JMO 1992

Let $x,\ y$ be relatively prime numbers such that $xy\neq 1$. For positive even integer $n$, prove that $x + y$ isn't a divisor of $x^n + y^n$.

Suppose that $D,\ E$ are points on $AB,\ AC$ of $\triangle{ABC}$ with area $1$ respectively and $P$ is $BE\cap CD$. When $D,\ E$ move on $AB,\ AC$ with satisfying the condition $[BCED] = 2\triangle{PBC}$, find the maximum area of $\triangle{PDE}$.

Prove the inequality $\sum_{k=1}^{n-1} \frac{n}{n-k}\cdot \frac{1}{2^{k-1}}<4\ (n\geq 2)$.

Let $A$ be a $m\times n\ (m\neq n)$ matrix with the entries $0$ and $1$. Suppose that if $f$ is injective such that $f: \{1,\ 2,\ \cdots ,\ m\}\longrightarrow \{1,\ 2,\ \cdots ,\ n\}$, then there exists $1\leq i\leq m$ such that $(i,\ f(i))$ element is $0$.

Prove that there exist $S\subseteq \{1,\ 2,\ \cdots ,\ m\}$ and $T\subseteq \{1,\ 2,\ \cdots ,\ n\}$ satisfying the condition:

$i)$ if $i\in{S},\ j\in{T}$, then $(i,\ j)$ entry is $0$.

$ii)$ $card\ S + card\ T > n$.

Suppose that $n\geq 2$ be integer and $a_1,\ a_2,\ a_3,\ a_4$ satisfy the following condition:

i) $\ n$ and $a_i\ (i = 1,\ 2,\ 3,\ 4)$ are relatively prime.

ii) $\ (ka_1)_n + (ka_2)_n + (ka_3)_n + (ka_4)_n = 2n$ for $k = 1,\ 2,\ \cdots ,\ n - 1$.

Note that $(a)_n$ expresses the divisor when $a$ is divided by $n$.

Prove that $(a_1)_n,\ (a_2)_n,\ (a_3)_n,\ (a_4)_n$ can be divided into two pair with sum $n$.