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

2016 Tournament Of Towns, 6
Tags: combinatorial game , algebra , polynomial , number theory , game theory , combinatorics
Petya and Vasya play the following game. Petya conceives a polynomial $P(x)$ having integer coefficients. On each move, Vasya pays him a ruble, and calls an integer $a$ of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equation $P(x)=a$. The game continues until Petya is forced to repeat an answer. What minimal amount of rubles must Vasya pay in order to win? [i](Anant Mudgal)[/i] (Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])

2007 Romania Team Selection Test, 4
Tags: algebra , polynomial , modular arithmetic , induction , number theory
i) Find all infinite arithmetic progressions formed with positive integers such that there exists a number $N \in \mathbb{N}$, such that for any prime $p$, $p > N$, the $p$-th term of the progression is also prime. ii) Find all polynomials $f(X) \in \mathbb{Z}[X]$, such that there exist $N \in \mathbb{N}$, such that for any prime $p$, $p > N$, $| f(p) |$ is also prime. [i]Dan Schwarz[/i]

2007 Junior Balkan Team Selection Tests - Romania, 2
Tags: number theory
Solve in positive integers: $(x^{2}+2)(y^{2}+3)(z^{2}+4)=60xyz$.

2020 Stars of Mathematics, 2
Tags: combinatorics , number theory
Given a positive integer $k,$ prove that for any integer $n \geq 20k,$ there exist $n - k$ pairwise distinct positive integers whose squares add up to $n(n + 1)(2n + 1)/6.$ [i]The Problem Selection Committee[/i]

2002 All-Russian Olympiad, 1
Tags: number theory
Determine the smallest natural number which can be represented both as the sum of $2002$ positive integers with the same sum of decimal digits, and as the sum of $2003$ integers with the same sum of decimal digits.

2001 Cono Sur Olympiad, 2
Tags: number theory
Find all positive integers $m$ for which $2001\cdot S (m) = m$ where $S(m)$ denotes the sum of the digits of $m$.

2010 Contests, 4
Tags: induction , number theory
Prove that for each positive integer n,the equation $x^{2}+15y^{2}=4^{n}$ has at least $n$ integer solution $(x,y)$

2015 China Second Round Olympiad, 4
Tags: number theory , legendre s theorem
Find all positive integers $k$ such that for any positive integer $n$, $2^{(k-1)n+1}$ does not divide $\frac{(kn)!}{n!}$.

2020 Macedonia Additional BMO TST, 3
Tags: number theory , divisibility
Does there exist a set of $2020$ distinct positive whole numbers with the property that the product of any $101$ of them is divisible by the sum of those $101$ numbers?

2003 Poland - Second Round, 1
Tags: binomial coefficient , number theory
Prove that exists integer $n > 2003$ that in sequence $\binom{n}{0}$, $\binom{n}{1}$, $\binom{n}{2}$, ..., $\binom{n}{2003}$ each element is a divisor of all elements which are after him.

2003 Canada National Olympiad, 2
Tags: modular arithmetic , euler , function , number theory
Find the last three digits of the number $2003^{{2002}^{2001}}$.

2024 Princeton University Math Competition, B2
Tags: number theory
Find the remainder when $$\sum_{x=1}^{2024} \sum_{y=1}^{2024} (xy)$$ is divided by $31.$

2012 Silk Road, 3
Tags: binomial coefficient , number theory , greatest common divisor
Let $n > 1$ be an integer. Determine the greatest common divisor of the set of numbers $\left\{ \left( \begin{matrix} 2n \\ 2i+1 \\ \end{matrix} \right):0 \le i \le n-1 \right\}$ i.e. the largest positive integer, dividing $\left( \begin{matrix} 2n \\ 2i+1 \\ \end{matrix} \right)$ without remainder for every $i = 0, 1, ..., n–1$ . (Here $\left( \begin{matrix} m \\ l \\ \end{matrix} \right)=\text{C}_{m}^{l}=\frac{m\text{!}}{l\text{!}\left( m-l \right)\text{!}}$ is binomial coefficient.)

