Olympiad problems

2015 Indonesia MO Shortlist, N8

The natural number $n$ is said to be good if there are natural numbers $a$ and $b$ that satisfy $a + b = n$ and $ab | n^2 + n + 1$. (a) Show that there are infinitely many good numbers. (b) Show that if $n$ is a good number, then $7 \nmid n$.

2001 Bosnia and Herzegovina Team Selection Test, 6

Tags: equation , infinitely many solutions , number theory

Prove that there exists infinitely many positive integers $n$ such that equation $(x+y+z)^3=n^2xyz$ has solution $(x,y,z)$ in set $\mathbb{N}^3$

1979 Bundeswettbewerb Mathematik, 4

Tags: last digit , sequence , infinitely many solutions , number theory

An infinite sequence $p_1, p_2, p_3, \ldots$ of natural numbers in the decimal system has the following property: For every $i \in \mathbb{N}$ the last digit of $p_{i+1}$ is different from $9$, and by omitting this digit one obtains number $p_i$. Prove that this sequence contains infinitely many composite numbers.

2002 IMO Shortlist, 4

Tags: infinitely many solutions , vieta jumping , pell equation , equation , algebra , number theory

Is there a positive integer $m$ such that the equation \[ {1\over a}+{1\over b}+{1\over c}+{1\over abc}={m\over a+b+c} \] has infinitely many solutions in positive integers $a,b,c$?

2013 Vietnam Team Selection Test, 2

Tags: infinitely many solutions , perfect square , divisibility , function , number theory

a. Prove that there are infinitely many positive integers $t$ such that both $2012t+1$ and $2013t+1$ are perfect squares. b. Suppose that $m,n$ are positive integers such that both $mn+1$ and $mn+n+1$ are perfect squares. Prove that $8(2m+1)$ divides $n$.

2002 IMO Shortlist, 3

Tags: polynomial , infinitely many solutions , equation , algebra

Let $P$ be a cubic polynomial given by $P(x)=ax^3+bx^2+cx+d$, where $a,b,c,d$ are integers and $a\ne0$. Suppose that $xP(x)=yP(y)$ for infinitely many pairs $x,y$ of integers with $x\ne y$. Prove that the equation $P(x)=0$ has an integer root.

2020 Latvia TST, 1.3

Tags: infinitely many solutions , fibonacci , number theory

Prove that equation $a^2 - b^2=ab - 1$ has infinitely many solutions, if $a,b$ are positive integers

2015 Bosnia Herzegovina Team Selection Test, 3

Tags: number theory , composite number , infinitely many solutions , divisibility

Prove that there exist infinitely many composite positive integers $n$ such that $n$ divides $3^{n-1}-2^{n-1}$.

1977 Bundeswettbewerb Mathematik, 3

Tags: number theory , infinitely many solutions , generalization , integer

Show that there are infinitely many positive integers $a$ that cannot be written as $a = a_{1}^{6}+ a_{2}^{6} + \ldots + a_{7}^{6},$ where the $a_i$ are positive integers. State and prove a generalization.

2016 Bundeswettbewerb Mathematik, 2

Tags: prime number , infinitely many solutions , number theory

Prove that there are infinitely many positive integers that cannot be expressed as the sum of a triangular number and a prime number.

2018 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: prime , sequence , composite , infinitely many solutions , number theory

We observe that number $10001=73\cdot137$ is not prime. Show that every member of infinite sequence $10001, 100010001, 1000100010001,...$ is not prime

2005 Austrian-Polish Competition, 9

Tags: diophantine equation , sum of cubes , infinitely many solutions , number theory

Consider the equation $x^3 + y^3 + z^3 = 2$. a) Prove that it has infinitely many integer solutions $x,y,z$. b) Determine all integer solutions $x, y, z$ with $|x|, |y|, |z| \leq 28$.

2010 Belarus Team Selection Test, 4.3

Tags: number theory , floor function , odd , infinitely many solutions

a) Prove that there are infinitely many pairs $(m, n)$ of positive integers satisfying the following equality $[(4 + 2\sqrt3)m] = [(4 -2\sqrt3)n]$ b) Prove that if $(m, n)$ satisfies the equality, then the number $(n + m)$ is odd. (I. Voronovich)

2003 German National Olympiad, 6

Tags: coprime , divisibility , pairs , infinitely many solutions , number theory

Prove that there are infinitely many coprime, positive integers $a,b$ such that $a$ divides $b^2 -5$ and $b$ divides $a^2 -5.$