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

2002 Tournament Of Towns, 1
Tags: modular arithmetic , algebra , polynomial , quadratic , number theory
Show that if the last digit of the number $x^2+xy+y^2$ is $0$ (where $x,y\in\mathbb{N}$ ) then last two digits are zero.

2015 China Second Round Olympiad, 3
Tags: number theory
Prove that there exist infinitely many positive integer triples $(a,b,c)(a,b,c>2015)$ such that $$ a|bc-1, b|ac+1, c|ab+1.$$

2023 Puerto Rico Team Selection Test, 5
Tags: number theory , algebra
Six fruit baskets contain peaches, apples and pears. The number of peaches in each basket is equal to the total number of apples in the other baskets. The number of apples in each basket is equal to the total number of pears in the other baskets. (a) Find a way to place $31$ fruits in the baskets, satisfying the conditions of the statement. (b) Explain why the total number of fruits must always be multiple of $31$.

VMEO III 2006 Shortlist, N6
Tags: number theory , divide
Find all sets of natural numbers $(a, b, c)$ such that $$a+1|b^2+c^2\,\, , b+1|c^2+a^2\,\,, c+1|a^2+b^2.$$

2016 India IMO Training Camp, 1
Tags: number theory
We say a natural number $n$ is perfect if the sum of all the positive divisors of $n$ is equal to $2n$. For example, $6$ is perfect since its positive divisors $1,2,3,6$ add up to $12=2\times 6$. Show that an odd perfect number has at least $3$ distinct prime divisors. [i]Note: It is still not known whether odd perfect numbers exist. So assume such a number is there and prove the result.[/i]

2014 Korea Junior Math Olympiad, 4
Tags: number theory , greatest common divisor , positive integer
Positive integers $p, q, r$ satisfy $gcd(a,b,c) = 1$. Prove that there exists an integer $a$ such that $gcd(p,q+ar) = 1$.

2014 All-Russian Olympiad, 1
Tags: number theory
Let $a$ be [i]good[/i] if the number of prime divisors of $a$ is equal to $2$. Do there exist $18$ consecutive good natural numbers?

2023 Turkey Team Selection Test, 3
Tags: number theory , equation
For all $n>1$, let $f(n)$ be the biggest divisor of $n$ except itself. Does there exists a positive integer $k$ such that the equality $n-f(n)=k$ has exactly $2023$ solutions?

2019 Saudi Arabia Pre-TST + Training Tests, 2.1
Tags: number theory
Let pairwise different positive integers $a,b, c$ with gcd$(a,b,c) = 1$ are such that $a | (b - c)^2, b | (c- a)^2, c | (a - b)^2$. Prove, that there is no non-degenerate triangle with side lengths $a, b$ and $c$.

2005 Switzerland - Final Round, 4
Tags: number theory , greatest common divisor , gcd
Determine all sets $M$ of natural numbers such that for every two (not necessarily different) elements $a, b$ from $M$ , $$\frac{a + b}{gcd(a, b)}$$ lies in $M$.

2005 Turkey MO (2nd round), 4
Tags: number theory
Find all triples of nonnegative integers $(m,n,k)$ satisfying $5^m+7^n=k^3$.

2022 IMO, 3
Tags: number theory , prime number
Let $k$ be a positive integer and let $S$ be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of $S$ around the circle such that the product of any two neighbors is of the form $x^2+x+k$ for some positive integer $x$.

2015 Tuymaada Olympiad, 2
Tags: number theory
We call number as funny if it divisible by sum its digits $+1$.(for example $ 1+2+1|12$ ,so $12$ is funny) What is maximum number of consecutive funny numbers ? [i] O. Podlipski [/i]

2025 Philippine MO, P2
Tags: pythagorean triple , number theory
A positive integer is written on a blackboard. Carmela can perform the following operation as many times as she wants: replace the current integer $x$ with another positive integer $y$, as long as $|x^2 - y^2|$ is a perfect square. For example, if the number on the blackboard is $17$, Carmela can replace it with $15$, because $|17^2 - 15^2| = 8^2$, then replace it with $9$, because $|15^2 - 9^2| = 12^2$. If the number on the blackboard is initially $3$, determine all integers that Carmela can write on the blackboard after finitely many operations.

