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

2009 Purple Comet Problems, 13
Tags: number theory , prime number
How many subsets of the set $\{1, 2, 3, \ldots, 12\}$ contain exactly one or two prime numbers?

2024 AMC 10, 11
Tags: number theory
How many ordered pairs of integers $(m, n)$ satisfy $\sqrt{n^2 - 49} = m$? $ \textbf{(A) }1 \qquad \textbf{(B) }2 \qquad \textbf{(C) }3 \qquad \textbf{(D) }4 \qquad \textbf{(E) } \text{Infinitely many} \qquad $

2015 Danube Mathematical Competition, 3
Tags: number theory
Solve in N $a^2 = 2^b3^c + 1$.

2024 Chile Junior Math Olympiad, 3
Tags: number theory
Determine all triples \( (a, b, c) \) of positive integers such that: \[ a + b + c = abc. \]

2007 Estonia National Olympiad, 4
Tags: number theory , greatest common divisor , divide
Let $a, b,c$ be positive integers such that $gcd(a, b, c) = 1$ and each product of two is divided by the third. a) Prove that each of these numbers is equal to the least two remaining numbers the quotient of the coefficient and the highest coefficient. b) Give an example of one of these larger numbers $a, b$ and $c$

2018 Canada National Olympiad, 4
Tags: number theory , polynomial , algebra
Find all polynomials $p(x)$ with real coefficients that have the following property: there exists a polynomial $q(x)$ with real coefficients such that $$p(1) + p(2) + p(3) +\dots + p(n) = p(n)q(n)$$ for all positive integers $n$.

2011 AIME Problems, 10
Tags: number theory , analytic geometry , geometry , circumcircle , trigonometry
A circle with center $O$ has radius 25. Chord $\overline{AB}$ of length 30 and chord $\overline{CD}$ of length 14 intersect at point $P$. The distance between the midpoints of the two chords is 12. The quantity $OP^2$ can be represented as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the remainder where $m+n$ is divided by 1000.

2025 Poland - First Round, 3
Tags: number theory , prime number
Let $n$ be a product of 2024 different prime numbers. Find the number of positive integers $k$, such that $$n+gcd(n, k)=k.$$

2022 Saudi Arabia IMO TST, 1
Tags: floor function , algebra , summation , number theory
Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

2023 Indonesia TST, N
Tags: number theory , primitive root , fermat's little theorem
Let $p,q,r$ be primes such that for all positive integer $n$, $$n^{pqr}\equiv n (\mod{pqr})$$ Prove that this happens if and only if $p,q,r$ are pairwise distinct and $LCM(p-1,q-1,r-1)|pqr-1$

1984 Putnam, B1
Tags: number theory , sequence , recurrence relation
Let $n$ be a positive integer, and define $f(n)=1!+2!+\ldots+n!$. Find polynomials $P$ and $Q$ such that $$f(n+2)=P(n)f(n+1)+Q(n)f(n)$$for all $n\ge1$.

2014 Federal Competition For Advanced Students, P2, 1
Tags: number theory , number of divisors , divisor
For each positive natural number $n$ let $d (n)$ be the number of its divisors including $1$ and $n$. For which positive natural numbers $n$, for every divisor $t$ of $n$, that $d (t)$ is a divisor of $d (n)$?

PEN A Problems, 105
Tags: number theory , least common multiple , divisibility theory
Find the smallest positive integer $n$ such that [list][*] $n$ has exactly $144$ distinct positive divisors, [*] there are ten consecutive integers among the positive divisors of $n$. [/list]

2004 India Regional Mathematical Olympiad, 6
Tags: modular arithmetic , relatively prime , number theory
Let $p_1, p_2, \ldots$ be a sequence of primes such that $p_1 =2$ and for $n\geq 1, p_{n+1}$ is the largest prime factor of $p_1 p_2 \ldots p_n +1$ . Prove that $p_n \not= 5$ for any $n$.

1997 Estonia National Olympiad, 1
Tags: diophantine , diophantine equation , number theory
Find: a) Any quadruple of positive integers $(a, k, l, m)$ such that $a^k = a^l + a^m,$ b) Any quintuple of positive integers $(a, k, l, m, n)$ for which $a^k = a^l + a^m+a^n$

2013 Baltic Way, 20
Tags: number theory , algebra , limit , function , polynomial
Find all polynomials $f$ with non-negative integer coefficients such that for all primes $p$ and positive integers $n$ there exist a prime $q$ and a positive integer $m$ such that $f(p^n)=q^m$.

2018 Balkan MO, 4
Tags: number theory
Find all primes $p$ and $q$ such that $3p^{q-1}+1$ divides $11^p+17^p$ Proposed by Stanislav Dimitrov,Bulgaria

2025 6th Memorial "Aleksandar Blazhevski-Cane", P5
Tags: number theory , divisibility , 2-adic valuation
Let $s < t$ be positive integers. Define a sequence by: $a_1 = s, a_2 = t$; $a_3$ is the smallest integer that's greater than $a_2$ and divisible by $a_1$; in general, $a_{n + 1}$ is the smallest integer greater than $a_n$ that's divisible by $a_1, a_2, ..., a_{n - 2}, a_{n - 1}$. [b]a)[/b] What is the maximum number of odd integers that can appear in such a sequence? (Justify your answer) [b]b)[/b] Prove that $a_{2025}$ is divisible by $2^{808}$, regardless of the choice of $s$ and $t$. Proposed by [i]Ilija Jovcevski[/i]

2000 Brazil Team Selection Test, Problem 2
Tags: number theory
For a positive integer $n$, let $A_n$ be the set of all positive numbers greater than $1$ and less than $n$ which are coprime to $n$. Find all $n$ such that all the elements of $A_n$ are prime numbers.

2001 Saint Petersburg Mathematical Olympiad, 10.6
Tags: least common multiple , greatest common divisor , algebra , gcd , number theory , inequalities
For any positive integers $n>m$ prove the following inequality: $$[m,n]+[m+1,n+1]\geq 2m\sqrt{n}$$ As usual, [x,y] denotes the least common multiply of $x,y$ [I]Proposed by A. Golovanov[/i]

2024 Cono Sur Olympiad, 1
Tags: number theory
Prove that there are infinitely many quadruplets of positive integers $(a,b,c,d)$, such that\\ $ab+1$, $bc+16$, $cd+4$, $ad+9$\\ are perfect squares

1980 Bundeswettbewerb Mathematik, 1
Tags: number theory , perfect cube , rational
Let $a$ and $b$ be integers. Prove that if $\sqrt[3]{a}+\sqrt[3]{b}$ is a rational number, then both $a$ and $b$ are perfect cubes.

2012 IFYM, Sozopol, 2
Tags: number theory , natural number
Find all natural numbers, which cannot be expressed in the form $\frac{a}{b}+\frac{a+1}{b+1}$ where $a,b\in \mathbb{N}$.

2022 Junior Balkan Team Selection Tests - Romania, P1
Tags: number theory , squarefree
Determine all squarefree positive integers $n\geq 2$ such that \[\frac{1}{d_1}+\frac{1}{d_2}+\cdots+\frac{1}{d_k}\]is a positive integer, where $d_1,d_2,\ldots,d_k$ are all the positive divisors of $n$.

1987 IMO Longlists, 54
Tags: number theory , modular arithmetic
Let $n$ be a natural number. Solve in integers the equation \[x^n + y^n = (x - y)^{n+1}.\]