2022 Czech-Polish-Slovak Junior Match, 2
Tags: number theory , divisible
The number $2022$ is written on the board. In each step, we replace one of the $2$ digits with the number $2022$. For example $$2022 \Rightarrow 2020222 \Rightarrow 2020220222 \Rightarrow ...$$ After how many steps can a number divisible by $22$ be written on the board? Specify all options.

2024 Serbia National Math Olympiad, 1
Tags: geometric sequence , number theory
Find all positive integers $n$, such that if their divisors are $1=d_1<d_2<\ldots<d_k=n$ for $k \geq 4$, then the numbers $d_2-d_1, d_3-d_2, \ldots, d_k-d_{k-1}$ form a geometric progression in some order.

2014 Turkey Team Selection Test, 2
Tags: number theory
$a_1=-5$, $a_2=-6$ and for all $n \geq 2$ the ${(a_n)^\infty}_{n=1}$ sequence defined as, \[a_{n+1}=a_n+(a_1+1)(2a_2+1)(3a_3+1)\cdots((n-1)a_{n-1}+1)((n^2+n)a_n+2n+1)).\] If a prime $p$ divides $na_n+1$ for a natural number n, prove that there is a integer $m$ such that $m^2\equiv5(modp)$

TNO 2008 Junior, 2
Tags: number theory
A cube of size $4 \times 4 \times 4$ is divided into 16 equal squares per face, with numbers from 1 to 96 randomly assigned to these squares. An operation consists of taking two squares that share a vertex, summing their numbers, and rewriting this sum in one of the squares while leaving the other blank. After performing several such operations, only one number remains. Prove that regardless of the order of operations, the final remaining number is always the same. Additionally, find this number.

2015 All-Russian Olympiad, 5
Tags: combinatorics , number theory , all naturals
An immortal flea jumps on whole points of the number line, beginning with $0$. The length of the first jump is $3$, the second $5$, the third $9$, and so on. The length of $k^{\text{th}}$ jump is equal to $2^k + 1$. The flea decides whether to jump left or right on its own. Is it possible that sooner or later the flee will have been on every natural point, perhaps having visited some of the points more than once?

2014 Grand Duchy of Lithuania, 4
Tags: number theory
Determine all positive integers $n > 1$ for which $n + D(n)$ is a power of $10$, where $D(n)$ denotes the largest integer divisor of $n$ satisfying $D(n) < n$.

2008 Iran Team Selection Test, 11
Tags: function , modular arithmetic , number theory , relatively prime , functional equation
$ k$ is a given natural number. Find all functions $ f: \mathbb{N}\rightarrow\mathbb{N}$ such that for each $ m,n\in\mathbb{N}$ the following holds: \[ f(m)\plus{}f(n)\mid (m\plus{}n)^k\]

2012 ELMO Shortlist, 5
Tags: modular arithmetic , number theory
Let $n>2$ be a positive integer and let $p$ be a prime. Suppose that the nonzero integers are colored in $n$ colors. Let $a_1,a_2,\ldots,a_{n}$ be integers such that for all $1\le i\le n$, $p^i\nmid a_i$ and $p^{i-1}\mid a_i$. In terms of $n$, $p$, and $\{a_i\}_{i=1}^{n}$, determine if there must exist integers $x_1,x_2,\ldots,x_{n}$ of the same color such that $a_1x_1+a_2x_2+\cdots+a_{n}x_{n}=0$. [i]Ravi Jagadeesan.[/i]

2022 New Zealand MO, 6
Tags: number theory , sum , divisible
Let a positive integer $n$ be given. Determine, in terms of $n$, the least positive integer $k$ such that among any $k$ positive integers, it is always possible to select a positive even number of them having sum divisible by $n$.

Russian TST 2014, P2
Tags: number theory , prime number
Let $p,q$ and $s{}$ be prime numbers such that $2^sq =p^y-1$ where $y > 1.$ Find all possible values of $p.$

2000 Portugal MO, 4
Tags: number theory , algebra
Calculates the sum of all numbers that can be formed using each of the odd digits once, that is, the numbers $13579$, $13597$, ..., $97531$.

2006 Stanford Mathematics Tournament, 14
Tags: floor function , factorial , function , number theory , modular arithmetic
Find the smallest nonnegative integer $n$ for which $\binom{2006}{n}$ is divisible by $7^3$.