2023 Durer Math Competition Finals, 1
Tags: number theory , algebra , prime number
Nüx has three moira daughters, whose ages are three distinct prime numbers, and the sum of their squares is also a prime number. What is the age of the youngest moira?

2016 Iran Team Selection Test, 3
Tags: number theory , prime number , modular arithmetic , quadratic residue
Let $p \neq 13$ be a prime number of the form $8k+5$ such that $39$ is a quadratic non-residue modulo $p$. Prove that the equation $$x_1^4+x_2^4+x_3^4+x_4^4 \equiv 0 \pmod p$$ has a solution in integers such that $p\nmid x_1x_2x_3x_4$.

2012 Brazil National Olympiad, 4
Tags: floor function , number theory
There exists some integers $n,a_1,a_2,\ldots,a_{2012}$ such that \[ n^2=\sum_{1 \leq i \leq 2012}{{a_i}^{p_i}} \] where $p_i$ is the i-th prime ($p_1=2,p_2=3,p_3=5,p_4=7,\ldots$) and $a_i>1$ for all $i$?

2006 Romania National Olympiad, 4
Tags: calculus , integration , number theory , greatest common divisor , least common multiple , relatively prime
Let $A$ be a set of positive integers with at least 2 elements. It is given that for any numbers $a>b$, $a,b \in A$ we have $\frac{ [a,b] }{ a- b } \in A$, where by $[a,b]$ we have denoted the least common multiple of $a$ and $b$. Prove that the set $A$ has [i]exactly[/i] two elements. [i]Marius Gherghu, Slatina[/i]

2004 Junior Balkan Team Selection Tests - Moldova, 1
Tags: number theory , inequalities
Determine all triplets of integers $(x, y, z)$ that validate the inequality $x^2 + y^2 + z^2 <xy + 3y + 2z$.

1992 Swedish Mathematical Competition, 1
Tags: number theory , integer
Is $\frac{19^{92} - 91^{29}}{90}$ an integer?

2011 Indonesia TST, 4
Tags: number theory , prime , sum of divisors , divide
Given $N = 2^ap_1p_2...p_m$, $m \ge 1$, $a \in N$ with $p_1, p_2,..., p_m$ are different primes. It is known that $\sigma (N) = 3N $ where $\sigma (N)$ is the sum of all positive integers which are factors of $N$. Show that there exists a prime number $p$ such that $2^p- 1$ is also a prime, and $2^p - 1|N$.

2003 China Team Selection Test, 3
Tags: analytic geometry , inequalities , graph theory , number theory
Given $S$ be the finite lattice (with integer coordinate) set in the $xy$-plane. $A$ is the subset of $S$ with most elements such that the line connecting any two points in $A$ is not parallel to $x$-axis or $y$-axis. $B$ is the subset of integer with least elements such that for any $(x,y)\in S$, $x \in B$ or $y \in B$ holds. Prove that $|A| \geq |B|$.

2021 Turkey Team Selection Test, 9
Tags: number theory , gcd
For which positive integer couples $(k,n)$, the equality $\Bigg|\Bigg\{{a \in \mathbb{Z}^+: 1\leq a\leq(nk)!, gcd \left(\binom{a}{k},n\right)=1}\Bigg\}\Bigg|=\frac{(nk)!}{6}$ holds?

2017 NMTC Junior, 1
Tags: number theory , algebra
(a) Find all prime numbers $p$ such that $4p^2+1$ and $6p^2+1$ are also primes. (b)Find real numbers $x,y,z,u$ such that \[xyz+xy+yz+zx+x+y+z=7\]\[yzu+yz+zu+uy+y+z+u=10\]\[zux+zu+ux+xz+z+u+x=10\]\[uxy+ux+xy+yu+u+x+y=10\]

2024 India Regional Mathematical Olympiad, 1
Tags: number theory
Find all positive integers $x,y$ such that $202x + 4x^2 = y^2$.