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Consider a drawing of a regular star, such as this:

http://i16.tinypic.com/64199b6.png

Notice that it has 5 distinct triangles, marked with green color in the drawing.

The problem: draw two straight lines in order to get 10 triangles. Note that only that if a triangle is cut in two pieces, only these pieces can be considered, the triangle overall now doesn't count.

For instance, in this drawing:

http://i15.tinypic.com/4uyklr5.png

there are only 7 triangles, not 11.

e^PI or PI^e?

There're 25 horses, numbered 1 through 25.

There's a race track.

Every horse's speed is constant: if horse #3 completes the track in 3:05 minutes, for instance, it will do so every time it races.

You can only race 5 horses at a time.

How many races will it take to find the fastest, second fastest, and third fastest horses?

There is an infinitely long staircase. It has stairs numbered 1, 2, 3, ..., n, n+1, ..... You are standing at the bottom of it.

There are two identical crystall balls in your hand.

Its known that if either of the balls is rolled down from any stair that's N or lower, it will not break. If it is however rolled down from N+1 stair, or any stair higher than that, it will break.

Design a way to find the number N in smallest possible number of steps.

For the sake of argument, assume N is large

There is an infinitely long staircase. It has stairs numbered 1, 2, 3, ..., n, n+1, ..... You are standing at the bottom of it.

There are two identical crystall balls in your hand.

Its known that if either of the balls is rolled down from any stair that's N or lower, it will not break. If it is however rolled down from N+1 stair, or any stair higher than that, it will break.

Design a way to find the number N in smallest possible number of steps.

For the sake of argument, assume N is large

There is an infinitely long staircase.

There are two identical crystall balls.

Its known that if either of the balls is rolled down from any stair that's N or lower, it will not break. If it is however rolled down from N+1 stair, or any stair higher than that, it will break.

Design a way to find the number N in smallest possible number of steps.

5 pirates have found 1000 gold coins, which they now need to split among themselves. Here's how they decided to split the money:

Pirate #1 proposes a division scheme, smth like "i get 250, B gets 150, C gets nothing", etc. All pirates vote on that scheme. If there's a majority (3 out of 5, 3 out of 4, 2 ot of 3, 2 out of 2, 1 out of 1), the scheme is implemented. If there is no majority, Pirate 1 is killed, and it's now up to Pirate 2 to suggest a scheme. If his scheme is adopted, good, if not - he gets killed, and Pirate 3 proposes a scheme, and so on.

All pirates' actions are guarded by three principles:

- above all else, they want to stay alive

- They're extremely greedy: they want to acquire as many gold coins as possible, and every last coin counts.

- They're extremely violent: other things being equal, they'd rather kill a fellow pirate that let him live.

So, which pirate gets how many coins?

There are two train stations: A and B, 180 miles apart.

First train leaves A and travels towards B at 25 miles per hour.

At the same instant, second train leaves B and travels towards A at 35 miles per hour.

There is a fly sitting on the train 1. The moment the train leaves, the fly takes off and flies towards train 2 at 45 miles per hour. As soon as the fly reaches train 2, it makes a U-turn and flies back towards train 1. It reaches train 1, makes a U-turn, and flies towards train 2, etc. etc., until the trains collide and the fly gets squashed.

Question: what's the total distance the fly has travelled?

Suppose you have a bunch of coins of 2 denominations: X and Y, such that X and Y are co-prime. What is the smallest sum S, starting from which you can pay any sum without change?

Note that X and Y are given. The goal is to calculate S given X and Y, and not to come up with X and Y that make S be small and cute.

Include the proof of the formula you find.

For example: x=3, y=5.

1 - cannot pay

2 - cannot

3 - can

4 - cannot

5 - can

6 - can: 3 + 3

7 - cannot

8 - can: 3 + 5

9 - can: 3 + 3 + 3

10 - can: 5 + 5

11 - can: 8 (which we've shown already) + 3

12 - can: 9 + 3

13 - can: 10 + 3

et cetera...

so, the answer here would be 8.

Suppose you have a bunch of coins of 2 denominations, say X and Y, such that X and Y are co-prime. What is the smallest sum S, starting from which you can pay any sum without change?

Include the proof of the formula you find.

For example: suppose x=3, y=5.

1 - cannot pay

2 - cannot

3 - can

4 - cannot

5 - can

6 - can: 3 + 3

7 - cannot

8 - can: 3 + 5

9 - can: 3 + 3 + 3

10 - can: 5 + 5

11 - can: 8 (which we've shown already) + 3

so, the answer here would be 8.

Suppose you have a bunch of coins of 2 denominations, say X and Y, such that X and Y are co-prime. What is the smallest sum S, starting from which you can pay any sum without change?

Include the proof of the formula you find.

For example: suppose x=3, y=5.

1 - cannot pay

2 - cannot

3 - can

4 - cannot

5 - can

6 - can: 3 + 3

7 - cannot

8 - can: 3 + 5

9 - can: 3 + 3 + 3

10 - can: 5 + 5

11 - can: 8 (which we've shown already) + 3

so, the answer here would be 8.

If it takes one woman 9 months to bear one child, how many women will it take to bear one child in 1 month?