Olympiad problems

1972 Spain Mathematical Olympiad, 7

Prove that for every positive integer $n$, the number $$A_n = 5^n + 2 \cdot 3^{n-1} + 1$$ is a multiple of $8$.

2009 QEDMO 6th, 1

Tags: number theory , diophantine , diophantine equation

Solve $y^5 - x^2 = 4$ in integers numbers $x,y$.

1983 IMO Longlists, 22

Tags: modular arithmetic , number theory

Does there exist an infinite number of sets $C$ consisting of $1983$ consecutive natural numbers such that each of the numbers is divisible by some number of the form $a^{1983}$, with $a \in \mathbb N, a \neq 1?$

1990 IMO Longlists, 98

Tags: number theory , decimal representation , prime number , digit

Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n \minus{} 1$ digits $ 1$ and one digit $ 7$ is prime.

2017 China Team Selection Test, 4

Tags: number theory , algebra

An integer $n>1$ is given . Find the smallest positive number $m$ satisfying the following conditions： for any set $\{a,b\}$ $\subset \{1,2,\cdots,2n-1\}$ ,there are non-negative integers $ x, y$ ( not all zero) such that $2n|ax+by$ and $x+y\leq m.$

2022 Princeton University Math Competition, B2

Tags: number theory

Find the sum of the $23$ smallest positive integers that are $4$ more than a multiple of $23$ and whose last two digits are $23.$

2018 Hanoi Open Mathematics Competitions, 11

Tags: number theory , diophantine equation

Find all pairs of nonnegative integers $(x, y)$ for which $(xy + 2)^2 = x^2 + y^2 $.

1989 Poland - Second Round, 4

Tags: number theory , divide , divisible

The given integers are $ a_1, a_2, \ldots , a_{11} $ . Prove that there exists a non-zero sequence $ x_1, x_2, \ldots, x_{11} $ with terms from the set $ \{-1,0,1\} $ such that the number $ x_1a_1 + \ldots x_{11}a_{ 11}$ is divisible by 1989.

2012 IFYM, Sozopol, 3

Tags: number theory , diophantine equation

Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$

2009 Moldova Team Selection Test, 2

Tags: ratio , number theory

[color=darkblue]Let $ M$ be a set of aritmetic progressions with integer terms and ratio bigger than $ 1$. [b]a)[/b] Prove that the set of the integers $ \mathbb{Z}$ can be written as union of the finite number of the progessions from $ M$ with different ratios. [b]b)[/b] Prove that the set of the integers $ \mathbb{Z}$ can not be written as union of the finite number of the progessions from $ M$ with ratios integer numbers, any two of them coprime.[/color]

2011 Indonesia MO, 6

Tags: number theory , modular arithmetic

Let a sequence of integers $a_0, a_1, a_2, \cdots, a_{2010}$ such that $a_0 = 1$ and $2011$ divides $a_{k-1}a_k - k$ for all $k = 1, 2, \cdots, 2010$. Prove that $2011$ divides $a_{2010} + 1$.

2022 Irish Math Olympiad, 1

Tags: number theory , factorial

1. For [i]n[/i] a positive integer, [i]n[/i]! = 1 $\cdot$ 2 $\cdot$ 3 $\dots$ ([i]n[/i] - 1) $\cdot$ [i]n[/i] is the product of the positive integers from 1 to [i]n[/i]. Determine, with proof, all positive integers [i]n[/i] for which [i]n[/i]! + 3 is a power of 3.

2010 Peru Iberoamerican Team Selection Test, P2

Tags: number theory

For each positive integer $k$, let $S(k)$ be the sum of the digits of $k$ in the decimal system. Find all positive integers N for which there exist positive integers $a$,$b$,$c$, coprime two by two, such that: $S(ab) = S(bc) = S(ca) = N$.

1987 Tournament Of Towns, (141) 1

Tags: number theory , sum of squares , perfect square , odd

Is it possible to represent the number $1986$ as the sum of squares of $6$ odd integers?

2022 New Zealand MO, 2

Tags: number theory , perfect power , power of number

Is it possible to pair up the numbers $0, 1, 2, 3,... , 61$ in such a way that when we sum each pair, the product of the $31$ numbers we get is a perfect f ifth power?

2019 ELMO Shortlist, N1

Tags: number theory

Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$, and let $c > 1$ be an integer. Define $x_0=0$ and $x_{i+1} = P(x_i)$ for all integers $i \ge 0$. Show that there are infinitely many positive integers $n$ such that $\gcd (x_n, n+c)=1$. [i]Proposed by Milan Haiman and Carl Schildkraut[/i]

1999 Canada National Olympiad, 3

Tags: number theory

Determine all positive integers $n$ with the property that $n = (d(n))^2$. Here $d(n)$ denotes the number of positive divisors of $n$.

2024 Singapore MO Open, Q3

Tags: number theory , induction

Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.

2009 Hanoi Open Mathematics Competitions, 3

Tags: number theory , perfect square

Let $a, b,c$ be positive integers with no common factor and satisfy the conditions $\frac1a +\frac1b=\frac1c$ Prove that $a + b$ is a square.

2014 Contests, 2

Tags: floor function , number theory

Let $M$ be the set of palindromic integers of the form $5n+4$ where $n\ge 0$ is an integer. [list=a] [*]If we write the elements of $M$ in increasing order, what is the $50^{\text{th}}$ number? [*]Among all numbers in $M$ with nonzero digits which sum up to $2014$ which is the largest and smallest one?[/list]

VI Soros Olympiad 1999 - 2000 (Russia), 8.1

Tags: inequalities , number theory

Let $p,q,r$ be prime numbers such that $2p>q$, $q > 2r$ and $q>p+r$. Prove that $p+q+r\ge 20$.

2002 South africa National Olympiad, 6

Tags: number theory

Find all rational numbers $a$, $b$, $c$ and $d$ such that \[ 8a^2 - 3b^2 + 5c^2 + 16d^2 - 10ab + 42cd + 18a + 22b - 2c - 54d = 42, \] \[ 15a^2 - 3b^2 + 21c^2 - 5d^2 + 4ab +32cd - 28a + 14b - 54c - 52d = -22. \]

2011 ELMO Shortlist, 4

Tags: modular arithmetic , number theory , quadratic

Let $p>13$ be a prime of the form $2q+1$, where $q$ is prime. Find the number of ordered pairs of integers $(m,n)$ such that $0\le m<n<p-1$ and \[3^m+(-12)^m\equiv 3^n+(-12)^n\pmod{p}.\] [i]Alex Zhu.[/i] [hide="Note"]The original version asked for the number of solutions to $2^m+3^m\equiv 2^n+3^n\pmod{p}$ (still $0\le m<n<p-1$), where $p$ is a Fermat prime.[/hide]

KoMaL A Problems 2019/2020, A. 776

Tags: number theory , sequence

Let $k > 1$ be a fixed odd number, and for non-negative integers $n$ let $$f_n=\sum_{\substack{0\leq i\leq n\\ k\mid n-2i}}\binom{n}{i}.$$ Prove that $f_n$ satisfy the following recursion: $$f_{n}^2=\sum_{i=0}^{n} \binom{n}{i}f_{i}f_{n-i}.$$

2019 Romania Team Selection Test, 4

Tags: number theory , diophantine

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?