Problem Solving Problems for Group 3(Due by EOC Feb. 4)

Exactly How Do You Want Your Million?

1. 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 1,000,000 1,000,000x x .}

Stand On Your Heads And Get It Together.

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

{Hint: 50x y , 25xy , so divide the first equation by the second equation.}

b) The sum of the squares of two numbers is 50, and their product is 25. Find the ratio of the

two numbers.

Mind Your Four’s And Two’s

3. What is the value of x if 20,000 20,0008 8 2x ?

{Hint: Factor 20,000 20,0008 8 and use the fact that 38 2 .}

Cover All Your Bases, If It’s Within Your Power.

4. Solve for x if 2 9 20

2 5 5 1x x

x x

.

{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.}

The Collapse Of Rationalism.

5. Find the exact value of 1 1 1 1

1 2 2 3 3 4 999,999 1,000,000

.

{Hint: Rationalize the denominators. For example:1 1 2 1 2

2 111 2 1 2

.}

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

6. If 1 1f , 2 3f , and 1 2f n f n f n for 3,4,5,n

a) What is the value of 12f ?

{Hint: 3 2 1 1 3 4f f f , 4 3 2 4 3 7f f f ,…Keep going.}

Amazingly, f can be represented as n nf n x y for 1,2,3,4,5,n

b) Find the values of x and y .

{Hint: 1 , 1 1f x y f , 2 22 , 2 3f x y f , this should be enough to find

values of x and y.}

c) Alicia always climbs steps 1, 2, or 4 at a time. For example, she climbs 4 steps by

1-1-1-1, 1-1-2, 1-2-1,2-1-1,2-2, or 4. In how many ways can she climb 10 steps?

{Hint: If f n represents the number of ways to climb to the nth step, then

1 2 4f n f n f n f n .}

Highs And Lows In The Classroom.

7. a) 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: 1 2 3 4 5 61 2 3 4 5 693 558

6

T T T T T TT T T T T T

1 2 3 4 5 6558T T T T T T , so 1T will be as small as possible when

2 3 4 5 6T T T T T is as large as possible.}

b) If the average for the six tests is 16, what is the highest possible grade on any one of the

tests?

Looky Here Son, This Is A Problem, Not A Chicken.

8. 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.}

The Last One Standing

9. Find the ones digit of 2421 378313 17 .

{Hint: Look for a pattern:

Powers of 13 One’s-digit Powers of 17 One’s digit 113 3 117 7 213 9 217 9 313 7 317 3 413 1 417 1 513 3 517 7 613 9 617 9

}

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

10. a) What is the largest amount of money in current 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.}

b) A collection of coins is made up of an equal number of pennies, nickels, dimes, and

quarters. What is the largest possible value of the collection which is less than $2?

c) Trina has two dozen coins, all dimes and nickels, worth between $1.72 and $2.11. What

is the least number of dimes she could have?

Solving Without Completely Solving.

11. If 2a b and 2 2 3a b , then find 8 8a b .

{Hint: Start squaring and substituting.}

Gerry Benzel’s Favorite Problems.

12. a) A bottle and a cork together cost $1.10. If the bottle costs $1.00 more than the cork, what

does the cork cost?

{Hint: Let x be the cost of the cork and y the cost of the bottle.}

b) A shirt and a tie sell for $9.50. The shirt costs $5.50 more than the tie. What is the cost

of the tie?

Two Smokin’ Good Problems In One.

13. a) 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.}

b) Hobo Hal makes his cigars by connecting 5 cigar butts. Hal smokes 4

5 of a cigar and

then stops, leaving a cigar butt. If Hal finds 625 cigar butts, how many cigars will he be

able to make and smoke?

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

14. a) 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?

{Hint:

}

b) Do the same calculation if the horse can graze at a distance of 22 feet from the corner of the

corral where the rope is tied.

18

18

8

8

Round And Round With Donald And Hillary

15. Donald and Hillary are racing cars around a track. Donald can make a complete circuit in

72 seconds, and Hillary completes a circuit in 68 seconds.

a) If they start together at the starting line, how many seconds will it take for Hillary to pass

Donald at the starting line for the first time.

{Hint: Every time Donald reaches the starting line must be a multiple of 72 seconds, and

every time Hillary reaches the starting line must be a multiple of 68 seconds. So

they will both be at the starting line at common multiples of 72 and 68.}

b) If they start together, how many laps will Donald have completed when Hillary has

completed one more lap than Donald?

