Olympiad problems

2020 Kazakhstan National Olympiad, 3

Let $p$ be a prime number and $k,r$ are positive integers such that $p>r$. If $pk+r$ divides $p^p+1$ then prove that $r$ divides $k$.

1916 Eotvos Mathematical Competition, 3

Tags: number theory , combinatorics

Divide the numbers $$1, 2,3, 4,5$$ into two arbitrarily chosen sets. Prove that one of the sets contains two numbers and their difference.

2023 BMT, 9

Tags: number theory

For positive integers $a$ and $b$, consider the curve $x^a + y^b = 1$ over real numbers $x$, $y$ and let $S(a, b)$ be the sum $P$ of the number of $x$-intercepts and $y$-intercepts of this curve. Compute $\sum^{10}_{a=1}\sum^5_{b=1} S(a, b).$

2012 Uzbekistan National Olympiad, 2

Tags: modular arithmetic , calculus , integration , number theory

For any positive integers $n$ and $m$ satisfying the equation $n^3+(n+1)^3+(n+2)^3=m^3$, prove that $4\mid n+1$.

2012 Baltic Way, 20

Tags: number theory

Find all integer solutions of the equation $2x^6 + y^7 = 11$.

Taiwan TST 2015 Round 1, 1

Tags: number theory , equation

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]

1999 Akdeniz University MO, 4

Tags: number theory

In a sequence ,first term is $2$ and after $2.$ term all terms is equal to sum of the previous number's digits' $5.$ power. (Like this $2.$term is $2^5=32$ , $3.$term is $3^5+2^5=243+32=275\dotsm$) Prove that, this infinite sequence has at least $2$ two numbers are equal.

2011 China Team Selection Test, 3

Tags: number theory , combinatorics

A positive integer $n$ is known as an [i]interesting[/i] number if $n$ satisfies \[{\ \{\frac{n}{10^k}} \} > \frac{n}{10^{10}} \] for all $k=1,2,\ldots 9$. Find the number of interesting numbers.

2006 Czech and Slovak Olympiad III A, 1

Tags: number theory

Define a sequence of positive integers $\{a_n\}$ through the recursive formula: $a_{n+1}=a_n+b_n(n\ge 1)$,where $b_n$ is obtained by rearranging the digits of $a_n$ (in decimal representation) in reverse order (for example,if $a_1=250$,then $b_1=52,a_2=302$,and so on). Can $a_7$ be a prime?

2005 Brazil National Olympiad, 6

Tags: modular arithmetic , induction , greatest common divisor , geometric sequence , number theory

Given positive integers $a,c$ and integer $b$, prove that there exists a positive integer $x$ such that \[ a^x + x \equiv b \pmod c, \] that is, there exists a positive integer $x$ such that $c$ is a divisor of $a^x + x - b$.

2018 India PRMO, 15

Tags: minimum , number theory , perfect square

Let $a$ and $b$ be natural numbers such that $2a-b$, $a-2b$ and $a+b$ are all distinct squares. What is the smallest possible value of $b$ ?

1984 All Soviet Union Mathematical Olympiad, 379

Tags: diophantine , diophantine equation , number theory , algebra , exponential

Find integers $m$ and $n$ such that $(5 + 3 \sqrt2)^m = (3 + 5 \sqrt2)^n$.

2014 Hanoi Open Mathematics Competitions, 5

Tags: sequence , integer , number theory

The first two terms of a sequence are $2$ and $3$. Each next term thereafter is the sum of the nearestly previous two terms if their sum is not greather than $10, 0$ otherwise. The $2014$th term is: (A): $0$, (B): $8$, (C): $6$, (D): $4$, (E) None of the above.

