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#### Extremal principle (other) , Partitions into pairs and groups bijections , Pigeonhole principle (other)

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

#### Convex solids , Extremal principle (other) , Pigeonhole principle (other) , Polyhedra and polygons in space (other) , Visual geometry in space

Prove that for every convex polyhedron there are two faces with the same number of sides.

#### Auxiliary simialr triangles , Extremal principle (other) , Pigeonhole principle (area and volume)

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 $\frac{1}{n}$.

#### Extremal principle (other) , Number sequences (other) , The fundamental theorm of arithmetic. Prime factorisation.

In a row there are 20 different natural numbers. The product of every two of them standing next to one another is the square of a natural number. The first number is 42. Prove that at least one of the numbers is greater than 16,000.

#### Equations in integer numbers , Extremal principle (other) , Geometric interpretations in algebra , Pigeonhole principle (other) , The greatest common divisor (GCD) and the least common multiplier (LCM). Mutually prime numbers

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$)$.

#### Extremal principle (other)

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

#### Extremal principle (other) , Pigeonhole principle (other) , Systems of points, lines, and line segments

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.

#### Extremal principle (other) , Mathematical induction (other) , Polynomial remainder theorem (little Bezout's theorem). Factorisation. , Polynomials with integer coefficients and integer values , Proof by contradiction , Rational and irrational numbers

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.

#### Extremal principle (other) , Mathematical induction (other) , Partitions into pairs and groups bijections , Pigeonhole principle (other)

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.

#### Extremal principle (other) , Pigeonhole principle (other) , Proof by contradiction

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?

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