This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 23

2014 IMO Shortlist, A4
Tags: function , IMO Shortlist , algebra , functional equation , IMO Shortlist 2014
Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$. [i]Proposed by Netherlands[/i]

2015 India IMO Training Camp, 3
Tags: floor function , IMO Shortlist , number theory , Sequence , Parity , IMO Shortlist 2014 , Hi
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

2015 India IMO Training Camp, 2
Tags: IMO Shortlist , number theory , primes , IMO Shortlist 2014 , N5
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

2015 India IMO Training Camp, 3
Tags: floor function , IMO Shortlist , number theory , Sequence , Parity , IMO Shortlist 2014 , Hi
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

2014 IMO Shortlist, N4
Tags: floor function , IMO Shortlist , number theory , Sequence , Parity , IMO Shortlist 2014 , Hi
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

Taiwan TST 2015 Round 1, 1
Tags: IMO Shortlist , number theory , equation , IMO Shortlist 2014
Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

2015 Ukraine Team Selection Test, 10
Tags: IMO Shortlist , number theory , equation , IMO Shortlist 2014
Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

2015 Peru IMO TST, 6
Tags: floor function , IMO Shortlist , number theory , Sequence , Parity , IMO Shortlist 2014 , Hi
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

2015 Belarus Team Selection Test, 2
Tags: IMO Shortlist , number theory , equation , IMO Shortlist 2014
Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

2015 Belarus Team Selection Test, 3
Tags: function , IMO Shortlist , algebra , functional equation , IMO Shortlist 2014
Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$. [i]Proposed by Netherlands[/i]

2015 Taiwan TST Round 2, 3
Tags: IMO Shortlist , number theory , primes , IMO Shortlist 2014 , N5
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

2022 Thailand TSTST, 3
Tags: floor function , IMO Shortlist , number theory , Sequence , Parity , IMO Shortlist 2014 , Hi
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

2015 India IMO Training Camp, 2
Tags: IMO Shortlist , number theory , primes , IMO Shortlist 2014 , N5
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

2015 Brazil Team Selection Test, 2
Tags: IMO Shortlist , number theory , equation , IMO Shortlist 2014
Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

2015 Taiwan TST Round 2, 2
Tags: function , IMO Shortlist , algebra , functional equation , IMO Shortlist 2014
Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$. [i]Proposed by Netherlands[/i]

2014 IMO Shortlist, N5
Tags: IMO Shortlist , number theory , primes , IMO Shortlist 2014 , N5
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

2015 Brazil Team Selection Test, 2
Tags: IMO Shortlist , number theory , primes , IMO Shortlist 2014 , N5
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

2015 Ukraine Team Selection Test, 3
Tags: IMO Shortlist , number theory , primes , IMO Shortlist 2014 , N5
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

2015 Germany Team Selection Test, 1
Tags: IMO Shortlist , number theory , equation , IMO Shortlist 2014
Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

2015 Brazil Team Selection Test, 2
Tags: floor function , IMO Shortlist , number theory , Sequence , Parity , IMO Shortlist 2014 , Hi
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

2015 Germany Team Selection Test, 1
Tags: IMO Shortlist , number theory , equation , IMO Shortlist 2014
Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

2015 Belarus Team Selection Test, 3
Tags: floor function , IMO Shortlist , number theory , Sequence , Parity , IMO Shortlist 2014 , Hi
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

2014 IMO Shortlist, N2
Tags: IMO Shortlist , number theory , equation , IMO Shortlist 2014
Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]