Olympiad problems

2022 Iran MO (3rd Round), 2

$m\times n$ grid is tiled by mosaics $2\times2$ and $1\times3$ (horizontal and vertical). Prove that the number of ways to choose a $1\times2$ rectangle (horizontal and vertical) such that one of its cells is tiled by $2\times2$ mosaic and the other cell is tiled by $1\times3$ mosaic [horizontal and vertical] is an even number.

2019 Junior Balkan Team Selection Tests - Romania, 4

Tags: combinatorics , parity

Let $n$ be a positive integer. $2n+1$ tokens are in a row, each being black or white. A token is said to be [i]balanced[/i] if the number of white tokens on its left plus the number of black tokens on its right is $n$. Determine whether the number of [i]balanced[/i] tokens is even or odd.

2015 India IMO Training Camp, 3

Tags: number theory , floor function , sequence , parity

Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

Kvant 2022, M2718

Tags: rectangle , combinatorics , parity

$m\times n$ grid is tiled by mosaics $2\times2$ and $1\times3$ (horizontal and vertical). Prove that the number of ways to choose a $1\times2$ rectangle (horizontal and vertical) such that one of its cells is tiled by $2\times2$ mosaic and the other cell is tiled by $1\times3$ mosaic [horizontal and vertical] is an even number.

2015 Brazil Team Selection Test, 2

Tags: floor function , number theory , sequence , parity

Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

2018 Spain Mathematical Olympiad, 2

Tags: combinatorics , parity

Let $n$ be a positive integer. $2n+1$ tokens are in a row, each being black or white. A token is said to be [i]balanced[/i] if the number of white tokens on its left plus the number of black tokens on its right is $n$. Determine whether the number of [i]balanced[/i] tokens is even or odd.

2019-IMOC, C1

Tags: partition , distributions , parity , combinatorics

Given a natural number $n$, if the tuple $(x_1,x_2,\ldots,x_k)$ satisfies $$2\mid x_1,x_2,\ldots,x_k$$ $$x_1+x_2+\ldots+x_k=n$$ then we say that it's an [i]even partition[/i]. We define [i]odd partition[/i] in a similar way. Determine all $n$ such that the number of even partitions is equal to the number of odd partitions.