Olympiad problems

2022 Switzerland - Final Round, 7

Let $n > 6$ be a perfect number. Let $p_1^{a_1} \cdot p_2^{a_2} \cdot ... \cdot p_k^{a_k}$ be the prime factorisation of $n$, where we assume that $p_1 < p_2 <...< p_k$ and $a_i > 0$ for all $ i = 1,...,k$. Prove that $a_1$ is even. Remark: An integer $n \ge 2$ is called a perfect number if the sum of its positive divisors, excluding $ n$ itself, is equal to $n$. For example, $6$ is perfect, as its positive divisors are $\{1, 2, 3, 6\}$ and $1+2+3=6$.

1983 Tournament Of Towns, (041) O4

Tags: combinatorics , Even

There are $K$ boys placed around a circle. Each of them has an even number of sweets. At a command each boy gives half of his sweets to the boy on his right. If, after that, any boy has an odd number of sweets, someone outside the circle gives him one more sweet to make the number even. This procedure can be repeated indefinitely. Prove that there will be a time at which all boys will have the same number of sweets. (A Andjans, Riga)

2002 Singapore Team Selection Test, 2

Tags: Sequence , combinatorics , odd , Even

Let $n$ be a positive integer and $(x_1, x_2, ..., x_{2n})$, $x_i = 0$ or $1, i = 1, 2, ... , 2n$ be a sequence of $2n$ integers. Let $S_n$ be the sum $S_n = x_1x_2 + x_3x_4 + ... + x_{2n-1}x_{2n}$. If $O_n$ is the number of sequences such that $S_n$ is odd and $E_n$ is the number of sequences such that $S_n$ is even, prove that $$\frac{O_n}{E_n}=\frac{2^n - 1}{2^n + 1}$$

1993 Swedish Mathematical Competition, 3

Tags: Even , number theory , diophantine , Diophantine equation

Assume that $a$ and $b$ are integers. Prove that the equation $a^2 +b^2 +x^2 = y^2$ has an integer solution $x,y$ if and only if the product $ab$ is even.

1986 Czech And Slovak Olympiad IIIA, 1

Tags: Even , Subsets , combinatorics

Given $n \in N$, let $A$ be a family of subsets of $\{1,2,...,n\}$. If for every two sets $B,C \in A$ the set $(B \cup C) -(B \cap C)$ has an even number of elements, find the largest possible number of elements of $A$ .

2009 Tournament Of Towns, 1

Tags: convex polygon , convex , diagonal , combinatorial geometry , Even , geometry

In a convex $2009$-gon, all diagonals are drawn. A line intersects the $2009$-gon but does not pass through any of its vertices. Prove that the line intersects an even number of diagonals.

1992 Austrian-Polish Competition, 1

Tags: Sum , positive , Divisors , number theory , Product , Even

For a natural number $n$, denote by $s(n)$ the sum of all positive divisors of n. Prove that for every $n > 1$ the product $s(n - 1)s(n)s(n + 1)$ is even.

2005 Thailand Mathematical Olympiad, 12

Tags: number theory , divides , divisible , Even

Find the number of even integers n such that $0 \le n \le 100$ and $5 | n^2 \cdot 2^{{2n}^2}+ 1$.

2005 Abels Math Contest (Norwegian MO), 1b

Tags: geometry , 3D geometry , pyramid , Even , Integers , Volume

In a pyramid, the base is a right-angled triangle with integer sides. The height of the pyramid is also integer. Show that the volume of the pyramid is even.

1998 Tournament Of Towns, 2

Tags: Chessboard , combinatorics , Even

A chess king tours an entire $8\times 8$ chess board, visiting each square exactly once and returning at last to his starting position. Prove that he made an even number of diagonal moves. (V Proizvolov)

2015 Danube Mathematical Competition, 2

Tags: graph theory , graph , combinatorics , Even

Show that the edges of a connected simple (no loops and no multiple edges) finite graph can be oriented so that the number of edges leaving each vertex is even if and only if the total number of edges is even

1989 All Soviet Union Mathematical Olympiad, 508

Tags: polyhedron , combinatorial geometry , combinatorics , 3D geometry , Even

A polyhedron has an even number of edges. Show that we can place an arrow on each edge so that each vertex has an even number of arrows pointing towards it (on adjacent edges).

1956 Moscow Mathematical Olympiad, 322

Tags: broken line , Even , combinatorics

A closed self-intersecting broken line intersects each of its segments once. Prove that the number of its segments is even.

1983 Swedish Mathematical Competition, 3

Tags: Even , number theory , system of equations , System , diophantine , Diophantine equation

The systems of equations \[\left\{ \begin{array}{l} 2x_1 - x_2 = 1 \\ -x_1 + 2x_2 - x_3 = 1 \\ -x_2 + 2x_3 - x_4 = 1 \\ -x_3 + 3x_4 - x_5 =1 \\ \cdots\cdots\cdots\cdots\\ -x_{n-2} + 2x_{n-1} - x_n = 1 \\ -x_{n-1} + 2x_n = 1 \\ \end{array} \right. \] has a solution in positive integers $x_i$. Show that $n$ must be even.