Olympiad problems

2021 Austrian MO Regional Competition, 3

The numbers $1, 2, ..., 2020$ and $2021$ are written on a blackboard. The following operation is executed: Two numbers are chosen, both are erased and replaced by the absolute value of their difference. This operation is repeated until there is only one number left on the blackboard. (a) Show that $2021$ can be the final number on the blackboard. (b) Show that $2020$ cannot be the final number on the blackboard. (Karl Czakler)

2005 Poland - Second Round, 1

Tags: algebra , polynomial , modular arithmetic , quadratic , number theory

The polynomial $W(x)=x^2+ax+b$ with integer coefficients has the following property: for every prime number $p$ there is an integer $k$ such that both $W(k)$ and $W(k+1)$ are divisible by $p$. Show that there is an integer $m$ such that $W(m)=W(m+1)=0$.

2021 Middle European Mathematical Olympiad, 4

Tags: number theory , meme

Let $n \ge 3$ be an integer. Zagi the squirrel sits at a vertex of a regular $n$-gon. Zagi plans to make a journey of $n-1$ jumps such that in the $i$-th jump, it jumps by $i$ edges clockwise, for $i \in \{1, \ldots,n-1 \}$. Prove that if after $\lceil \tfrac{n}{2} \rceil$ jumps Zagi has visited $\lceil \tfrac{n}{2} \rceil+1$ distinct vertices, then after $n-1$ jumps Zagi will have visited all of the vertices. ([i]Remark.[/i] For a real number $x$, we denote by $\lceil x \rceil$ the smallest integer larger or equal to $x$.)

2024 Malaysian APMO Camp Selection Test, 1

Tags: number theory

Let $a_1<a_2< \cdots$ be a strictly increasing sequence of positive integers. Suppose there exist $N$ such that for all $n>N$, $$a_{n+1}\mid a_1+a_2+\cdots+a_n$$ Prove that there exist $M$ such that $a_{m+1}=2a_m$ for all $m>M$. [i]Proposed by Ivan Chan Kai Chin[/i]

2011 ELMO Shortlist, 3

Tags: number theory

Let $n>1$ be a fixed positive integer, and call an $n$-tuple $(a_1,a_2,\ldots,a_n)$ of integers greater than $1$ [i]good[/i] if and only if $a_i\Big|\left(\frac{a_1a_2\cdots a_n}{a_i}-1\right)$ for $i=1,2,\ldots,n$. Prove that there are finitely many good $n$-tuples. [i]Mitchell Lee.[/i]

2017 Regional Olympiad of Mexico Southeast, 6

Tags: fibonacci , number theory

Consider $f_1=1, f_2=1$ and $f_{n+1}=f_n+f_{n-1}$ for $n\geq 2$. Determine if exists $n\leq 1000001$ such that the last three digits of $f_n$ are zero.

2000 Moldova National Olympiad, Problem 2

Tags: number theory , algebra

Thirty numbers are arranged on a circle in such a way that each number equals the absolute difference of its two neighbors. Given that the sum of the numbers is $2000$, determine the numbers.

1984 IMO Longlists, 43

Tags: number theory , equation , divisibility , power of 2

Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.

2011 AIME Problems, 9

Tags: number theory , relatively prime

Let $x_1,x_2,\dots ,x_6$ be nonnegative real numbers such that $x_1+x_2+x_3+x_4+x_5+x_6=1$, and $x_1x_3x_5+x_2x_4x_6 \geq \frac{1}{540}$. Let $p$ and $q$ be positive relatively prime integers such that $\frac{p}{q}$ is the maximum possible value of $x_1x_2x_3+x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_6 + x_5x_6x_1 + x_6x_1x_2$. Find $p+q$.

2000 Estonia National Olympiad, 3

Tags: number theory , diophantine , diophantine equation

Are there any (not necessarily positive) integers $m$ and $n$ such that a) $\frac{1}{m}-\frac{1}{n}=\frac{1}{m-n}$ ? b) $\frac{1}{m}-\frac{1}{n}=\frac{1}{n-m}$

2025 Kyiv City MO Round 1, Problem 5

Tags: number theory

Find all quadruples of positive integers $ (a, p, q, r) $, where $ p, q, r $ are prime numbers, such that the following equation holds: \[ p^2q^2 + q^2r^2 + r^2p^2 + 3 = 4 \cdot 13^a. \] [i]Proposed by Oleksii Masalitin[/i]

2023 Romania Team Selection Test, P2

Tags: number theory

Find all positive integers, such that there exist positive integers $a, b, c$, satisfying $\gcd(a, b, c)=1$ and $n=\gcd(ab+c, ac-b)=a+b+c$.

