Problem Solving

Caution, This Induction May Induce Vomiting

1. Observe that , , and

.

Use inductive reasoning to make a conjecture about the value of .

Use your conjecture to determine the value 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 .}

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 is the five-digit number, then.}

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 is the 21-digit number then ,

and ,

and }

6. The sum of two numbers is 50, and their product is 25. Find the sum of their reciprocals.

{Hint: , , so divide the first equation by the second equation.}

The Last Two Standing

7. What are the final two digits of ?

{Hint: Look for a pattern:

Power of 7 / Final two digits
/ 49
/ 43
/ 01
/ 07
/ 49

}

Don’t Give Up; Don’t Ever Give Up!

8. Given that and for all , find. First find .

{Hint: Look for a pattern:

n / f(n)
11 / 11
14 /
17 /
20 /
23 / 11

}

9. What is the value of if ?

{Hint: Factor and use the fact that .}

A Lot Of Weeks, But How Many Days Left Over?

10. What is the remainder when is divided by 7?

{Hint: Look for a pattern in the remainders:

Power of 2 / Remainder when divided by 7
/ 2
/ 4
/ 1
/ 2
/ 4

}

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 / Sum of the factors
1,1,72 / 74
1,2,36 / 39
1,3,24 / 28
1,4,18 / 23
1,6,12 / 19
1,8,9 / 18
2,2,18 / 22
2,3,12 / 17
2,4,9 / 15
2,6,6 / 14
3,3,8 / 14
3,4,6 / 13

}

12. Solve for if .

{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 and , then find the value of .

{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 .}

I Cannot Tell A Fib(onacci), My Name Is Lucas.

15. If , , and for

a) What is the value of ?

{Hint: ,,…Keep going.}

Amazingly, can be represented as for

b) Find the values of and .

{Hint: , , 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!.

{Hint: The multiples of 7 occurring in the expansion of 343! are .

The multiples of occurring in the expansion of 343! are

The multiple of 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: .

This means that

. So 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?

{Hint:

, so will be as small as possible when is as large as possible.}

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 has already earned on the tests. Then .}

A Whole Lotta Zeroes

20. How many zeroes are at the end of the number ?

{Hint: Zeroes come from factors of 10. Factors of 10 come from 5’s and 2’s. See the hint for problem #16.}

The Last One Standing

21. Find the ones digit of .

{Hint: Look for a pattern:

Powers of 13 / One’s-digit / Powers of 17 / One’s digit
/ 3 / / 7
/ 9 / / 9
/ 7 / / 3
/ 1 / / 1
/ 3 / / 7
/ 9 / / 9

}

Happy 2009!

22. Find the 2009th digit in the decimal representation of .

{Hint: , so use a pattern.}

Zero, The Something That Stands For Nothing.

23. How many zeroes are at the end of the number ?

{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 / 12:00 / 12:00:34 / 12:01:08 / 12:01:42 /
Foghorn D / 12:00 / 12:00:38 / 12:01:16 /
sound together /

{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.}

I Hope That You Are A Digital Computer?

26. What is the ones digit of ?

{Hint: See what happens to the one’s digits:

Factorial / One’s-digit / One’s digit of the sum
1! / 1 / 1
2! / 2 / 3
3! / 6 / 9
4! / 4 / 3

}

The Collapse Of Rationalism

27. Find the exact value of .

{Hint: Rationalize the denominators. For example:.}

The Beast With Many Fingers And Toes

28. How many digits does the number have?

{Hint: Zeroes come from factors of 10. Factors of 10 come from 5’s and 2’s.}

Don’t Get Stumped; Use The Fundamental Theorem Of Arithmetic.

29. Forrest stump heard that there are only two numbers between 2 and 300,000,000,000,000 which are perfect squares, perfect cubes, and perfect fifth powers. He decided to look for them, and so far he has checked out every number up to about 100,000 and is beginning to get discouraged. What are the numbers he is trying to find?

{Hint: Every positive whole number greater than 1 can be written as a product of prime factors. If N is a positive whole number greater than 2, then . In order for N to be a perfect square, all the positive exponents would have to be multiples of 2; in order for N to be a perfect cube, all the positive exponents would have to be multiples of 3; and in order for N to be a perfect fifth power, all the positive exponents would have to be multiples of 5. So all the positive exponents would have to be common multiples of 2, 3, and 5.}

Sorry, I Can’t Give You Change For A Dollar.

30. What is the largest amount of money in U.S. coins(pennies, nickels, dimes, quarters, but no half-dollars or dollars) you can have and still not have change for a dollar?

{Hint: It’s more than 99 cents. For instance: 3 quarters and 3 dimes is \$1.05, but you can’t make change for a dollar.}

Destination Cancellation.

31. Express as a fraction, in lowest terms, the value of the following product of 1,999,999 factors .

{Hint: Look for a pattern:

}

Officer, I Got The License Plate Number, But I Was Lying On My Back.

32. The number on a license plate consists of five digits. When the license plate is turned upside-down, you can still read a number, but the upside-down number is 78,633 greater than the original license number. What is the original license number?

{Hint: The digits that make sense when viewed upside-down are 0, 1, 6, 8, and 9.

1st digit / 2nd digit / 3rd digit / 4th digit / 5th digit
Upside-down plate
Original plate
Difference of the plates / 7 / 8 / 6 / 3 / 3

}

One Smokin’ Good Problem

33. Mrs. Puffem, a heavy smoker for many years finally decided to stop smoking altogether. “I’ll finish the 27 cigarettes I have left,” she said to herself, “and never smoke another one.” It was Mrs. Puffem’s practice to smoke exactly two-thirds of each complete cigarette(the cigarettes are filterless). It did not take her long to discover that with the aid of some tape, she could stick three butts together to make a new complete cigarette. With 27 cigarettes on hand, how many complete cigarettes can she smoke before she gives up smoking forever, and what portion of a cigarette will remain?

{Hint: With 27 complete cigarettes, she can smoke 27 complete ones and assemble 9 new complete ones…, keep going.}

Just gimme an A.

34. A class of fewer than 45 students took a test. The results were mixed. One-third of the class received a B, one-fourth received a C, one-sixth received a D, one-eight of the class received an F, and the rest of the class received an A. How many students in the class got an A?

{Hint: The number of students in the class must be a multiple of 3, 4, 6, and 8, and must be smaller than 45.}

Working Backwards In Notsuoh

35. A castle in the far away land of Notsuoh was surrounded by four moats. One day the castle was attacked and captured by a fierce tribe from the north. Guards were stationed at each bride over the moats. Johann, from the castle, was allowed to take a number of bags of gold as he went into exile. However, the guard at the first bridge took half of the bags of gold plus one more bag. The guards at the second third and fourth bridges made identical demands, all of which Johann met. When Johann finally crossed all the bridges, he had just one bag of gold left. With how many bags of gold did Johann start?

{Hint: Sometimes working backwards is a good idea. If Johann has 1 bag of gold left, then how many did he have when he approached the fourth guard?

}

Grazin’ In The Grass Is A Gas, Baby, Can You Dig It?

36. A horse is tethered by a rope to a corner on the outside of a square corral that is 10 feet on each side. The horse can graze at a distance of 18 feet from the corner of the corral where the rope is tied. What is the total grazing area for the horse?