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

A conference was attended by a group of scientists, some of whom in this group were friends. It turned out that every two of them, having an equal number of friends at the conference, do not have friends in common. Prove that there is a scientist who has exactly one friend among the conference attendees.

Several pieces of carpet are laid along a corridor. Pieces cover the entire corridor from end to end without omissions and even overlap one another, so that over some parts of the floor lie several layers of carpet. Prove that you can remove a few pieces, perhaps by taking them out from under others and leaving the rest exactly in the same places they used to be, so that the corridor will still be completely covered and the total length of the pieces left will be less than twice the length corridor.

A unit square is divided into $n$ triangles. Prove that one of the triangles can be used to completely cover a square with side length $\frac{1}{n}$.

Let a, b, c be integers; where a and b are not equal to zero.

Prove that the equation ax + by = c has integer solutions if and only if c is divisible by d = GCD $($a, b$)$.

In how many ways can you rearrange the numbers from 1 to 100 so that the neighbouring numbers differ by no more than 1?

On a line, there are 50 segments. Prove that either it is possible to find some 8 segments all of which have a shared intersection, or there can be found 8 segments, no two of which intersect.

Prove that if the irreducible rational fraction p/q is a root of the polynomial $P (x)$ with integer coefficients, then $P (x) = (qx – p) Q (x)$, where the polynomial $Q (x)$ also has integer coefficients.

The order of books on a shelf is called * wrong * if no three adjacent books are arranged in order of height (either increasing or decreasing). How many wrong orders is it possible to construct from $n$ books of different heights, if:

a) $n = 4$;

b) $n = 5$?

We are given $n+1$ different natural numbers, which are less than $2n$ $(n>1)$. Prove that among them there will always be three numbers, where the sum of two of them is equal to the third.

2011 numbers are written on a blackboard. It turns out that the sum of any of these written numbers is also one of the written numbers. What is the minimum number of zeroes within this set of 2011 numbers?

Each of the 1994 deputies in parliament slapped exactly one of his colleagues. Prove that it is possible to draw up a parliamentary commission of 665 people whose members did not clarify the relationship between themselves in the manner indicated above.

One term a school ran 20 sessions of an after-school Astronomy Club. Exactly five pupils attended each session and no two students encountered one another over all of the sessions more than once. Prove that no fewer than 20 pupils attended the Astronomy Club at some point during the term.