1 2  2  3  3  4   100,000 100,001.
of
Oh Brother! No, Oh Sister!
2. A boy has twice as many sisters as brothers, and each sister has two more sisters than
brothers. How many brothers and sisters are in the family?
{Hint: Let b be the number of boys and g the number of girls. Now write down some
equations.}
Exactly How Do You Want Your Million?
3. Find a positive number that you can add to 1,000,000 that will give you a larger value than if
you multiplied this number by 1,000,000? Find all such numbers.
{Hint: Let the positive number be x, and solve x  1,000,000  1,000,000x .}
Interesting Is In The Eye Of The Beholder
4. There is an interesting five-digit number. With a 1 after it, it is three times as large as with a
1 before it. What is the number?
{Hint: If x is the five-digit number, then x  abcde, abcde1  10 x  1,1abcde  100,000  x .}
Twenty-one, But Not Blackjack
5. Find the 21-digit number so that when you write the digit 1 in front and behind, the new
number is 99 times the original number.
{Hint: If x is the 21-digit number then x  abcdefghijklmnopqrstu ,
and abcdefghijklmnopqrstu1  10 x  1 ,
and 1abcdefghijklmnopqrstuv1  10,000,000,000,000,000,000,000  10 x  1 }
Stand On Your Heads And Get It Together
6. The sum of two numbers is 50, and their product is 25. Find the sum of their reciprocals.
{Hint: x  y  50 , xy  25 , so divide the first equation by the second equation.}
The Last Two Standing
7. What are the final two digits of 71997 ?
{Hint: Look for a pattern:
Power of 7
7 2  49
73  343
74  2401
75  16807
76  117649
Final two digits
49
43
01
07
49
}
Don’t Give Up; Don’t Ever Give Up!
f  x 1
8. Given that f 11  11 and f  x  3 
for all x , find f  2000  . First find
f  x  1
f 14  , f 17  , f  20  ,
.
{Hint: Look for a pattern:
n
11
f(n)
11
14
5
6
1
11
17

20

23
6
5
11
}
Mind Your Four’s And Two’s
9. What is the value of x if 4  4200  2x ?
200
{Hint: Factor 4200  4200 and use the fact that 4  22 .}
A Lot Of Weeks, But How Many Days Left Over?
10. What is the remainder when 215,110 is divided by 7?
{Hint: Look for a pattern in the remainders:
Power of 2 Remainder when divided by 7
2
21  2
4
22  4
3
1
2 8
2
24  16
5
4
2  32
}
Can You Just Tell Me How Old Your Children Are!
11. A student asked his math teacher, “How many children do you have, and how old are they?”
“I have 3 girls,” replied the teacher. “The product of their ages is 72, and the sum of their
ages is the same as the room number of this classroom.” Knowing that number, the student
did some calculations and said, “There are two solutions.” “Yes, that is so,” said the
teacher, “but I still hope that the oldest will some day win a math prize at this school.” The
student then gave the ages of the three girls. What are the ages?
{Hint:
Triple factors of 72
1,1,72
1,2,36
1,3,24
1,4,18
1,6,12
1,8,9
2,2,18
2,3,12
2,4,9
2,6,6
3,3,8
3,4,6
Sum of the factors
74
39
28
23
19
18
22
17
15
14
14
13
}
Cover All Your Bases, If It’s Within Your Power.
12. Solve for x if  x  5 x  5
2
x 2 9 x  20
 1.
{Hint: Any number raised to the zero power, except zero itself, equals 1. 1 raised to any
power is equal to 1. -1 raised to an even power is equal to 1.}
Who Needs Logarithms?
13. If 2  15 and 15  32 , then find the value of xy .
x
y
{Hint: Substitute the first equation into the second equation, and use an exponent property.}
I Refuse To Join Any Club That Would Have Me As A Member.
14. A club found that it could achieve a membership ratio of 2 Aggies for each Longhorn either
by inducting 24 Aggies or by expelling x Longhorns. Find x.
{Hint: Let L be the number of Longhorns and A be the number of Aggies, to get
2 A  24 2
A
.}

