Olympiad problems

2013 Grand Duchy of Lithuania, 4

A positive integer $n \ge 2$ is called [i]peculiar [/i] if the number $n \choose i$ + $n \choose j $ $-i-j$ is even for all integers $i$ and $j$ such that $0 \le i \le j \le n$. Determine all peculiar numbers.

2021 China National Olympiad, 5

Tags: graph theory , combinatorics , polyhedron , even , parity

$P$ is a convex polyhedron such that: [b](1)[/b] every vertex belongs to exactly $3$ faces. [b](1)[/b] For every natural number $n$, there are even number of faces with $n$ vertices. An ant walks along the edges of $P$ and forms a non-self-intersecting cycle, which divides the faces of this polyhedron into two sides, such that for every natural number $n$, the number of faces with $n$ vertices on each side are the same. (assume this is possible) Show that the number of times the ant turns left is the same as the number of times the ant turn right.

2005 Thailand Mathematical Olympiad, 12

Tags: divisible , number theory , divide , even

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

2009 Tournament Of Towns, 1

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

2011 QEDMO 10th, 2

Tags: combinatorics , odd , even , sum

Let $n$ be a positive integer. Let $G (n)$ be the number of $x_1,..., x_n, y_1,...,y_n \in \{0,1\}$, for which the number $x_1y_1 + x_2y_2 +...+ x_ny_n$ is even, and similarly let $U (n)$ be the number for which this sum is odd. Prove that $$\frac{G(n)}{U(n)}= \frac{2^n + 1}{2^n - 1}.$$

1989 All Soviet Union Mathematical Olympiad, 508

Tags: combinatorial geometry , combinatorics , polyhedron , 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).

1998 Tournament Of Towns, 2

Tags: combinatorics , chessboard , 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)

2000 Czech And Slovak Olympiad IIIA, 1

Tags: divisible , number theory , even , divide

Let $n$ be a natural number. Prove that the number $4 \cdot 3^{2^n}+ 3 \cdot4^{2^n}$ is divisible by $13$ if and only if $n$ is even.

2015 Brazil Team Selection Test, 1

Tags: function , odd , even , periodic , algebra

Let's call a function $f : R \to R$ [i]cool[/i] if there are real numbers $a$ and $b$ such that $f(x + a)$ is an even function and $f(x + b)$ is an odd function. (a) Prove that every cool function is periodic. (b) Give an example of a periodic function that is not cool.

1992 Austrian-Polish Competition, 1

Tags: divisor , number theory , sum , positive , 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.

2013 Costa Rica - Final Round, 2

Tags: number theory , composite number , odd , even , sum

Determine all even positive integers that can be written as the sum of odd composite positive integers.

1983 Swedish Mathematical Competition, 3

Tags: diophantine , number theory , even , system of equations , system , 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.

1999 Singapore MO Open, 1

Tags: combinatorics , even , coloring , point

Let $n$ be a positive integer. A square $ABCD$ is divided into $n^2$ identical small squares by drawing $(n-1)$ equally spaced lines parallel to the side $AB$ and another $(n- 1)$ equally spaced lines parallel to $BC$, thus giving rise to $(n+1)^2$ intersection points. The points $A, C$ are coloured red and the points $B, D$ are coloured blue. The rest of the intersection points are coloured either red or blue. Prove that the number of small squares having exactly $3$ vertices of the same colour is even.

1996 Greece Junior Math Olympiad, 4a

Tags: number theory , fraction , simplify , even

If the fraction $\frac{an + b}{cn + d}$ may be simplified using $2$ (as a common divisor ), show that the number $ad - bc$ is even. ($a, b, c, d, n$ are natural numbers and the $cn + d$ different from zero).