{Hint: Let n be the number of laps completed by Donald, then 72 68 1n n .}

It Squares; It Cubes; It Does It All!

16. If 3 3 10a b and 5a b , then what is the value of 2 2a ab b ?

{Hint: 2 2 3 3a b a ab b a b .}

It’s Just Gotta Be 6.

17. a) Show that for any integer n, 6 divides 3n n .

{Hint: 3 2 1 1 1n n n n n n n }

b) Show that for any integer n, 6 divides 5n n .

Real-valued Function, What’s Your Function?

18. a) Suppose that f is a real-valued function with 20142 3x

f x f x for all 0x . Find

2f . {Hint: Plug in 2 and 1007, and solve for 2f .}

b) Suppose that f is a real-valued function with 21

23

x

f xf x

x for all 0x . Find

2f .

On The Mark, Off The Mark, Or Bull’s Eye?

19. In the following figure, the curves are concentric circles with the indicated radii. Which

shaded region has the larger area, the inner circle or the outer ring?

Calculate the area of each region and check your visual estimation ability.

Don’t Put All Of Your Eggs In One Basket!

20. If eggs are taken from a basket two at a time, then one egg remains in the basket. If eggs are

taken three at a time from the same basket, then two eggs remain in the basket. If eggs are

taken four at a time from the same basket, then three eggs remain in the basket. If eggs are

taken five at a time from the same basket, then four eggs remain in the basket. If eggs are

taken six at a time from the same basket, then five eggs remain in the basket. If eggs are

taken seven at a time from the same basket, then no eggs remain in the basket. What is the

fewest possible number of eggs in the basket?

{Hint: If N is the number of eggs in the basket, then 1N must be a multiple of 2, 1N

must be a multiple of 3, 1N must be a multiple of 4, 1N must be a multiple of 5, and

1N must be a multiple of 6. So 1N must be a multiple of the LCM of 2, 3, 4, 5, and 6.

Also, N must be a multiple of 7.}

3 5

4

Sometimes Reduction Can Get In The Way Of Induction.

21. Observe that

2

2 2

2 2 2

2 2 2 2

1 31

2 4

1 1 41 1

2 3 6

1 1 1 51 1 1

2 3 4 8

1 1 1 1 61 1 1 1

2 3 4 5 10

.

Use inductive reasoning to make a conjecture about the value of

2 2 2 2

1 1 1 11 1 1 1

2 3 4 n

.

Use your conjecture to find the value of 2 2 2 2

1 1 1 11 1 1 1

2 3 4 1,000,000

.

Middleaged At 40?

22. The counting numbers are written in the pattern below. Find the middle number of the 40 th

row.

1

2 3 4

5 6 7 8 9

10 11 12 13 14 15 16

Shifty Four.

23. The leftmost digit of a 6 digit number is 4. If it’s shifted to be the rightmost digit, the new

number is one-fourth of the original number. What’s the original number?

{Hint: If the original number is 4x abcde , then 10 4 4 4x abcde , and if you subtract the

right number from 10 4x , you’ll have the new number.}

Zero Is My Hero.

24. If a and b are two unequal numbers, and ax bx , then find the exact numerical value of

3x

a b .

Slide Your Way Into An Answer.

25. What fractional part of the figure is shaded(assuming that line segments that appear parallel

actually are, all angles are right angles, and the vertical line segments are equally spaced.)?

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.

Sister Act.

27. Three sisters leave home on the same day. One returns every 5 days, another returns every

4 days, and the third returns every 3 days. How many days until all three sisters meet at

home again for the first time?

The People Under The Stairs.

28. Three rectangles are connected as in the figure. The first rectangle is 2 by 1; the second

rectangle is 4 by 2; the third rectangle is 8 by 4. A line is drawn from a vertex of the

smallest rectangle to a vertex of the largest rectangle. Find the area of the shaded region.

A B C D

A 0 ? 21 9

B ? 0 5 7

C 21 5 0 12

D 9 7 12 0

2 4 8

4

1 2

What’s The Difference?

29. Consider the two overlapping rectangles below:

What is the difference between the areas of the non-overlapping regions of the rectangles?

{Hint: The area of the first non-overlapping region is 80 xy . Find the area of the second

non-overlapping region, and subtract it from the first.}

Mission: Impossible/Possible Perimeters.

30. A rectangle has the following sides. Find all possible numerical values of the perimeter of

the rectangle.

8 feet

10 feet

5 feet

7 feet

x

y

2x y

4 8y

20 16x

Mary, Mary, Not So Contrary.

