Olympiad problems

2016 Saudi Arabia IMO TST, 3

Tags: SAU , miscellaneous

Find the number of permutations $ ( a_1, a_2, . \ . \ , a_{2016}) $ of the first $ 2016 $ positive integers satisfying the following two conditions: 1. $ a_{i+1} - a_i \leq 1$ for all $i = 1, 2, . \ . \ . , 2015$, and 2. There are exactly two indices $ i < j $ with $ 1 \leq i < j \leq 2016 $ such that $ a_i = i $ and $ a_j = j$.

2016 Saudi Arabia BMO TST, 3

Tags: SAU , Divisibility

Let $ m $ and $ n $ be odd integers such that $n^2 - 1 $ is divisible by $m^2 + 1 - n^2 $. Prove that $ |m^2 + 1 - n^2 | $ is a perfect square.

2016 Saudi Arabia IMO TST, 1

Tags: SAU , Diophantine Equations

Let $k$ be a positive integer. Prove that there exist integers $x$ and $y$, neither of which divisible by $7$, such that \begin{align*} x^2 + 6y^2 = 7^k. \end{align*}

2016 Saudi Arabia BMO TST, 1

Tags: SAU , Divisibility

Let $ a > b > c > d $ be positive integers such that \begin{align*} a^2 + ac - c^2 = b^2 + bd - d^2 \end{align*} Prove that $ ab + cd $ is a composite number.

2016 Saudi Arabia IMO TST, 2

Tags: SAU , Divisibility

Let $a$ be a positive integer. Find all prime numbers $ p $ with the following property: there exist exactly $ p $ ordered pairs of integers $ (x, y)$, with $ 0 \leq x, y \leq p - 1 $, such that $ p $ divides $ y^2 - x^3 - a^2x $.