1. - soduku
2. - tangram
The following tangram can become a square.

A group of four people has to cross a bridge. It is dark, and they have to light the path with a torch. No more than two people can cross the bridge simultaneously, and the group has only one torch. It takes different time for the people in the group to cross the bridge:

Annie crosses the bridge in 1 minute, Bob crosses the bridge in 2 minutes,
Fred crosses the bridge in 5 minutes, Dorothy crosses the bridge in 10 minutes.
How can the group cross the bridge in 17 minutes?

(see over)

The distance between the towns A and B is 1000 miles. There is 3000 apples in A, and the apples have to be delivered to B. The available car can take 1000 apples at most. The car driver has developed an addiction to apples: when he has apples aboard he eats 1 apple with each mile made. Figure out the strategy that yields the largest amount of apples to be delivered to B. Generalize the strategy for an arbitrary amount of apples.

A galaxy consists of three planets, each of them moving along a straight line with its own constant speed. If the centers of all three planets happen to lie on a straight line (some kind of eclipse) the inhabitants of each planet go nuts (they cannot see their two neighbor planets all at once), start talking about the end of the world, and the stock market crashes. Show that there will be no more than two such market crashes on each of these planets.

Only answer question 10.

A computer scientist claims that he proved somehow that the Fermat theorem is correct for the following 3 numbers:

x=2233445566,
y=7788990011,
z=9988776655
He announces these 3 numbers and calls for a press conference where he is going to present the value of N (to show that
x^N + y^N = z^N
and that the guy from Princeton was wrong). As the press conference starts, a 10-years old boy raises his hand and says that the respectable scientist has made a mistake and the Fermat theorem cannot hold for those 3 numbers. The scientist checks his computer calculations and finds a bug.
How did the boy figure out that the scientist was wrong?

9. Dragons have to meet for a brainstorm in a convention center. The delegates have to be selected to provide the maximum efficiency of the brainstorm session. A dragon has any amount of heads, and for any N, any amount of N-headed dragons is available if needed. The problem is that the size of the convention center is limited so no more than 1000 heads can fit into the assembly hall. The intellectual power of a dragon pack is the product of head numbers of dragons in the pack. How should an optimum pack look like (the total number of dragons, the head number distribution)?

10. Write your name on this sheet of paper and hand it back to the examiner.

## Read the paper before starting.

No conferring.1. - soduku

2. - tangram

The following tangram can become a square.

Annie crosses the bridge in 1 minute,Bob crosses the bridge in 2 minutes,Fred crosses the bridge in 5 minutes,

Dorothy crosses the bridge in 10 minutes.How can the group cross the bridge in 17 minutes?

- (see over)
- The distance between the towns A and B is 1000 miles. There is 3000 apples in A, and the apples have to be delivered to B. The available car can take 1000 apples at most. The car driver has developed an addiction to apples: when he has apples aboard he eats 1 apple with each mile made. Figure out the strategy that yields the largest amount of apples to be delivered to B. Generalize the strategy for an arbitrary amount of apples.
- A galaxy consists of three planets, each of them moving along a straight line with its own constant speed. If the centers of all three planets happen to lie on a straight line (some kind of eclipse) the inhabitants of each planet go nuts (they cannot see their two neighbor planets all at once), start talking about the end of the world, and the stock market crashes. Show that there will be no more than two such market crashes on each of these planets.
- A computer scientist claims that he proved somehow that the Fermat theorem is correct for the following 3 numbers:

x=2233445566,## Only answer question 10.

y=7788990011,

z=9988776655

He announces these 3 numbers and calls for a press conference where he is going to present the value of N (to show that

x^N + y^N = z^N

and that the guy from Princeton was wrong). As the press conference starts, a 10-years old boy raises his hand and says that the respectable scientist has made a mistake and the Fermat theorem cannot hold for those 3 numbers. The scientist checks his computer calculations and finds a bug.

How did the boy figure out that the scientist was wrong?

9. Dragons have to meet for a brainstorm in a convention center. The delegates have to be selected to provide the maximum efficiency of the brainstorm session. A dragon has any amount of heads, and for any N, any amount of N-headed dragons is available if needed. The problem is that the size of the convention center is limited so no more than 1000 heads can fit into the assembly hall. The intellectual power of a dragon pack is the product of head numbers of dragons in the pack. How should an optimum pack look like (the total number of dragons, the head number distribution)?

## 10. Write your name on this sheet of paper and hand it back to the examiner.