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100 cars are parked along the right hand side of a road. Among them there are 30 red, 20 yellow, and 20 pink Mercedes. It is known that no two Mercedes of different colours are parked next to one another. Prove that there must be three Mercedes cars parked next to one another of the same colour somewhere along the road.

Ben noticed that all 25 of his classmates have a different number of friends in this class. How many friends does Ben have?

The product of a group of 48 natural numbers has exactly 10 prime factors. Prove that the product of some four of the numbers in the group will always give a square number.

The product of 1986 natural numbers has exactly 1985 different prime factors. Prove that either one of these natural numbers, or the product of several of them, is the square of a natural number.

The sum of 100 natural numbers, each of which is no greater than 100, is equal to 200. Prove that it is possible to pick some of these numbers so that their sum is equal to 100.

Two identical gears have 32 teeth. They were combined and 6 pairs of teeth were simultaneously removed. Prove that one gear can be rotated relative to the other so that in the gaps in one gear where teeth were removed there will be whole teeth of the second gear.

Harry thought of two positive numbers x and y. He wrote down the numbers x + y, x – y, xy and x/y on a board and showed them to Sam, but did not say which number corresponded to which operation.

Prove that Sam can uniquely figure out x and y.

Carry out the following experiment 10 times: first, toss a coin 10 times in a row and record the number of heads, then toss the coin 9 times in a row and again, record the number of heads. We call the experiment successful, if, in the first case, the number of heads is greater than in the second case. After conducting a series of 10 such experiments, record the number of successful and unsuccessful experiments. Collect the statistics in the form of a table.

a) Anton throws a coin 3 times, and Tina throws it two times. What is the probability that Anton gets more heads than Tina?

b) Anton throws a coin n + 1 times, and Tanya throws it n times. What is the probability that Anton gets more heads than Tina?

A pack of 36 cards was placed in front of a psychic face down. He calls the suit of the top card, after which the card is opened, shown to him and put aside. After this, the psychic calls out the suit of the next card, etc. The task of the psychic is to guess the suit as many times as possible. However, the card backs are in fact asymmetrical, and the psychic can see in which of the two positions the top card lies. The deck is prepared by a bribed employee. The clerk knows the order of the cards in the deck, and although he cannot change it, he can prompt the psychic by having the card backs arranged in a way according to a specific arrangement. Can the psychic, with the help of such a clue, ensure the guessing of the suit of

a$)$ more than half of the cards;

b$)$ no less than 20 cards?

Hannah placed 101 counters in a row which had values of 1, 2 and 3 points. It turned out that there was at least one counter between every two one point counters, at least two counters lie between every two two point counters, and at least three counters lie between every two three point counters. How many three point counters could Hannah have?

Prove that amongst any 11 different decimal fractions of infinite length, there will be two whose digits in the same column – 10ths, 100s, 1000s, etc – coincide $($are the same$)$ an infinite number of times.

You are given 1002 different integers that are no greater than 2000. Prove that it is always possible to choose three of the given numbers so that the sum of two of them is equal to the third.

Will this still always be possible if we are given 1001 integers rather than 1002?

We are given 51 two-digit numbers – we will count one-digit numbers as two-digit numbers with a leading 0. Prove that it is possible to choose 6 of these so that no two of them have the same digit in the same column.

From the set of numbers 1 to 2n, n + 1 numbers are chosen. Prove that among the chosen numbers there are two, one of which is divisible by another.

26 numbers are chosen from the numbers $1, 2, 3,…, 49, 50$. Will there always be two numbers chosen whose difference is 1?

The key of the cipher, called the “swivelling grid”, is a stencil made from a square sheet of chequered paper of size $n \times n$ $($where n is even$)$. Some of the cells are cut out. One side of the stencil is marked. When this stencil is placed onto a blank sheet of paper in four possible ways $($marked side up, right, down or left$)$, its cut-outs completely cover the entire area of the square, where each cell is found under the cut-out exactly once. The letters of the message, that have length $n^2$, are successively written into the cut-outs of the stencil, where the sheet of paper is placed on a blank sheet of paper with the marked side up. After filling in all of the cut-outs of the stencil with the letters of the message, the stencil is placed in the next position, etc. After removing the stencil from the sheet of paper, there is an encrypted message.

Find the number of different keys for an arbitrary even number n.

2001 vertices of a regular 5000-gon are painted. Prove that there are three coloured vertices lying on the vertices of an isosceles triangle.

In a square which has sides of length 1 there are 100 figures, the total area of which sums to more than 99. Prove that in the square there is a point which belongs to all of these figures.

Two people toss a coin: one tosses it 10 times, the other – 11 times.

What is the probability that the second person’s coin showed heads more times than the first?

Is it possible to find 57 different two digit numbers, such that no sum of any two of them was equal to 100?