Olympiad problems

2011 Morocco TST, 1

Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

2011 Belarus Team Selection Test, 2

Tags: equation , lte lemma , number theory , modular arithmetic

Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

2010 IMO Shortlist, 2

Tags: equation , number theory , lte lemma , modular arithmetic

Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

2011 India IMO Training Camp, 2

Tags: modular arithmetic , equation , lte lemma , number theory

Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

2011 Ukraine Team Selection Test, 7

Tags: equation , lte lemma , modular arithmetic , number theory

Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

2022 IMO Shortlist, N4

Tags: divisibility , lte lemma , diophantine equation , number theory

Find all triples $(a,b,p)$ of positive integers with $p$ prime and \[ a^p=b!+p. \]

2014 Chile TST IMO, 2

Tags: lte lemma , number theory

Given $n, k \in \mathbb{N}$, prove that $(n-1)^2$ divides $n^k - 1$ if and only if $n-1 \mid k$.

2017 SG Originals, Q4

Tags: lte lemma , lifting the exponent , number theory

Call a rational number $r$ [i]powerful[/i] if $r$ can be expressed in the form $\dfrac{p^k}{q}$ for some relatively prime positive integers $p, q$ and some integer $k >1$. Let $a, b, c$ be positive rational numbers such that $abc = 1$. Suppose there exist positive integers $x, y, z$ such that $a^x + b^y + c^z$ is an integer. Prove that $a, b, c$ are all [i]powerful[/i]. [i]Jeck Lim, Singapore[/i]

2017 APMO, 4

Tags: lte lemma , lifting the exponent , number theory

Call a rational number $r$ [i]powerful[/i] if $r$ can be expressed in the form $\dfrac{p^k}{q}$ for some relatively prime positive integers $p, q$ and some integer $k >1$. Let $a, b, c$ be positive rational numbers such that $abc = 1$. Suppose there exist positive integers $x, y, z$ such that $a^x + b^y + c^z$ is an integer. Prove that $a, b, c$ are all [i]powerful[/i]. [i]Jeck Lim, Singapore[/i]

2011 India IMO Training Camp, 2

Tags: number theory , modular arithmetic , equation , lte lemma

Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

2017 Brazil Team Selection Test, 4

Tags: number theory , lte lemma , lifting the exponent

Call a rational number $r$ [i]powerful[/i] if $r$ can be expressed in the form $\dfrac{p^k}{q}$ for some relatively prime positive integers $p, q$ and some integer $k >1$. Let $a, b, c$ be positive rational numbers such that $abc = 1$. Suppose there exist positive integers $x, y, z$ such that $a^x + b^y + c^z$ is an integer. Prove that $a, b, c$ are all [i]powerful[/i]. [i]Jeck Lim, Singapore[/i]

2022 IMO, 5

Tags: number theory , divisibility , lte lemma , diophantine equation

Find all triples $(a,b,p)$ of positive integers with $p$ prime and \[ a^p=b!+p. \]

2020 Macedonia Additional BMO TST, 2

Tags: number theory , modular arithmetic , lte lemma

Given are a prime $p$ and a positive integer $a$. Let $q$ be a prime divisor of $\frac{a^{p^3}-1}{a^{p^2}-1}$ and $q\neq p$. Prove that $q\equiv 1 ( \mod p^3)$.