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

1990 China Team Selection Test, 3
Tags: modular arithmetic , induction , number theory
Prove that for every integer power of 2, there exists a multiple of it with all digits (in decimal expression) not zero.

1981 All Soviet Union Mathematical Olympiad, 306
Tags: product , consecutive , number theory
Let us say, that a natural number has the property $P(k)$ if it can be represented as a product of $k$ succeeding natural numbers greater than $1$. a) Find k such that there exists n which has properties $P(k)$ and $P(k+2)$ simultaneously. b) Prove that there is no number having properties $P(2)$ and $P(4)$ simultaneously

2014 Contests, 2
Tags: limit , number theory , floor function
$a)$ Let $n$ a positive integer. Prove that $gcd(n, \lfloor n\sqrt{2} \rfloor)<\sqrt[4]{8}\sqrt{n}$. $b)$ Prove that there are infinitely many positive integers $n$ such that $gcd(n, \lfloor n\sqrt{2} \rfloor)>\sqrt[4]{7.99}\sqrt{n}$.

2016 Regional Olympiad of Mexico Northeast, 1
Tags: diophantine , diophantine equation , number theory
Determine if there is any triple of nonnegative integers, not necessarily different, $(a, b, c)$ such that: $$a^3 + b^3 + c^3 = 2016$$

2019 CHMMC (Fall), 7
Tags: number theory
Let $S$ be the set of all positive integers $n$ satisfying the following two conditions: $\bullet$ $n$ is relatively prime to all positive integers less than or equal to $\frac{n}{6}$ $\bullet$ $2^n \equiv 4$ mod $n$ What is the sum of all numbers in $S$?

2000 Czech And Slovak Olympiad IIIA, 6
Tags: digit , number theory
Find all four-digit numbers $\overline{abcd}$ (in decimal system) such that $\overline{abcd}= (\overline{ac}+1).(\overline{bd} +1)$

1997 Romania National Olympiad, 1
Tags: integer , number theory
Let $n_1 = \overline{abcabc}$ and $n_2= \overline{d00d}$ be numbers represented in the decimal system, with $a\ne 0$ and $d \ne 0$. a) Prove that $\sqrt{n_1}$ cannot be an integer. b) Find all positive integers $n_1$ and $n_2$ such that $\sqrt{n_1+n_2}$ is an integer number. c) From all the pairs $(n_1,n_2)$ such that $\sqrt{n_1 n_2}$ is an integer find those for which $\sqrt{n_1 n_2}$ has the greatest possible value

2019 Brazil Team Selection Test, 1
Tags: number of divisors , arithmetic function , divisor , number theory
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

KoMaL A Problems 2021/2022, A. 808
Tags: number theory
Find all triples of positive integers $a, b, c$ such that they are pairwise relatively prime and $a^2+3b^2c^2=7^c$.

1997 Turkey MO (2nd round), 1
Tags: quadratic , number theory , modular arithmetic
Find all pairs of integers $(x, y)$ such that $5x^{2}-6xy+7y^{2}=383$.

2022 MOAA, 5
Tags: number theory
Find the smallest positive integer that is equal to the sum of the product of its digits and the sum of its digits.

2006 IberoAmerican Olympiad For University Students, 1
Tags: number theory , modular arithmetic
Let $m,n$ be positive integers greater than $1$. We define the sets $P_m=\left\{\frac{1}{m},\frac{2}{m},\cdots,\frac{m-1}{m}\right\}$ and $P_n=\left\{\frac{1}{n},\frac{2}{n},\cdots,\frac{n-1}{n}\right\}$. Find the distance between $P_m$ and $P_n$, that is defined as \[\min\{|a-b|:a\in P_m,b\in P_n\}\]

2022 Iran MO (3rd Round), 3
Tags: number theory
We call natural number $m$ [b]ziba[/b], iff every natural number $n$ with the condition $1\le n\le m$ can be shown as sum of [some of] positive and [u]distinct[/u] divisors of $m$. Prove that infinitely ziba numbers in the form of $(k\in\mathbb{N})k^2+k+2022$ exist.

