Problem Solving Problems for Group 1(Due by EOC Sep. 13)

Caution, This Induction May Induce Vomiting!

1. a) Observe that 2 3 4

1 2 2 33

,

3 4 51 2 2 3 3 4

3

, and

4 5 6

1 2 2 3 3 4 4 53

.

Use inductive reasoning to make a conjecture about the value of

1 2 2 3 3 4 1n n .

Use your conjecture to determine the value of 1 2 2 3 3 4 100,000 100,001 .

b) Observe that 2

3 12

1 3 , 2

5 12

1 3 5 , and 2

7 12

1 3 5 7 .

Use inductive reasoning to make a conjecture about the value of 1 3 5 2 1n .

Use your conjecture to determine the value of 1 3 5 1,999,999 .

c) Observe that 1 1

1 2 1 1

,

1 1 2

1 2 2 3 2 1

, and

1 1 1 3

1 2 2 3 3 4 3 1

.

Use inductive reasoning to make a conjecture about the value of

1 1 1 1

1 2 2 3 3 4 1n n

.

Use your conjecture to determine the value of 1 1 1 1

1 2 2 3 3 4 999 1,000

.

Interesting Is In The Eye Of The Beholder

2. 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 , 1 10 1,1 100,000x abcde abcde x abcde x .}

Who Needs Logarithms?

3. If 2 15x and 15 32y , then find the value of xy .

{Hint: Substitute the first equation into the second equation, and use an exponent property.}

The Last Two Standing

4. What are the final two digits of 20187 ?

{Hint: Look for a pattern:

Power of 7 Final two digits 17 07 07 27 49 49

37 343 43 47 2401 01

57 16807 07 67 117649 49

}

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

5. What is the remainder when 15,1102 is divided by 7?

{Hint: Look for a pattern in the remainders:

Power of 2 Remainder when divided by 7 12 2 2 22 4 4 32 8 1

42 16 2 52 32 4

}

Seven Heaven or Seven…

6. 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 27 49 occurring in the expansion of 343! are 49,98, ,49 7

The multiple of 37 343 occurring in the expansion of 343! is just 343.

There are no multiples of higher powers of 7 occurring in the expansion of 343!}

Just Your Average Joe.

7. If Joe gets 97 on his next math quiz, his average will be 90. If he gets 73, his average will be

87. How many quizzes has Joe already taken?

{Hint: Let n be the number of quizzes he has already taken, and T the total number of points

he has already earned on the quizzes. Then 97 73

90, 871 1

T T

n n

.}

Happy 2018!

8. Find the 2018th digit in the decimal representation of 1

7.

{Hint: 1

70.142857 , so use a pattern.}

A European Sampler.

9. A box contains 8 French books, 12 Spanish books, 9 German books, 15 Russian books, 18

Italian books, and 10 Chinese 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.}

The Beast With Many Fingers And Toes.

10. How many digits does the number 6,666 20,0008 5 have?

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

Odds, Evens, What’s The Difference?

11. a) What do you get if the sum of the first 8,000,000,000 positive odd integers is subtracted

from the sum of the first 8,000,000,000 positive even integers?

{Hint: 2 4 6 8 16,000,000,000 1 3 5 7 15,999,999,999 }

b) What do you get if you subtract 8,000,000,000 from the sum of the first 8,000,000,000

even numbers?

Destination Cancellation.

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

factors 1 1 1 1

1 1 1 12 3 4 2,000,000

.

{Hint: Look for a pattern:

11

2

1

2

1 1 1 21 1

2 3 2 3

1

3

1 1 1 1 2 31 1 1

2 3 4 2 3 4

1

4

}

Just Gimme An A While I’m Hanging Out In The Library.

13. a) 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.}

b) The library in Johnson City has between 1000 and 2000 books. Of these, 25% are

fiction, 1

13 are biographies, and

1

17 are atlases. How many books are either biographies

or atlases?

Cogswell Cogs Or Spacely Sprockets?

14. In a machine, a small gear with 45 teeth is engaged with a large gear with 96 teeth. How

many more revolutions will the smaller gear have made than the larger gear the first time

the two gears are in their starting position?

{Hint: A revolution of the smaller gear is a multiple of 45 teeth, and a revolution of the

larger gear is a multiple of 96 teeth. So the gears are again in the starting positions

at common multiples of 45 and 96.}

Life Is Like A Box Of Chocolate Covered Cherries.

15. a) Assume that chocolate covered cherries come in boxes of 5, 7, and 10. What is the

largest number of chocolate covered cherries that cannot be ordered exactly?

{Hint: If you can get five consecutive amounts of cherries, then you can get all amounts

larger. Here’s why: Suppose you can get the amounts 23,24,25,26,27 , then by the

addition of the box of size 5, you can also get 28,29,30,31,32 , and another addition

of the box of size 5 produces 33,34,35,36,37 and so on. This would also be true of

seven consecutive amounts and ten consecutive amounts, but five consecutive

amounts would occur first. So look for amounts smaller than the first five

consecutive amounts.}

b) Do the same problem, except the cherries come in boxes of 6, 9, and 20.

