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

2008 Romania Team Selection Test, 2
Tags: modular arithmetic , number theory
Let $ m, n \geq 1$ be two coprime integers and let also $ s$ an arbitrary integer. Determine the number of subsets $ A$ of $ \{1, 2, ..., m \plus{} n \minus{} 1\}$ such that $ |A| \equal{} m$ and $ \sum_{x \in A} x \equiv s \pmod{n}$.

2024 ELMO Shortlist, N4
Tags: number theory
Find all pairs $(a,b)$ of positive integers such that $a^2\mid b^3+1$ and $b^2\mid a^3+1$. [i]Linus Tang[/i]

2007 Purple Comet Problems, 5
Tags: number theory , relatively prime
The repeating decimal $0.328181818181...$ can equivalently be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2011 Princeton University Math Competition, A5 / B8
Tags: number theory
Let $d(n)$ denote the number of divisors of $n$ (including itself). You are given that \[\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}.\] Find $p(6)$, where $p(x)$ is the unique polynomial with rational coefficients satisfying \[p(\pi) = \sum_{n=1}^{\infty} \frac{d(n)}{n^2}.\]

2003 Bulgaria Team Selection Test, 6
Tags: number theory
In natural numbers $m,n$ Solve : $n(n+1)(n+2)(n+3)=m(m+1)^2(m+2)^3(m+3)^4$

2012 Kazakhstan National Olympiad, 3
Tags: number theory
Consider the equation $ax^{2}+by^{2}=1$, where $a,b$ are fixed rational numbers. Prove that either such an equation has no solutions in rational numbers, or it has infinitely many solutions.

2008 Romania Team Selection Test, 2
Tags: prime number , number theory
Are there any sequences of positive integers $ 1 \leq a_{1} < a_{2} < a_{3} < \ldots$ such that for each integer $ n$, the set $ \left\{a_{k} \plus{} n\ |\ k \equal{} 1, 2, 3, \ldots\right\}$ contains finitely many prime numbers?

PEN A Problems, 114
Tags: number theory , greatest common divisor , modular arithmetic , divisibility theory
What is the greatest common divisor of the set of numbers \[\{{16}^{n}+10n-1 \; \vert \; n=1,2,\cdots \}?\]

2001 All-Russian Olympiad Regional Round, 11.1
Tags: number theory , diophantine , diophantine equation
Find all prime numbers $p$ and $q$ such that $p + q = (p -q)^3.$

1998 VJIMC, Problem 2
Tags: arithmetic sequence , algebra , number theory , palindrome
Decide whether there is a member in the arithmetic sequence $\{a_n\}_{n=1}^\infty$ whose first member is $a_1=1998$ and the common difference $d=131$ which is a palindrome (palindrome is a number such that its decimal expansion is symmetric, e.g., $7$, $33$, $433334$, $2135312$ and so on).

2016 CMIMC, 1
Tags: number theory
David, when submitting a problem for CMIMC, wrote his answer as $100\tfrac xy$, where $x$ and $y$ are two positive integers with $x<y$. Andrew interpreted the expression as a product of two rational numbers, while Patrick interpreted the answer as a mixed fraction. In this case, Patrick's number was exactly double Andrew's! What is the smallest possible value of $x+y$?

2000 Singapore Team Selection Test, 2
Tags: number theory , prime , perfect square
Find all prime numbers $p$ such that $5^p + 12^p$ is a perfect square

2000 Mexico National Olympiad, 3
Tags: number theory , combinatorics , set
Given a set $A$ of positive integers, the set $A'$ is composed from the elements of $A$ and all positive integers that can be obtained in the following way: Write down some elements of $A$ one after another without repeating, write a sign $+ $ or $-$ before each of them, and evaluate the obtained expression. The result is included in $A'$. For example, if $A = \{2,8,13,20\}$, numbers $8$ and $14 = 20-2+8$ are elements of $A'$. Set $A''$ is constructed from $A'$ in the same manner. Find the smallest possible number of elements of $A$, if $A''$ contains all the integers from $1$ to $40$.

