# Problem of the Week

**1. Fractions**

a) Six friends were eating cake at a birthday party. The first one got 1/6 of a cake, the second 1/5 of the remaining part. The third one got 1/4 of the part left after the first two, the fourth one got 1/3 of what remained after the first three. The rest was split evenly between the last two friends. Who got the biggest piece?

b) How will you divide 5 apples equally between 6 kids so that no apple is cut in more than three pieces?

**2. Weighings**

There are 9 coins among which one is false (it is lighter then normal coins). How one can find false coin in two attempts by using just a balance without weights? (Here by the balance we mean 2-cup balance used in old times)

**3. Four numbers**

Four integers *a*, *b*, *c*, *d* produce 6 pairwise sums 2, 4, 9, 9, 14, 16. Is that possible? If *a*, *b*, *c*, *d* are not necessarily integers then what are their values?

**4. Cube**

There is a solid cube with an edge length 6 *cm* and painted red on the outside. Imagine that you cut it in small cubes each with an egde length 1 *cm*. How many of small cubes have exactly (a) one; (b) two; (c) three red sides? (d) not a sigle red side?

**5. Wastage**

A product weighed 100 pounds. Originally, 99% of the weight was composed of water. After a time the product became drier and now 98% of the product is composed of water. What is the current weight of this product?

**6. Travelling**

Anne and Bob travel from *A* to *B*. Anne rides a bicycle, while Bob drives a car. The car is two times faster than the bicycle. At the two thirds way point the car breaks, and the remaining one third way Bob walks with the speed two times smaller than that of Anne. Who arrives first to *B*?

**7. The Traveling Bee**

Two trains leave two towns that are 50 miles apart. They travel toward each other at rates of 30 *mph* and 20 *mph* respectively. A bumblebee flying at the rate of 50 *mph* starts out just as the faster train departs the train station, and flies toward the slower train. The bee then turns around and goes back to meet the faster train. Then it turns around again, etc., and it keeps flying back and forth between the trains until the trains meet. How far does the tired bumblebee fly?

**8. Three digits**

Attribute three more digits from the right to the number 1000 in such a way that the resulting number will be divisible by 7, 8 and 9.

**9. Sticks**

Each figure is made up of sticks. The number of sticks in the first figure is 4, in the second figure is 10, and in the third figure is 18. Continuing in the same pattern, how many sticks are in the 6^{th} figure? How many sticks are in the 100^{th} figure?

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**10. The Walk**

Father and son are workers, and every day they walk from home to the plant. The father covers the distance in 40 minutes, while the son covers the same distance in 30 minutes. In how many minutes will the son overtake the father if the latter leaves home 5 minutes earlier than the son?

**11. Searching and Sorting**

(a) I have a friend who answers only YES and NO to all questions. He lives in a 16 story building and I want to find out what floor does he live on. My goal is to ask as few questions as possible. How can I do that?

(b) Is it possible to sort an array of 4 numbers with just 5 comparison operations? and with 4 comparison operations?

**12. Two-digit Number**

A two-digit number AB has A as its ten's digit, which is less than or equal to the unit's digit B. When AB is multiplied by BA, the result is 2701. What is the two-digit number AB?

**13. Fractions**

Each of the equations below is missing numerators:

a) x/7 - y/5 = 1/35;

b) x/5 - y/7 = 1/35.

Find the possible pairs of numerators. Don't forget about mixed fractions.

**14. Rectangle**

A rectangle is cut into several rectangles. Perimeter of each smaller rectangle is an integer number of meters (perimeter of a rectangle is a sum of lengths of its sides). Is it necessary that perimeter of the original rectangle is an integer number of meters?

**15. Painting**

A fence consists of 20 boards. Each is to be painted in blue, green or yellow, and any two adjacent boards must have different colors. In how many ways can the fence be painted?

**16. Tourists**

A group of tourists was planning to cover a given distance in 5 hours. However, after the first two hours they have reduced their speed by 0.2 miles per hour and were 20 minutes late. What was their original speed?

**17. Archery contest**

An archery contest was held in two days. In the first day every participant has received as many points as all the other participants have received in the second day. Prove that each participant received the same number of points for the whole tournament.

**18. Pieces**

Anne tears a sheet of paper into four pieces, then she tears one of the pieces into four pieces, and so on. Can she end up with 2007 pieces?

**19. The difference**

By how much is the sum of all even numbers no greater than a hundred is greater than the sum of all odd numbers smaller than a hundred?

**20. Prime Numbers**

Put 5 prime numbers in a sequence in such a way that differences between any two neighbours would be the same (one can assume that difference = right_neighbour - left_neighbour).

