Olympiad problems

2022 Kyiv City MO Round 1, Problem 2

Tags: number theory

You are given $2n$ distinct integers. What's the largest integer $C$ such that you can always form at least $C$ pairs from them, so that no integer is in more than one pair, and the sum of integers in each pair is a composite number? [i](Proposed by Anton Trygub)[/i]

2018 Lusophon Mathematical Olympiad, 4

Tags: diophantine equation , positive integer , number theory

Determine the pairs of positive integer numbers $m$ and $n$ that satisfy the equation $m^2=n^2 +m+n+2018$.

2011 India IMO Training Camp, 1

Tags: 3d geometry , diophantine equation , number theory , geometry

Find all positive integer $n$ satisfying the conditions $a)n^2=(a+1)^3-a^3$ $b)2n+119$ is a perfect square.

2003 Tournament Of Towns, 4

Tags: number theory

Each term of a sequence of positive integers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence?

1996 Estonia Team Selection Test, 1

Tags: divisibility , perfect cube , number theory

Suppose that $x,y$ and $\frac{x^2+y^2+6}{xy}$ are positive integers . Prove that $\frac{x^2+y^2+6}{xy}$ is a perfect cube.

1998 Baltic Way, 3

Tags: number theory

Find all positive integer solutions to $2x^2+5y^2=11(xy-11)$.

2009 Silk Road, 4

Tags: prime , perfect square , number theory

Prove that for any prime number $p$ there are infinitely many fours $(x, y, z, t)$ pairwise distinct natural numbers such that the number $(x^2+p t^2)(y^2+p t^2)(z^2+p t^2)$ is a perfect square.

1984 Tournament Of Towns, (056) O4

Tags: sum of digits , product of digits , digit , number theory

The product of the digits of the natural number $N$ is denoted by $P(N)$ whereas the sum of these digits is denoted by $S(N)$. How many solutions does the equation $P(P(N)) + P(S(N)) + S(P(N)) + S(S(N)) = 1984$ have?

2018 239 Open Mathematical Olympiad, 8-9.1

Tags: number theory

Given a prime number $p$. A positive integer $x$ is divided by $p$ with a remainder, and the number $p^2$ is divided by $x$ with a remainder. The remainders turned out to be equal. Find them [i]Proposed by Sergey Berlov[/i]

2016 Macedonia National Olympiad, Problem 3

Tags: number theory

Solve the equation in the set of natural numbers $xyz+yzt+xzt+xyt = xyzt + 3$

2008 Princeton University Math Competition, B5

Tags: number theory

How many integers $n$ are there such that $0 \le n \le 720$ and $n^2 \equiv 1$ (mod $720$)?

2006 Cono Sur Olympiad, 5

Tags: floor function , number theory

Find all positive integer number $n$ such that $[\sqrt{n}]-2$ divides $n-4$ and $[\sqrt{n}]+2$ divides $n+4$. Note: $[r]$ denotes the integer part of $r$.

2012 Indonesia MO, 1

Tags: least common multiple , greatest common divisor , number theory

Show that for any positive integers $a$ and $b$, the number \[n=\mathrm{LCM}(a,b)+\mathrm{GCD}(a,b)-a-b\] is an even non-negative integer. [i]Proposer: Nanang Susyanto[/i]

2011 Grand Duchy of Lithuania, 3

Tags: diophantine , diophantine equation , number theory

Find all primes $p,q$ such that $p ^3-q^7=p-q$.

2023 Grosman Mathematical Olympiad, 4

Tags: quadratic , prime number , number theory

Let $q$ be an odd prime number. Prove that it is impossible for all $(q-1)$ numbers \[1^2+1+q, 2^2+2+q, \dots, (q-1)^2+(q-1)+q\] to be products of two primes (not necessarily distinct).

2012 IFYM, Sozopol, 2

Tags: number theory , modulo

Let $p$ and $q=4p+1$ be prime numbers. Determine the least power $i$ of 2 for which $2^i\equiv 1\,(mod\, q)$.

1936 Eotvos Mathematical Competition, 1

Tags: algebra , number theory , sum

Prove that for all positive integers $n$, $$\frac{1}{1 \cdot 2}+\frac{1}{3 \cdot 4}+ ...+ \frac{1}{(2n - 1)2n}=\frac{1}{n + 1}\frac{1}{n + 2}+ ... +\frac{1}{2n}$$

2022 Iran Team Selection Test, 10

Tags: number theory

We call an infinite set $S\subseteq\mathbb{N}$ good if for all parwise different integers $a,b,c\in S$, all positive divisors of $\frac{a^c-b^c}{a-b}$ are in $S$. for all positive integers $n>1$, prove that there exists a good set $S$ such that $n \not \in S$. Proposed by Seyed Reza Hosseini Dolatabadi

2014 Cuba MO, 4

Tags: integer , number theory

Find all positive integers $a, b$ such that the numbers $\frac{a^2b + a}{a^2 + b}$ and $\frac{ab^2 + b}{b^2 - a}$ are integers.

2008 VJIMC, Problem 3

Tags: number theory

Find all pairs of natural numbers $(n,m)$ with $1<n<m$ such that the numbers $1$, $\sqrt[n]n$ and $\sqrt[m]m$ are linearly dependent over the field of rational numbers $\mathbb Q$.

2013 Federal Competition For Advanced Students, Part 1, 1

Tags: modular arithmetic , number theory

Show that if for non-negative integers $m$, $n$, $N$, $k$ the equation \[(n^2+1)^{2^k}\cdot(44n^3+11n^2+10n+2)=N^m\] holds, then $m = 1$.

2018 Baltic Way, 17

Tags: number theory

Prove that for any positive integers $p,q$ such that $\sqrt{11}>\frac{p}{q}$, the following inequality holds: \[\sqrt{11}-\frac{p}{q}>\frac{1}{2pq}.\]

1997 China National Olympiad, 3

Tags: number theory

Prove that there are infinitely many natural numbers $n$ such that we can divide $1,2,\ldots ,3n$ into three sequences $(a_n),(b_n)$ and $(c_n)$, with $n$ terms in each, satisfying the following conditions: i) $a_1+b_1+c_1= a_2+b_2+c_2=\ldots =a_n+b_n+c_n$ and $a_1+b_1+c_1$ is divisible by $6$; ii) $a_1+a_2+\ldots +a_n= b_1+b_2+\ldots +b_n=c_1+c_2+\ldots +c_n,$ and $a_1+a_2+\ldots +a_n$ is divisible by $6$.

1992 IMO Shortlist, 1

Tags: algebra , polynomial , vieta , quadratic , number theory

Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $ (x, y)$ such that [i](i)[/i] $ x$ and $ y$ are relatively prime; [i](ii)[/i] $ y$ divides $ x^2 \plus{} m$; [i](iii)[/i] $ x$ divides $ y^2 \plus{} m.$ [i](iv)[/i] $ x \plus{} y \leq m \plus{} 1\minus{}$ (optional condition)

2021 Belarusian National Olympiad, 9.4

Tags: number theory , combinatorics

In the table $n \times n$ numbers from $1$ to $n$ are written in a spiral way. For which $n$ all the numbers on the main diagonal are distinct?