, 
1
L
1 Lx
I Cannot Tell A Fib(onacci), My Name Is Lucas.
15. If f 1  1 , f  2   2 , and f  n   f  n  1  f  n  2  for n  3,4,5,
a) What is the value of f 12  ?
{Hint: f  3  f  2   f 1  1  2  3 , f  4   f  3  f  2   3  2  5 ,…Keep going.}
Amazingly, f can be represented as f  n   x n  y n for n  1,2,3,4,5,
b) Find the values of x and y .
{Hint: f 1  x  y, f 1  1 , f  2   x 2  y 2 , f  2   2 , this should be enough to find values
of x and y.}
Seven Heaven or Seven…
16. Find the largest power of 7 that divides 343!. 343!  1 2  3  4 
 342  343
{Hint: The multiples of 7 occurring in the expansion of 343! are 7,14,21,28, ,7  49 .
The multiples of 7 2  49 occurring in the expansion of 343! are 49,98, ,49  7
The multiple of 73  343 occurring in the expansion of 343! is just 343.
There are no multiples of higher powers of 7 occurring in the expansion of 343!}
A Special Case Of The Chinese Remainder Theorem
17. The positive integer n, when divided by 3, 4, 5, 6, and 7, leaves remainders of 2, 3, 4, 5, and
6, respectively. Find the smallest possible value of n.
{Hint: n  3a  2, n  4b  3, n  5c  4, n  6d  5, n  7e  6 .
This means that
n  1  3 a  1 , n  1  4  b  1 , n  1  5  c  1 , n  1  6  d  1 , n  1  7  e  1 .
So
n  1 is a common multiple of 3, 4, 5, 6, and 7. What’s the least common multiple?}
How Low Can It Go?
18. The grades on six tests all range from 0 to 100 inclusive. If the average for the six tests is
93, what is the lowest possible grade on any one of the tests?
T1  T2  T3  T4  T5  T6
 T1  T2  T3  T4  T5  T6  558
6
 T1  558  T2  T3  T4  T5  T6  , so T1 will be as small as possible when
T2  T3  T4  T5  T6 is as large as possible.}
{Hint: 93 
Just Your Average Joe.
19. If Joe gets 97 on his next math test, his average will be 90. If he gets 73, his average will be
87. How many tests has Joe already taken?
{Hint: Let n be the number of tests he has already taken, and T the total number of points he
T  97
T  73
has already earned on the tests. Then
 90,
 87 .}
n 1
n 1
A Whole Lotta Zeroes
20. How many zeroes are at the end of the number 127! ? 127!  1  2  3  4 
126 127
{Hint: Zeroes come from factors of 10. Factors of 10 come from 5’s and 2’s. See the hint
for problem #16.}
2421
21. Find the ones digit of 13
{Hint: Look for a pattern:
Powers of 13
131
132
133
134
135
136
The Last One Standing
 173783 .
One’s-digit
3
9
7
1
3
9
Powers of 17
171
17 2
173
17 4
175
176
One’s digit
7
9
3
1
7
9
}
Happy 2009!
22. Find the 2009th digit in the decimal representation of
{Hint:
1
.
7
1
 0.142857 , so use a pattern.}
7
Zero, The Something That Stands For Nothing.
23. How many zeroes are at the end of the number 2300  5600  4400 ?
{Hint: Zeroes come from factors of 10. Factors of 10 come from 5’s and 2’s.}
Looky Here Son, This Is A Problem, Not A Chicken.
24. Foghorn C sounds every 34 seconds, and foghorn D sounds every 38 seconds. If they sound
together at noon, what time will it be when they next sound together?
Foghorn C
Foghorn D
12:00
12:00
sound
together
12:00:34
12:01:08
12:00:38
12:01:42
12:01:16
{Hint: Every time they sound together after noon will have to be both a multiple of 34
seconds after noon and a multiple of 38 seconds after noon.}
A European Sampler
25. A box contains 8 French books, 12 Spanish books, 9 German books, 15 Portuguese books,
and 18 Italian books. What is the fewest number of books you can select from the box
without looking to be guaranteed of selecting at least 10 books of the same language?
{Hint: What is the largest number of books you can select and still not have 10 books of the
same language? The answer to the problem is 1 more than the answer to the
previous question.}
.