Two Squares Don’t Get Along – A Difference Of Squares!

16. If 20,000 20,000 5x y , 10,000 10,000 4x y , 5,000 5,000 3x y , 2,500 2,500 2x y , and 2,500 2,500 1x y , then find the value of 40,000 40,000x y .

Hint: 2 2a b a b a b .

I Hate This Problem To The Nth Degree.

17. Use the following properties of exponents to find the exact value of the given expressions.

n m n mx x x , n n nxy x y ,

mn nmx x

a) 90,000 90,000

89,999 90,000

3 6

2 9

b)

100,000 110,000

100,000 5000

2 3

6 9

Getting Solutions Without Actually Solving.

18. Notice that

2x a x b x a b x ab

3 2x a x b x c x a b c x ab ac bc x abc

4 3 2x a x b x c x d x a b c d x ab ac ad bc bd cd x

abc abd acd bcd x abcd .

a) Use inductive reasoning to determine the value of the coefficient of 1nx and the constant

term in the expansion of the following product: 1 2 nx a x a x a .

b) Use the previous result to determine the sum of the seventeenth powers of the 17

solutions of the equation 17 3 1 0x x .

{Hint: The Fundamental Theorem of Algebra guarantees that the equation 17 3 1 0x x has seventeen solutions(counting duplicates). The seventeen

solutions of 17

1 2 17 3 1 0x a x a x a x x are 1 2 17, , ,a a a . So

adding the seventeen equations together yields:

17

1 1

17

2 2

17

17 17

17 17 17

1 2 17 1 2 17

3 1 0

3 1 0

3 1 0

3 17 0

a a

a a

a a

a a a a a a

. Now use the previous result.}

Even So, It’s Odd.

19. Show that for every positive integer n, 2 3 8n n is even.

{Hint: n is either even or odd, so take it from here.}

An Odd Product.

20. Find the value of the following product:3 5 7 9 4021

1 1 3 5 4017

.

{Hint: What factors can you cancel out?}

Call It Like You See It.

21. Consider the following figure:

a) What fraction of the large square is shaded ?

b) What fraction of the large square is shaded ?

c) What fraction of the large square is shaded ?

Lucky 13, I Repeat, Lucky 13. Lucky 7, I Repeat….

22. a) Show that every six-digit number of the form abc,abc (for example 281,281 or 435,435)

is divisible by 13.

b) Show that every six-digit number of the form abc,abc (for example 281,281 or 435,435)

is divisible by 7.

c) Show that every six-digit number of the form abc,abc (for example 281,281 or 435,435)

is divisible by 11.

A Whole Lotta Eights; A Whole Lotta…

23. a) What is the smallest whole number that when multiplied by 9 gives a number whose

digits are all 8’s?

b) What is the smallest whole number that when multiplied by 9 gives a number whose

digits are all 5’s?

c) What is the smallest whole number that when multiplied by 9 gives a number whose

digits are all 3’s?

You’re A Real Square, Man.

24. A man born in the year 2x died, on his 87th birthday, in the year 2

1x . In what year was

he born?

It’s As Easy As 123… . 25. If the digit 9 is written to the right of a certain number, that number is increased by

111,111,111. Find the number.

{Hint: If the original number is x , then 10 9x is the new number.}

See How Everything Lines Up.

26. Given the following incomplete distance chart for 4 points in a plane, find the distance from

A to B.

Stick puzzles involve rearranging, removing, or adding sticks in order to accomplish the

requirements of a problem. In the following problems, you might want to use the following

suggestions:

This arrangement of three sticks can be used to represent a square root:

A stick representing an over bar can be used to multiply a Roman numeral by 1,000:

represents 5,000

A B C D

A 0 ? 21 9

B ? 0 5 7

C 21 5 0 12

D 9 7 12 0

Just Stick It.

27.

a) Move one stick to make a true equation.

b) Remove two sticks to make a true equation.

c) Move one stick to make a square.

d) Move one stick to make a true equation.

e) Move one stick to make a true equation.

What An Intersecting Little Problem.

28. Find the area of the following shaded region formed by the two perpendicular intersecting

rectangles.

Who Said Holes Have To Be Round?

29. Find the area of the square hole in the middle of the square.

{Hint: Find the areas of the four right triangles, and subtract it from the area of the large

square.}

The Arc Of Triangle.

30. In right triangle ABC with legs of 5 and 12, arcs of circles are drawn, one with center A and

radius 12, the other with center B and radius 5. What is the length of MN?

6

3

A

B

C

N

M

I Go Cuckoo For Coconuts.

31. Five sailors were stranded on a desert island, and their only food was coconuts. One day

they gathered all the coconuts on the island together, and the next day they would divide

them evenly. The first sailor woke up early and gave one coconut to a monkey and hid his

fifth of the remaining coconuts. Then the second sailor woke up and gave one coconut to

the same monkey and hid his fifth of the remaining coconuts. The third, fourth, and fifth

sailors all did the same. Upon arising the next day, one coconut was given to the monkey,

and the remaining coconuts were divided equally among the five sailors. What is the

smallest starting number of coconuts possible?