2019 Nigeria Senior MO Round 2, 4
Tags: algebra , number theory , greatest common divisor
Let $h(t)$ and $f(t)$ be polynomials such that $h(t)=t^2$ and $f_n(t)=h(h(h(h(h...h(t))))))-1$ where $h(t)$ occurs $n$ times. Prove that $f_n(t)$ is a factor of $f_N(t)$ whenever $n$ is a factor of $N$

2013 Nordic, 3
Tags: sequence , rational , number theory
Define a sequence ${(n_k)_{k\ge 0}}$ by ${n_{0 }= n_{1} = 1}$, and ${n_{2k} = n_k + n_{k-1} }$ and ${n_{2k+1} = n_k}$ for ${k \ge 1}$. Let further ${q_k = n_k }$ / ${ n_{k-1} }$ for each ${k \ge 1}$. Show that every positive rational number is present exactly once in the sequence ${(q_k)_{k\ge 1}}$

2009 Baltic Way, 7
Tags: quadratic , modular arithmetic , number theory
Suppose that for a prime number $p$ and integers $a,b,c$ the following holds: \[6\mid p+1,\quad p\mid a+b+c,\quad p\mid a^4+b^4+c^4.\] Prove that $p\mid a,b,c$.

1983 Federal Competition For Advanced Students, P2, 4
Tags: function , number theory
The sequence $ (x_n)_{n \in \mathbb{N}}$ is defined by $ x_1\equal{}2, x_2\equal{}3,$ and $ x_{2m\plus{}1}\equal{}x_{2m}\plus{}x_{2m\minus{}1}$ for $ m \ge 1;$ $ x_{2m}\equal{}x_{2m\minus{}1}\plus{}2x_{2m\minus{}2}$ for $ m \ge 2.$ Determine $ x_n$ as a function of $ n$.

2007 Iran MO (2nd Round), 1
Tags: number theory
Prove that for every positive integer $n$, there exist $n$ positive integers such that the sum of them is a perfect square and the product of them is a perfect cube.

1999 Tournament Of Towns, 3
Tags: number theory , divide , divisible , sum of squares
Find all pairs $(x, y)$ of integers satisfying the following condition: each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$ . (S Zlobin)

Kvant 2021, M2650
Tags: number theory
For which $n{}$ is it possible that a product of $n{}$ consecutive positive integers is equal to a sum of $n{}$ consecutive (not necessarily the same) positive integers? [i]Boris Frenkin[/i]

2016 Macedonia National Olympiad, Problem 1
Tags: equation , number theory
Solve the equation in the set of natural numbers $1+x^z + y^z = LCM(x^z,y^z)$

2011 Regional Competition For Advanced Students, 1
Tags: number theory
Let $p_1, p_2, \ldots, p_{42}$ be $42$ pairwise distinct prime numbers. Show that the sum \[\sum_{j=1}^{42}\frac{1}{p_j^2+1}\] is not a unit fraction $\frac{1}{n^2}$ of some integer square number.

2013 Polish MO Finals, 2
Tags: modular arithmetic , number theory
There are given integers $a$ and $b$ such that $a$ is different from $0$ and the number $3+ a +b^2$ is divisible by $6a$. Prove that $a$ is negative.

1997 Korea National Olympiad, 4
Tags: number theory , prime number
For any prime number $p>2,$ and an integer $a$ and $b,$ if $1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{(p-1)^3}=\frac{a}{b},$ prove that $a$ is divisible by $p.$

2018 Bosnia And Herzegovina - Regional Olympiad, 2
Tags: number theory , equal sets , set
Determine all triplets $(a,b,c)$ of real numbers such that sets $\{a^2-4c, b^2-2a, c^2-2b \}$ and $\{a-c,b-4c,a+b\}$ are equal and $2a+2b+6=5c$. In every set all elements are pairwise distinct