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: 15460

2022 ELMO Revenge, Bonus
Tags: number theory
Determine, with proof, if there exists an odd prime $p$ such that the following equation holds: $$\sum_{n = 1}^{\frac{p-1}{2}} \cot\left(\frac{\pi n^2}{p}\right) = 69\sqrt{p}$$ [i]Proposed by Chris Bao[/i]

1996 Baltic Way, 8
Tags: least common multiple , greatest common divisor , number theory
Consider the sequence: $x_1=19,x_2=95,x_{n+2}=\text{lcm} (x_{n+1},x_n)+x_n$, for $n>1$, where $\text{lcm} (a,b)$ means the least common multiple of $a$ and $b$. Find the greatest common divisor of $x_{1995}$ and $x_{1996}$.

2023 Kyiv City MO, Problem 3
Tags: number theory
Prove that there don't exist positive integer numbers $k$ and $n$ which satisfy equation $n^n+(n+1)^{n+1}+(n+2)^{n+2} = 2023^k$. [i]Proposed by Mykhailo Shtandenko[/i]

Kvant 2021, M2645
Tags: number theory
Vitya wrote down $n{}$ different natural numbers in his notebook. For each pair of numbers from the notebook, he wrote out their smallest common multiple on the board. Could it happen for some $n>100$ that $n(n-1)/2$ numbers on the board are (in some order) consecutive terms of a non-constant arithmetic progression? [i]Proposed by S. Berlov[/i]

1998 Slovenia National Olympiad, Problem 1
Tags: divisibility , number theory
Show that for any integter $a$, the number $\frac{a^5}5+\frac{a^3}3+\frac{7a}{15}$ is an integer.

2008 Gheorghe Vranceanu, 2
Tags: kronecker , fractional part , inequalities , number theory
Show that there is a natural number $ n $ that satisfies the following inequalities: $$ \sqrt{3} -\frac{1}{10}<\{ n\sqrt 3\} +\{ (n+1)\sqrt 3 \} <\sqrt 3. $$

1972 Bulgaria National Olympiad, Problem 1
Tags: number theory
Prove that there are don't exist integers $a,b,c$ such that for every integer $x$ the number $A=(x+a)(x+b)(x+c)-x^3-1$ is divisible by $9$. [i]I. Tonov[/i]

2007 Iran MO (3rd Round), 2
Tags: graph theory , number theory
Let $ m,n$ be two integers such that $ \varphi(m) \equal{}\varphi(n) \equal{} c$. Prove that there exist natural numbers $ b_{1},b_{2},\dots,b_{c}$ such that $ \{b_{1},b_{2},\dots,b_{c}\}$ is a reduced residue system with both $ m$ and $ n$.

2003 Indonesia MO, 1
Tags: number theory
Prove that $a^9 - a$ is divisible by $6$ for all integers $a$.

2025 Portugal MO, 5
Tags: number theory
An integer number $n \geq 2$ is called [i]feirense[/i] if it is possible to write on a sheet of paper some integers such that every positive divisor of $n$ less than $n$ is the difference between two numbers on the sheet, and no other positive number is. Find all the feirense numbers.

2021 Israel TST, 1
Tags: sum of digits , number theory
A pair of positive integers $(a,b)$ is called an [b]average couple[/b] if there exist positive integers $k$ and $c_1, \dots, c_k$ for which \[\frac{c_1+c_2+\cdots+c_k}{k}=a\qquad \text{and} \qquad \frac{s(c_1)+s(c_2)+\cdots+s(c_k)}{k}=b\] where $s(n)$ denotes the sum of digits of $n$ in decimal representation. Find the number of average couples $(a,b)$ for which $a,b<10^{10}$.