2013 Online Math Open Problems, 34

Tags: number theory , relatively prime

For positive integers $n$, let $s(n)$ denote the sum of the squares of the positive integers less than or equal to $n$ that are relatively prime to $n$. Find the greatest integer less than or equal to \[ \sum_{n\mid 2013} \frac{s(n)}{n^2}, \] where the summation runs over all positive integers $n$ dividing $2013$. [i]Ray Li[/i]

2014 Balkan MO Shortlist, N4

Tags: number theory

A [i]special number[/i] is a positive integer $n$ for which there exists positive integers $a$, $b$, $c$, and $d$ with \[ n = \frac {a^3 + 2b^3} {c^3 + 2d^3}. \] Prove that i) there are infinitely many special numbers; ii) $2014$ is not a special number. [i]Romania[/i]

1973 Poland - Second Round, 6

Tags: number theory , greatest common divisor

Prove that for every non-negative integer $m$ there exists a polynomial w with integer coefficients such that $2^m$ is the greatest common divisor of the numbers $$ a_n = 3^n + w(n), n = 0, 1, 2, ....$$

2019 Regional Olympiad of Mexico Southeast, 1

Tags: number theory

Found the smaller multiple of $2019$ of the form $abcabc\dots abc$, where $a,b$ and $c$ are digits.

2016 Belarus Team Selection Test, 3

Tags: number theory , divisibility

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

2005 Hong kong National Olympiad, 3

Tags: search , number theory

Show that there exist infinitely many square-free positive integers $n$ that divide $2005^n-1$.

2019 Mathematical Talent Reward Programme, MCQ: P 2

Tags: number theory

What is the number of integral solutions of the equation $a^{b^2}=b^{2a}$, where a > 0 and $|b|>|a|$ [list=1] [*] 3 [*] 4 [*] 6 [*] 8 [/list]

2023 Singapore Senior Math Olympiad, 2

Tags: diophantine equation , number theory

Find all positive integers $k$ such that there exists positive integers $a, b$ such that \[a^2+4=(k^2-4)b^2.\]

1960 IMO, 1

Tags: algebra , number theory , divisibility , decimal representation

Determine all three-digit numbers $N$ having the property that $N$ is divisible by 11, and $\dfrac{N}{11}$ is equal to the sum of the squares of the digits of $N$.

2023 Myanmar IMO Training, 5

Tags: function , floor function , modular arithmetic , relatively prime , number theory

For a real number $x$, let $\lfloor x\rfloor$ stand for the largest integer that is less than or equal to $x$. Prove that \[ \left\lfloor{(n-1)!\over n(n+1)}\right\rfloor \] is even for every positive integer $n$.

2008 India National Olympiad, 3

Tags: number theory

Let $ A$ be a set of real numbers such that $ A$ has at least four elements. Suppose $ A$ has the property that $ a^2 \plus{} bc$ is a rational number for all distinct numbers $ a,b,c$ in $ A$. Prove that there exists a positive integer $ M$ such that $ a\sqrt{M}$ is a rational number for every $ a$ in $ A$.

2008 Brazil Team Selection Test, 1

Tags: divisibility , number theory

Find all pairs of natural numbers $ (a, b)$ such that $ 7^a \minus{} 3^b$ divides $ a^4 \plus{} b^2$. [i]Author: Stephan Wagner, Austria[/i]

2008 Putnam, B5

Tags: function , number theory , greatest common divisor , calculus , derivative , integration

Find all continuously differentiable functions $ f: \mathbb{R}\to\mathbb{R}$ such that for every rational number $ q,$ the number $ f(q)$ is rational and has the same denominator as $ q.$ (The denominator of a rational number $ q$ is the unique positive integer $ b$ such that $ q\equal{}a/b$ for some integer $ a$ with $ \gcd(a,b)\equal{}1.$) (Note: $ \gcd$ means greatest common divisor.)