31. Lottie and Lucy Hill are both 90 years old. Mary Jones, on the other hand, is half again as

old as she was when she was half again as old as she was when she lacked 5 years of being

half as old as she is now. How old is Mary?

Hint: 3 3 12 2 2

5M is half again as old as she was when she was half again as old as she

was when she lacked 5 years of being half as old as she is now, and M is her age now.

Which One Are You Rooting For?

32. Which number is larger, the 10,000,000th root of 10 or the 3,000,000th root of 2?

Hint: Raise both numbers to the 30,000,000th power to decide.

Sometimes You Got To Kiss A Lot Of Frogs.

33. There are some princes and frogs in a fairy-tale. As a group, they have 35 heads and 94

feet. How many princes, and how many frogs are there?

I Guess I Have To Spell It Out For You.

34. How can 9 horses be used to fill the following row of 10 horse stalls?

{Hint: See the title of the problem.}

Irrationally Yours.

35. Show that there is an irrational number x so that 2x is a rational number.

{Hint: Give 2 a try for x.}

Eight Is Not Enough.

36. What is the fewest number of cards that can be selected without looking from a standard 52-

card deck to guarantee that at least 2 cards from each of the 4 suits will be selected?

{Hint: What if you get 13 hearts followed by 13 diamonds followed by…?}

Don’t Bank On The Old Switcheroo.

37. An absentminded bank teller switches the dollars and cents when he cashed a check for Mr.

Spencer, giving him dollars instead of cents, and cents instead of dollars. After buying a

five cent newspaper, Mr. Spencer discovered he had left exactly twice as much as his

original check. What was the original amount of the check?

{Hint: If D = original number of dollars and C = original number of cents, then the

original amount of the check in cents is 100X D C and the incorrect amount of

the check in cents is 100Y C D . This leads to the equation

100 5 2 100C D D C , which can be rearranged into 98 199 5C D or

199 5

98

DC

. We need to find D so that C is a whole number of at most 2-digits.}

D

199 5

98

DC

1 2.081632653

2 4.112244898

3 6.142857143

4 8.173469388

5 10.20408163

6 12.23469388

7 14.26530612

8 16.29591837

9 18.32653061

49 99.55102041

Pyramid Power.

38. In the following pyramid, the number in each stone is found by adding together the numbers

in the two stones below it. Complete the pyramid.

2 5 8 6

7 14

You May Be A King Of Comedy, But Do You Know How A Clock Works?

39. On a recent episode of Who Wants to Be a Millionaire with Cedric the Entertainer, the

following question appeared.

For which of the following times will the minute and hour hands of a clock form a right

angle?

a) 4:05 b) 5:20

c) 3:35 d) 11:50

The contestant chose answer a) and he was told that he was correct. He wasn’t correct, in

fact, none of the options are correct. For t measured in minutes after midnight, 6M t t

represents the cumulative angle of the minute hand in degrees, and 12

H t t represents

the cumulative angle of the hour hand in degrees. In order for the two hands to form a

right angle, the difference between the cumulative angle of the minute hand and the

cumulative angle of the hour hand must be an odd multiple of 90 . So we get that

12

112

2 1 90; 1,2,

6 2 1 90; 1,2,

2 1 90; 1,2,

180 2 1; 1,2,

11

M t H t n n

t t n n

t n n

nt n

a) Use the previous formula to find the number of times from one midnight to the next that

the minute and hour hands form a right angle.

{Hint: 180 2 1

# of minutes in a 24 hour period11

n .}

b) Use the same reasoning to find a formula for the times(in minutes after midnight) from

one midnight to the next(inclusive) that the minute and hour hands point in exactly the

same direction, and the number of times that it occurs.

c) Use the same reasoning to find a formula for the times(in minutes after midnight) from

one midnight to the next that the minute and hour hands point in exactly opposite

directions, and the number of times that it occurs.

Triangulate Your Answer.

40. What number is missing from the third triangle? Why?

6

51 7 9

9

88 8 16

8

71 6

Take It Step-By-Step, ‘Cause It’s All There In Black And White.

41. A "stair-step" figure is made up of alternating black and white squares in each row.

Rows 1 through 4 are shown. All rows begin and end with a white square. How many

black squares are in the 37th row?

It Takes Two To Know One.

42. A list of 8 numbers is formed by beginning with two given numbers. Each new number in

the list is the product of the two previous numbers. Find the first number in the list.