2023 CMIMC Team, 3

Tags: team , number theory , relatively prime

Find the number of ordered triples of positive integers $(a,b,c),$ where $1 \leq a,b,c \leq 10,$ with the property that $\gcd(a,b), \gcd(a,c),$ and $\gcd(b,c)$ are all pairwise relatively prime. [i]Proposed by Kyle Lee[/i]

2004 Iran MO (3rd Round), 19

Tags: euler , number theory

Find all integer solutions of $ p^3\equal{}p^2\plus{}q^2\plus{}r^2$ where $ p,q,r$ are primes.

2010 Contests, 3

Tags: modular arithmetic , relatively prime , number theory

Prove that for every given positive integer $n$, there exists a prime $p$ and an integer $m$ such that $(a)$ $p \equiv 5 \pmod 6$ $(b)$ $p \nmid n$ $(c)$ $n \equiv m^3 \pmod p$

2017 Balkan MO Shortlist, N2

Tags: perfect square , functional equation , functional equation in n , number theory

Find all functions $f :Z_{>0} \to Z_{>0}$ such that the number $xf(x) + f ^2(y) + 2xf(y)$ is a perfect square for all positive integers $x,y$.

1997 Estonia Team Selection Test, 3

Tags: number theory

It is known that for every integer $n > 1$ there is a prime number among the numbers $n+1,n+2,...,2n-1.$ Determine all positive integers $n$ with the following property: Every integer $m > 1$ less than $n$ and coprime to $n$ is prime.

2018 CMIMC Number Theory, 3

Tags: number theory

Let $S$ be the set of natural numbers that cannot be written as the sum of three squares. Legendre's three-square theorem states that $S$ consists of precisely the integers of the form $4^a(8b+7)$ where $a$ and $b$ are nonnegative integers. Find the smallest $n\in\mathbb N$ such that $n$ and $n+1$ are both in $S$.

2025 Ukraine National Mathematical Olympiad, 8.6

Tags: number theory

Given $2025$ positive integer numbers such that the least common multiple (LCM) of all these numbers is not a perfect square. Mykhailo consecutively hides one of these numbers and writes down the LCM of the remaining $2024$ numbers that are not hidden. What is the maximum number of the $2025$ written numbers that can be perfect squares? [i]Proposed by Oleksii Masalitin[/i]

2014 Ukraine Team Selection Test, 9

Tags: diophantine equation , number theory , prime number , parameterization , diophantine , system of equations

Let $m, n$ be odd prime numbers. Find all pairs of integers numbers $a, b$ for which the system of equations: $x^m+y^m+z^m=a$, $x^n+y^n+z^n=b$ has many solutions in integers $x, y, z$.

1941 Moscow Mathematical Olympiad, 072

Tags: number theory , divisible

Find the number $\overline {523abc}$ divisible by $7, 8$ and $9$.

2021 Alibaba Global Math Competition, 17

Tags: algebra , polynomial , number theory , finite field

Let $p$ be a prime number and let $\mathbb{F}_p$ be the finite field with $p$ elements. Consider an automorphism $\tau$ of the polynomial ring $\mathbb{F}_p[x]$ given by \[\tau(f)(x)=f(x+1).\] Let $R$ denote the subring of $\mathbb{F}_p[x]$ consisting of those polynomials $f$ with $\tau(f)=f$. Find a polynomial $g \in \mathbb{F}_p[x]$ such that $\mathbb{F}_p[x]$ is a free module over $R$ with basis $g,\tau(g),\dots,\tau^{p-1}(g)$.

2013 Abels Math Contest (Norwegian MO) Final, 3

Tags: number theory , prime , divide

A prime number $p \ge 5$ is given. Write $\frac13+\frac24+... +\frac{p -3}{p - 1}=\frac{a}{b}$ for natural numbers $a$ and $b$. Show that $p$ divides $a$.

2011 Brazil National Olympiad, 4

Tags: number theory , number theory gcd , induction

Do there exist $2011$ positive integers $a_1 < a_2 < \ldots < a_{2011}$ such that $\gcd(a_i,a_j) = a_j - a_i$ for any $i$, $j$ such that $1 \le i < j \le 2011$?