**21. Ice Cream and Rolls**

6 scoops of Ice Cream are more expensive than 10 rolls, but cheaper than 5 chocolate bars. However 10 scoops of Ice Cream are more expensive than 8 chocolate bars. What is more expensive: 2 scoops of Ice Cream or 3 rolls?

**22. The Daughters' Ages**

A census taker came to a house where a farmer lived with three daughters. ``What are your daughters' ages?'' he asked. The man replied, ``The product of their ages is 72, and the sum of their ages is my house number.'' ``But that's not enough information,'' the census taker insisted. ``All right,'' answered the farmer, ``the oldest loves chocolate''. What are the daughters' ages?

**23. The difference of two squares**

Show that any odd number can be expressed as the difference of two squares, where each square is an integer squared.

**24. Grasshopper**

A grasshopper makes 25 jumps along the straight line in either direction. The length of the first jump is 1 *cm*, of the second jump — 2 *cm*, then 3 *cm*, etc. Can the grasshopper end up at the same point at which it started?

**25. Work together**

If it takes Anne 2 hours to paint a room, while it takes Bob 3 hours to do the same job, how long will it take if they work together yet independently?

**26. Plus Signs**

Insert plus signs between digits of the following number so that correct equality will be: 12345=33.

**Example**. Given: 12345=69. Solution: 1+23+45=69.

**27. Three piggy banks**

One millionaire has three piggy banks. In the first one he keeps just 20$ bills, in the second one he keeps just 10$ bills and in the last one he keeps only 5$ bills. There is the same number of bills in each of the three banks. How much money are there in each one of them if there is 1400$ in all of them together?

**28. Four straight lines**

Into how many parts can the plane be divided by four different straight lines? Give an example for each possible case.

**29. Is it possible to connect?**

Is it possible to connect 16 points by arcs so that each point will be connected with exactly 4 other points?

**30. Change**

A cashier has only 5 cent and 10 cent coins. In how many different ways can he give 50 cents of change?

**31. The flight**

A plane flied 300 miles south from Washington (DC), then 300 miles west, then 300 miles north and finally 300 miles east. Will the plane land to the south, north, east or west of Washington, or it will land at Washington?

**32. Four people**

Among 4 people no three have the same first, middle or last name. However any two of them have the same first or middle or last name. How is that possible?

**33. 100 pieces**

Anne and Bob are tearing up a sheet of paper. Anne only tears a piece of paper into 3 smaller pieces while Bob only tears a piece of paper into 5 smaller pieces. After a few minutes can there be exactly 100 pieces of paper?

**34. Barrels**

There are 21 barrels such that 7 of them are empty, 7 are half full with orange juice and 7 are full with orange juice. One needs to load this barrels into 3 trucks so that barrels and juice are divided equally between trucks. How can he do that?

**35. The sum of all digits**

What is the sum of all the digits (not the numbers) in the sequence 1, 2, 3, 4, 5, 6, 7, …, 999, 1000?

**36. The smallest number**

What is the smallest number divisible by each of the numbers 1 to 20?

**37. Cookies**

A boy ate 100 cookies in five days. Each day he ate 6 more than the day before. How many cookies did he eat on the first day?

**38. Straight Lines**

There are 10 points on a plane and no three of them lay on the same straight line. Through each 2 points a straight line passes. How many straight lines are there?

**39. The new average**

The average of seven numbers is 6. If 1 is added to the first number, 2 is added to the second number, 3 is added to the third number and so on up to the seventh number, what is the new average?

**40. The last digit**

What is the last digit of 7^{1000} ?

**41. To cut without a ruler**

How can one cut off a piece of ribbon 1/2 of a yard long from the piece of ribbon 2/3 of a yard long without a ruler?

**42. Checkerboard with removed cells**

Prove that an 8×8 checkerboard with two opposite corner cells removed cannot be covered without overlapping by 1×2 dominos.

**Hint.** Two cells covered by a domino are of opposite colors.

**43. The difference of the two sums**

The sum of the first 100 multiples of 4 is: 4+8+12+…+400. The sum of the first 100 multiples of 3 is: 3+6+9+…+300. What number is equal to the difference of the two sums?

**44. Continuation of the sequence**

In the following sequence 2, 6, 12, 20, 30, … find a number standing in the 2008^{th} place.

**45. Cub with a hole**

Is it possible to build a cube 3×3×3 with a hole 1×1×1 in the center using 13 bricks 1×1×2 ?

**Hint.** Paint bricks and cube in two colors, count number of small cubes 1×1×1 of each color.

**46. Ostriches and giraffes**

A zoo has several ostriches and several giraffes. They have 30 eyes and 44 legs. How many ostriches and how many giraffes are in the zoo?

**47. Area**

On an infinite grid, draw a square whose area is 5 times that of a square on the grid. Use a straight-edge but not a compass.