2016 Iran MO (3rd Round), 3
Tags: number theory
A sequence $P=\left \{ a_{n} \right \}$ is called a $ \text{Permutation}$ of natural numbers (positive integers) if for any natural number $m,$ there exists a unique natural number $n$ such that $a_n=m.$ We also define $S_k(P)$ as: $S_k(P)=a_{1}+a_{2}+\cdots +a_{k}$ (the sum of the first $k$ elements of the sequence). Prove that there exists infinitely many distinct $ \text{Permutations}$ of natural numbers like $P_1,P_2, \cdots$ such that$:$ $$\forall k, \forall i<j: S_k(P_i)|S_k(P_j)$$

2016 NIMO Problems, 1
Tags: divisor , remainder , number theory
Let $m$ be a positive integer less than $2015$. Suppose that the remainder when $2015$ is divided by $m$ is $n$. Compute the largest possible value of $n$. [i] Proposed by Michael Ren [/i]

2008 Argentina Iberoamerican TST, 1
Tags: number theory
Find all integers $ x$ such that $ x(x\plus{}1)(x\plus{}7)(x\plus{}8)$ is a perfect square It's a nice problem ...hope you enjoy it! Daniel

2019 Saudi Arabia IMO TST, 1
Tags: divide , number theory , divisible
Let $a_0$ be an arbitrary positive integer. Let $(a_n)$ be infinite sequence of positive integers such that for every positive integer $n$, the term $a_n$ is the smallest positive integer such that $a_0 + a_1 +... + a_n$ is divisible by $n$. Prove that there exist $N$ such that $a_{n+1} = a_n$ for all $n \ge N$

2022 Princeton University Math Competition, A6 / B8
Tags: number theory
Given a positive integer $\ell,$ define the sequence $\{a^{(\ell)}\}_{n=1}^{\infty}$ such that $a_n^{(\ell)}=\lfloor n + \sqrt[\ell]{n}+\tfrac{1}{2}\rfloor$ for all positive integers $n.$ Let $S$ denote the set of positive integers that appear in all three of the sequences $\{a_n^{(2)} \}_{n=1}^{\infty},$ $\{a_n^{(3)} \}_{n=1}^{\infty},$ and $\{a_n^{(4)} \}_{n=1}^{\infty}.$ Find the sum of the elements of $S$ that lie in the interval $[1,100].$

2019 All-Russian Olympiad, 5
Tags: prime number , rectangle , number theory
In a kindergarten, a nurse took $n$ congruent cardboard rectangles and gave them to $n$ kids, one per each. Each kid has cut its rectangle into congruent squares(the squares of different kids could be of different sizes). It turned out that the total number of the obtained squares is a prime number. Prove that all the initial squares were in fact squares.

2012 Balkan MO Shortlist, N1
Tags: sequence , consecutive , perfect square , number theory
A sequence $(a_n)_{n=1}^{\infty}$ of positive integers satisfies the condition $a_{n+1} = a_n +\tau (n)$ for all positive integers $n$ where $\tau (n)$ is the number of positive integer divisors of $n$. Determine whether two consecutive terms of this sequence can be perfect squares.

2022 Dutch Mathematical Olympiad, 2
Tags: number theory
A set consisting of at least two distinct positive integers is called [i]centenary [/i] if its greatest element is $100$. We will consider the average of all numbers in a centenary set, which we will call the average of the set. For example, the average of the centenary set $\{1, 2, 20, 100\}$ is $\frac{123}{4}$ and the average of the centenary set $\{74, 90, 100\}$ is $88$. Determine all integers that can occur as the average of a centenary set.

2019 Poland - Second Round, 4
Tags: positive integer , divisibility , number theory
Let $a_1, a_2, \ldots, a_n$ ($n\ge 3$) be positive integers such that $gcd(a_1, a_2, \ldots, a_n)=1$ and for each $i\in \lbrace 1,2,\ldots, n \rbrace$ we have $a_i|a_1+a_2+\ldots+a_n$. Prove that $a_1a_2\ldots a_n | (a_1+a_2+\ldots+a_n)^{n-2}$.

2009 India IMO Training Camp, 8
Tags: modular arithmetic , number theory
Let $ n$ be a natural number $ \ge 2$ which divides $ 3^n\plus{}4^n$.Prove That $ 7\mid n$.

2023 Baltic Way, 19
Tags: number theory
Show that $S(2^{2^{2 \cdot 2023}})>2023$, where $S(m)$ denotes the digit sum of $m$.

2009 China Team Selection Test, 3
Tags: modular arithmetic , number theory , inequalities
Prove that for any odd prime number $ p,$ the number of positive integer $ n$ satisfying $ p|n! \plus{} 1$ is less than or equal to $ cp^\frac{2}{3}.$ where $ c$ is a constant independent of $ p.$