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

1966 Dutch Mathematical Olympiad, 3
Tags: perfect square , number theory
How many natural numbers are there whose square is a thirty-digit number which has the following curious property: If that thirty-digit number is divided from left to right into three groups of ten digits, then the numbers given by the middle group and the right group formed numbers are both four times the number formed by the left group?

2006 Junior Balkan Team Selection Tests - Romania, 3
Tags: sum of digits , divisor , number theory
For any positive integer $n$ let $s(n)$ be the sum of its digits in decimal representation. Find all numbers $n$ for which $s(n)$ is the largest proper divisor of $n$.

2008 Regional Olympiad of Mexico Center Zone, 5
Tags: number theory , product , digit , divide
Each positive integer number $n \ ge 1$ is assigned the number $p_n$ which is the product of all its non-zero digits. For example, $p_6 = 6$, $p_ {32} = 6$, $p_ {203} = 6$. Let $S = p_1 + p_2 + p_3 + \dots + p_ {999}$. Find the largest prime that divides $S $.

1998 Slovenia National Olympiad, Problem 2
Tags: quadratic , number theory
Find all pairs $(p,q)$ of real numbers such that $p+q=1998$ and the solutions of the equation $x^2+px+q=0$ are integers.

2018 Latvia Baltic Way TST, P14
Tags: integer sequence , number theory
Let $a_1,a_2,...$ be a sequence of positive integers with $a_1=2$. For each $n \ge 1$, $a_{n+1}$ is the biggest prime divisor of $a_1a_2...a_n+1$. Prove that the sequence does not contain numbers $5$ and $11$.

2022 Switzerland Team Selection Test, 6
Tags: integer , number theory , divisibility
Let $n \geq 2$ be an integer. Prove that if $$\frac{n^2+4^n+7^n}{n}$$ is an integer, then it is divisible by 11.

2005 IMO, 4
Tags: number theory , fermat's little theorem
Determine all positive integers relatively prime to all the terms of the infinite sequence \[ a_n=2^n+3^n+6^n -1,\ n\geq 1. \]

2013 International Zhautykov Olympiad, 1
Tags: quadratic , number theory , relatively prime , algebra
A quadratic trinomial $p(x)$ with real coefficients is given. Prove that there is a positive integer $n$ such that the equation $p(x) = \frac{1}{n}$ has no rational roots.

2005 Thailand Mathematical Olympiad, 2
Tags: number theory , divide , divisible
Let $S $ be a set of three distinct integers. Show that there are $a, b \in S$ such that $a \ne b$ and $10 | a^3b - ab^3$.

2009 Tournament Of Towns, 7
Tags: gcd , number theory , prime
Initially a number $6$ is written on a blackboard. At $n$-th step an integer $k$ on the blackboard is replaced by $k+gcd(k,n)$. Prove that at each step the number on the blackboard increases either by $1$ or by a prime number.

2013 Chile TST Ibero, 2
Tags: number theory
Let $a \in \mathbb{N}$ such that $a + n^2$ can be expressed as the sum of two squares for all $n \in \mathbb{N}$. Prove that $a$ is the square of a natural number.

2006 All-Russian Olympiad, 2
Tags: geometry , 3d geometry , modular arithmetic , relatively prime , number theory
If an integer $a > 1$ is given such that $\left(a-1\right)^3+a^3+\left(a+1\right)^3$ is the cube of an integer, then show that $4\mid a$.

2023 AIME, 6
Tags: probability , expected value , number theory , relatively prime
Alice knows that $3$ red cards and $3$ black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2004 China National Olympiad, 3
Tags: induction , number theory
Prove that every positive integer $n$, except a finite number of them, can be represented as a sum of $2004$ positive integers: $n=a_1+a_2+\cdots +a_{2004}$, where $1\le a_1<a_2<\cdots <a_{2004}$, and $a_i \mid a_{i+1}$ for all $1\le i\le 2003$. [i]Chen Yonggao[/i]

2019 Junior Balkan Team Selection Tests - Romania, 2
Tags: number theory , coprime , divide
Let $n$ be a positive integer and $A$ a set containing $8n + 1$ positive integers co-prime with $6$ and less than $30n$. Prove that there exist $a, b \in A$ two different numbers such that $a$ divides $b$.

2001 239 Open Mathematical Olympiad, 5
Tags: polynomial , number theory
Let $P(x)$ be a monic polynomial with integer coefficients of degree $10$. Prove that there exist distinct positive integers $a,b$ not exceeding $101$ such that $P(a)-P(b)$ is divisible by $101$.

2007 Pre-Preparation Course Examination, 21
Tags: number theory
Find all primes $p,q$ such that \[p^q-q^p=pq^2-19\]

2018 Romania Team Selection Tests, 2
Tags: number theory
Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$. [i]Proposed by Warut Suksompong, Thailand[/i]

2014 AMC 12/AHSME, 18
Tags: algebra , function , domain , number theory , logarithm
The domain of the function $f(x)=\log_{\frac12}(\log_4(\log_{\frac14}(\log_{16}(\log_{\frac1{16}}x))))$ is an interval of length $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? $\textbf{(A) }19\qquad \textbf{(B) }31\qquad \textbf{(C) }271\qquad \textbf{(D) }319\qquad \textbf{(E) }511\qquad$

2020 Thailand TST, 4
Tags: induction , number theory
For any $h = 2^{r}$ ($r$ is a non-negative integer), find all $k \in \mathbb{N}$ which satisfy the following condition: There exists an odd natural number $m > 1$ and $n \in \mathbb{N}$, such that $k \mid m^{h} - 1, m \mid n^{\frac{m^{h}-1}{k}} + 1$.

2020 Dutch IMO TST, 3
Tags: number theory , relatively prime
Find all pairs $(a, b)$ of positive integers for which $a + b = \phi (a) + \phi (b) + gcd (a, b)$. Here $ \phi (n)$ is the number of numbers $k$ from $\{1, 2,. . . , n\}$ with $gcd (n, k) = 1$.

1987 China National Olympiad, 6
Tags: analytic geometry , number theory
Sum of $m$ pairwise different positive even numbers and $n$ pairwise different positive odd numbers is equal to $1987$. Find, with proof, the maximum value of $3m+4n$.

2011 IFYM, Sozopol, 2
Tags: number theory , prime number
Let $k>1$ and $n$ be natural numbers and $p=\frac{((n+1)(n+2)…(n+k))}{k!}-1$. Prove that, if $p$ is prime, then $n|k!$.

2006 Singapore Junior Math Olympiad, 1
Tags: diophantine , diophantine equation , number theory
Find all integers $x,y$ that satisfy the equation $x+y=x^2-xy+y^2$

2025 Junior Balkan Team Selection Tests - Romania, P4
Tags: number theory
Find all positive integers $n$ such that $2^n-n^2+1$ is a perfect square.