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

2017 Bundeswettbewerb Mathematik, 4
Tags: sequence , recursion , prime number , integer , number theory
The sequence $a_0,a_1,a_2,\dots$ is recursively defined by \[ a_0 = 1 \quad \text{and} \quad a_n = a_{n-1} \cdot \left(4-\frac{2}{n} \right) \quad \text{for } n \geq 1. \] Prove for each integer $n \geq 1$: (a) The number $a_n$ is a positive integer. (b) Each prime $p$ with $n < p \leq 2n$ is a divisor of $a_n$. (c) If $n$ is a prime, then $a_n-2$ is divisible by $n$.

2024 Iberoamerican, 5
Tags: combinatorics , number theory
Let $n \ge 2$ be an integer and let $a_1, a_2, \cdots a_n$ be fixed positive integers (not necessarily all distinct) in such a way that $\gcd(a_1, a_2 \cdots a_n)=1$. In a board the numbers $a_1, a_2 \cdots a_n$ are all written along with a positive integer $x$. A move consists of choosing two numbers $a>b$ from the $n+1$ numbers in the board and replace them with $a-b,2b$. Find all possible values of $x$, with respect of the values of $a_1, a_2 \cdots a_n$, for which it is possible to achieve a finite sequence of moves (possibly none) such that eventually all numbers written in the board are equal.

2012 Greece JBMO TST, 2
Tags: number theory , least common multiple , coprime , diophantine equation
Find all pairs of coprime positive integers $(p,q)$ such that $p^2+2q^2+334=[p^2,q^2]$ where $[p^2,q^2]$ is the leact common multiple of $p^2,q^2$ .

2018 Malaysia National Olympiad, A2
Tags: prime number , number theory
Let $a$ and $b$ be prime numbers such that $a+b = 10000$. Find the sum of the smallest possible value of $a$ and the largest possible value of $a$.

2017 Dutch BxMO TST, 5
Tags: number theory
Determine all pairs of prime numbers $(p; q)$ such that $p^2 + 5pq + 4q^2$ is the square of an integer.

2018 BMT Spring, 4
Tags: number theory
What is the remainder when $201820182018... $ [$2018$ times] is divided by $15$?

2012 Online Math Open Problems, 19
Tags: geometry , trapezoid , relatively prime , number theory
In trapezoid $ABCD$, $AB < CD$, $AB\perp BC$, $AB\parallel CD$, and the diagonals $AC$, $BD$ are perpendicular at point $P$. There is a point $Q$ on ray $CA$ past $A$ such that $QD\perp DC$. If \[\frac{QP} {AP}+\frac{AP} {QP} = \left( \frac{51}{14}\right)^4 - 2,\]then $\frac{BP} {AP}-\frac{AP}{BP}$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m,n$. Compute $m+n$. [i]Ray Li.[/i]

2015 Belarus Team Selection Test, 1
Tags: number theory , diophantine equation , diophantine
Solve the equation in nonnegative integers $a,b,c$: $3^a+2^b+2015=3c!$ I.Gorodnin

2007 Peru Iberoamerican Team Selection Test, P2
Tags: number theory
Find all positive integer solutions of the equation $n^5+n^4=7^{m}-1$

2016 IFYM, Sozopol, 8
Tags: number theory
Find all triples of natural numbers $(x,y,z)$ for which: $xyz=x!+y^x+y^z+z!$.

2020 Princeton University Math Competition, A2/B4
Tags: number theory
How many ordered triples of nonzero integers $(a, b, c)$ satisfy $2abc = a + b + c + 4$?

2000 All-Russian Olympiad, 6
Tags: number theory
A perfect number, greater than $6$, is divisible by $3$. Prove that it is also divisible by $9$.

2010 Thailand Mathematical Olympiad, 6
Tags: divide , number theory , prime
Show that no triples of primes $p, q, r$ satisfy $p > r, q > r$, and $pq | r^p + r^q$

2013 QEDMO 13th or 12th, 2
Tags: binomial , number theory , divisible , divide
Let $p$ be a prime number and $n, k$ and $q$ natural numbers, where $q\le \frac{n -1}{p-1}$ should be. Let $M$ be the set of all integers $m$ from $0$ to $n$, for which $m-k$ is divisible by $p$. Show that $$\sum_{m \in M} (-1) ^m {n \choose m}$$ is divisible by $p^q$.

2010 Balkan MO Shortlist, N2
Tags: calculus , integration , number theory
Solve the following equation in positive integers: $x^{3} = 2y^{2} + 1 $

2008 Cuba MO, 3
Tags: combinatorics , number theory
A boy write three times the natural number $n$ in a blackboard. He then performed an operation of the following type several times: He erased one of the numbers and wrote in its place the sum of the two others minus $1$. After several moves, one of the three numbers in the blackboard is $900$. Find all the posible values of $n$.

1996 VJIMC, Problem 2
Tags: binomial coefficient , summation , binomial , number theory , algebra
Let $\{x_n\}^\infty_{n=0}$ be the sequence such that $x_0=2$, $x_1=1$ and $x_{n+2}$ is the remainder of the number $x_{n+1}+x_n$ divided by $7$. Prove that $x_n$ is the remainder of the number $$4^n\sum_{k=0}^{\left\lfloor\frac n2\right\rfloor}2\binom n{2k}5^k$$

2022 ISI Entrance Examination, 5
Tags: number theory
For any positive integer $n$, and $i=1,2$, let $f_{i}(n)$ denote the number of divisors of $n$ of the form $3 k+i$ (including $1$ and $n$ ). Define, for any positive integer $n$, $$f(n)=f_{1}(n)-f_{2}(n)$$ Find the value of $f\left(5^{2022}\right)$ and $f\left(21^{2022}\right)$.

1985 IMO Longlists, 58
Tags: number theory
Prove that there are infinitely many pairs $(k,N)$ of positive integers such that $1 + 2 + \cdots + k = (k + 1) + (k + 2)+\cdots + N.$

2012 Paraguay Mathematical Olympiad, 1
Tags: number theory
Define a list of number with the following properties: - The first number of the list is a one-digit natural number. - Each number (since the second) is obtained by adding $9$ to the number before in the list. - The number $2012$ is in that list. Find the first number of the list.

1981 Putnam, B3
Tags: prime number , number theory
Prove that there are infinitely many positive $n$ that for all prime divisors $p$ of $n^2 + 3, \exists 0 \leq k \leq \sqrt{n}$ and $p \mid k^2+3$

2014 Junior Balkan Team Selection Tests - Moldova, 5
Tags: number theory , perfect square , floor function
Show that for any natural number $n$, the number $A = [\frac{n + 3}{4}] + [ \frac{n + 5}{4} ] + [\frac{n}{2} ] +n^2 + 3n + 3$ is a perfect square. ($[x]$ denotes the integer part of the real number x.)

2018 China Second Round Olympiad, 4
Tags: number theory
Prove that for any integer $a \ge 2$ and positive integer $n,$ there exist positive integer $k$ such that $a^k+1,a^k+2,\ldots,a^k+n$ are all composite numbers.

2018 Saudi Arabia GMO TST, 2
Tags: number theory , digit , divisible
Two positive integers $m$ and $n$ are called [i]similar [/i] if one of them can be obtained from the other one by swapping two digits (note that a $0$-digit cannot be swapped with the leading digit). Find the greatest integer $N$ such that N is divisible by $13$ and any number similar to $N$ is not divisible by $13$.

2011 USAJMO, 1
Tags: modular arithmetic , quadratic , number theory
Find, with proof, all positive integers $n$ for which $2^n + 12^n + 2011^n$ is a perfect square.