Olympiad problems

2021 Dutch IMO TST, 3

Prove that for every positive integer $n$ there are positive integers $a$ and $b$ exist with $n | 4a^2 + 9b^2 -1$.

2013 Saudi Arabia Pre-TST, 4.2

Tags: number theory , divide , divisible

Let $x, y$ be two integers. Prove that if $2013$ divides $x^{1433} + y^{1433}$ then $2013$ divides $x^7 + y^7$.

2000 Chile National Olympiad, 3

Tags: number theory , divide , divisible , periodic

A number $N_k$ is defined as [i]periodic[/i] if it is composed in number base $N$ of a repeated $k$ times . Prove that $7$ divides to infinite periodic numbers of the set $N_1, N_2, N_3,...$

2023 Grand Duchy of Lithuania, 4

Tags: number theory , divide , factorial

Note that $k\ge 1$ for an odd natural number $$k! ! = k \cdot (k - 2) \cdot ... \cdot 1.$$ Prove that $2^n$ divides $(2^n -1)!! -1$ for all $n \ge 3$.

2015 Thailand Mathematical Olympiad, 1

Tags: divide , recurrence relation , divisible , number theory

Let $p$ be a prime, and let $a_1, a_2, a_3, . . .$ be a sequence of positive integers so that $a_na_{n+2} = a^2_{n+1} + p$ for all positive integers $n$. Show that $a_{n+1}$ divides $a_n + a_{n+2}$ for all positive integers $n$.

2015 Finnish National High School Mathematics Comp, 3

Tags: number theory , factorial , factor , exponential , divide

Determine the largest integer $k$ for which $12^k$ is a factor of $120! $

2011 Saudi Arabia Pre-TST, 4.4

Tags: number theory , divisor , divide

Let $a, b, c, d$ be positive integers such that $a+b+c+d = 2011$. Prove that $2011$ is not a divisor of $ab - cd$.

2000 Kazakhstan National Olympiad, 5

Tags: prime divisor , divide , divisible , number theory

Let the number $ p $ be a prime divisor of the number $ 2 ^ {2 ^ k} + 1 $. Prove that $ p-1 $ is divisible by $ 2 ^ {k + 1} $.

1993 Swedish Mathematical Competition, 1

Tags: sum of digits , divisible , divide , number theory

An integer $x$ has the property that the sums of the digits of $x$ and of $3x$ are the same. Prove that $x$ is divisible by $9$.

2000 Tournament Of Towns, 2

Tags: number theory , divide , divisible

Positive integers $a, b, c, d$ satisfy the inequality $ad - bc > 1$. Prove that at least one of the numbers $a, b, c, d$ is not divisible by $ad - bc$. (A Spivak)

1991 Bundeswettbewerb Mathematik, 2

Tags: number theory , divide

Let $g$ be an even positive integer and $f(n) = g^n + 1$ , $(n \in N^* )$. Prove that for every positive integer $n$ we have: a) $f(n)$ divides each of the numbers $f(3n), f(5n), f(7n)$ b) $f(n)$ is relative prime to each of the numbers $f(2n), f(4n),f(6n),...$

2003 Estonia National Olympiad, 4

Tags: infinity , number theory , integer , floor function , divisible , divide

Prove that there exist infinitely many positive integers $n$ such that $\sqrt{n}$ is not an integer and $n$ is divisible by $[\sqrt{n}] $.