2021-IMOC, N4
Tags: number theory
There are $m \geq 3$ positive integers, not necessarily distinct, that are arranged in a circle so that any positive integer divides the sum of its neighbours. Show that if there is exactly one $1$, then for any positive integer $n$, there are at most $\phi(n)$ copies of $n$. [i]Proposed By- (usjl, adapted from 2014 Taiwan TST)[/i]

1982 IMO Longlists, 7
Tags: algorithm , euclidean algorithm , greatest common divisor , floor function , number theory , modular arithmetic
Find all solutions $(x, y) \in \mathbb Z^2$ of the equation \[x^3 - y^3 = 2xy + 8.\]

2011 ELMO Shortlist, 1
Tags: modular arithmetic , number theory
Prove that $n^3-n-3$ is not a perfect square for any integer $n$. [i]Calvin Deng.[/i]

2018 Iran MO (1st Round), 15
Tags: combinatorics , number theory
Let $a_1, a_2, a_3, \dots, a_{20}$ be a permutation of the numbers $1, 2, \dots, 20$. How many different values can the expression $a_1-a_2+a_3-\dots - a_{20}$ have?

2020 USEMO, 6
Tags: number theory
Prove that for every odd integer $n > 1$, there exist integers $a, b > 0$ such that, if we let $Q(x) = (x + a)^ 2 + b$, then the following conditions hold: $\bullet$ we have $\gcd(a, n) = gcd(b, n) = 1$; $\bullet$ the number $Q(0)$ is divisible by $n$; and $\bullet$ the numbers $Q(1), Q(2), Q(3), \dots$ each have a prime factor not dividing $n$.

2001 Portugal MO, 3
Tags: digit , number theory
How many consecutive zeros are there at the end of the number $2001! = 2001 \times 2000 \times ... \times 3 \times 2 \times 1$ ?

2021 Harvard-MIT Mathematics Tournament., 3
Tags: algebra , polynomial , number theory
Among all polynomials $P(x)$ with integer coefficients for which $P(-10) = 145$ and $P(9) = 164$, compute the smallest possible value of $|P(0)|.$

2003 Estonia Team Selection Test, 2
Tags: divisible , number theory
Let $n$ be a positive integer. Prove that if the number overbrace $\underbrace{\hbox{99...9}}_{\hbox{n}}$ is divisible by $n$, then the number $\underbrace{\hbox{11...1}}_{\hbox{n}}$ is also divisible by $n$. (H. Nestra)

2021 Saudi Arabia Training Tests, 38
Tags: perfect square , number theory
Prove that the set of all divisors of a positive integer which is not a perfect square can be divided into pairs so that in each pair, one number is divided by another.

2023 German National Olympiad, 6
Tags: cubic equation , number theory , algebra , ceiling function
The equation $x^3-3x^2+1=0$ has three real solutions $x_1<x_2<x_3$. Show that for any positive integer $n$, the number $\left\lceil x_3^n\right\rceil$ is a multiple of $3$.

2023 Malaysian IMO Training Camp, 1
Tags: number theory
Does there exist a positive integer, $x$, such that $(x+2)^{2023}-x^{2023}$ has exactly $2023^{2023}$ factors? [i]Proposed by Wong Jer Ren[/i]

2020 South East Mathematical Olympiad, 4
Tags: combinatorics , maximum value , number theory
Let $a_1,a_2,\dots, a_{17}$ be a permutation of $1,2,\dots, 17$ such that $(a_1-a_2)(a_2-a_3)\dots(a_{17}-a_1)=n^{17}$ .Find the maximum possible value of $n$ .

2013 ITAMO, 1
Tags: number theory
A model car is tested on some closed circuit $600$ meters long, consisting of flat stretches, uphill and downhill. All uphill and downhill have the same slope. The test highlights the following facts: [list] (a) The velocity of the car depends only on the fact that the car is driving along a stretch of uphill, plane or downhill; calling these three velocities $v_s, v_p$ and $v_d$ respectively, we have $v_s <v_p <v_d$; (b) $v_s,v_p$ and $v_d$, expressed in meter per second, are integers. (c) Whatever may be the structure of the circuit, the time taken to complete the circuit is always $50$ seconds. [/list] Find all possible values of $v_s, v_p$ and $v_d$.

2011 Saudi Arabia Pre-TST, 2.2
Tags: power of 2 , sum , number theory
Consider the sequence $x_n = 2^n-n$, $n = 0,1 ,2 ,...$. Find all integers $m \ge 0$ such that $s_m = x_0 + x_1 + x_2 + ... + x_m$ is a power of $2$.