N is the starting number of coconuts, and R is the remaining coconuts after the five sailors

have done their secret removals.

Or start with a smaller problem: If one sailor gives a coconut to the monkey, takes one-fifth

of the remaining coconuts, and then another coconut is given to the monkey and the rest are

divided among the five sailors then, the number of remaining coconuts would have to be a

multiple of 5: 45

1 1 5N k . So 5 5 1

14

kN

, which means that 5 1k must be a

multiple of 4, and 16 is the smallest possible multiple of 4 that works. This means that the

smallest number of coconuts in this case would be 21.

If one sailor gives a coconut to the monkey, takes one-fifth of the remaining coconuts, and

another sailor gives a coconut to the monkey, takes one-fifth of the remaining coconuts, and

Sailor Remaining coconuts

1 45

1N

2 4 45 5

1 1N

3 4 4 45 5 5

1 1 1N

4 4 4 4 45 5 5 5

1 1 1 1N

5 4 4 4 4 45 5 5 5 5

1 1 1 1 1N

Sailor Remaining coconuts

1 5 5 5 5 625 3694 4 4 4 256 64

1 1 1 1R R

2 5 5 5 125 614 4 4 64 16

1 1 1R R

3 5 5 25 94 4 16 4

1 1R R

4 54

1R

5 R

another coconut is given to the monkey, and the rest are divided among the five sailors then,

the number of remaining coconuts would have to be a multiple of 5:

4 45 5

1 1 1 5N k . So 125 61

16

kN

, which means that 125 61k must be a

multiple of 16, and 1936 is the smallest possible multiple of 16 that works. This means that

the smallest number of coconuts in this case would be 121. Keep going!

I Don’t Give A Square’s S!

32. A right triangle with leg measurements of a and b has an inscribed square with side

measurement s as shown in the figure. Find the value of s.

{Hint: The areas of the square and two little right triangles must equal the area of the big

right triangle.}

The Age Of Man.

33. A man lived one-sixth of his life in childhood, one-twelfth in youth, and one-seventh as a

bachelor. Five years after his marriage, a son was born who died four years before his

father at half his father’s final age. What was the man’s final age?

Hint: 1 1 1 16 12 7 2

5M M M M is the age of the man up to the death of his son, and this is

4 years less than the man’s final age.

This Game’s Just A Pile Of Shi-llings.

34. A game involves a pile of 11 coins and two players who alternately take turns removing 1,

2, 3, 4, or 5 coins from the pile. The player who removes the last coin(s) wins the game.

How many coins should the first player remove in order to guarantee that he can win on

his next turn?

b

a

s

s

s s

Crazy 8.

35. Show that the difference of the squares of two odd numbers is divisible by 8.

Hint: Suppose the two odd numbers are 2 1x n and 2 1y m .

Then 2 22 2 2 22 1 2 1 4 4 1 4 4 1 4 1x y n m n n m m n m n m .

If you can show that 1n m n m must be divisible by 2, then you’re done.

n m n m 1n m 1n m n m

even even even odd ?

odd odd even odd ?

even odd odd even ?

odd even odd even ?

Where There’s A Will.

36. A father in his will left all his money to his children in the following manner:

$1000 to the first born and 1/10 of what then remains, then

$2000 to the second born and 1/10 of what then remains, then

$3000 to the third born and 1/10 of what then remains, and so on.

When this was done each child had the same amount.

a) How much money did the father leave in his will?

b) How many children were there?

{Hint: If S is the amount in the will, then the first born gets 1

1,000 1,00010

S , and the

second born gets 1 9

2,000 1,000 2,00010 10

S

. Set them equal to each other, and

solve for S.}

Finally, A Light A The End Of The Tunnel.

37. A train which is 1 mile long is traveling at a steady speed of 20 miles per hour. It enters a

tunnel 1 mile long at noon. At what time does the rear of the train emerge from the tunnel?

(Both ends of the tunnel are in the same time zone!)

{Hint: How far does the front of the train have to travel?}

If The Glove Don’t Fit, Then I Quit!

38. There are 20 gloves in a drawer: 5 pairs of black gloves(5 left and 5 right), 3 pairs of brown

gloves(3 left and 3 right), and 2 pairs of grey gloves(2 left and 2 right). You will select

gloves from the drawer in the dark, and you may check them only after the selection has

been made. What is the fewest number of gloves you need to select in order to be

guaranteed of selecting at least

a) one matching pair of gloves?(left and right of the same color)

b) one matching pair of each color?

Making The Most And The Least Of Them.

39. Using the four digits 5, 6, 2, and 9,

a) Make two 2-digit numbers that have the largest possible sum.

b) Make two 2-digit numbers that have the smallest possible positive difference.

c) Make two 2-digit numbers that have the largest possible product.

d) Make two 2-digit numbers that have the smallest possible product.

It’s More Magical!

40. The numbers in each row, column, and diagonal add up to 34. All the numbers from 1 to 16

appear, but only once each. Fill in the missing numbers.

16 6

4 10

14 8

2 12