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

1977 Yugoslav Team Selection Test, Problem 2
Tags: diophantine equation , number theory
Determine all $6$-tuples $(p,q,r,x,y,z)$ where $p,q,r$ are prime, and $x,y,z$ natural numbers such that $p^{2x}=q^yr^z+1$.

2009 Thailand Mathematical Olympiad, 4
Tags: diophantine , diophantine equation , number theory
Let $k$ be a positive integer. Show that there are infinitely many positive integer solutions $(m, n)$ to $(m - n)^2 = kmn + m + n$.

2019 Denmark MO - Mohr Contest, 4
Tags: game , number theory , combinatorics
Georg writes a positive integer $a$ on a blackboard. As long as there is a number on the blackboard, he does the following each day: $\bullet$ If the last digit in the number on the blackboard is less than or equal to $5$, he erases that last digit. (If there is only this digit, the blackboard thus becomes empty.) $\bullet$ Otherwise he erases the entire number and writes $9$ times the number. Can Georg choose $a$ in such a way that the blackboard never becomes empty?

2016 PUMaC Number Theory A, 8
Tags: number theory
Let $n = 2^8 \cdot 3^9 \cdot 5^{10} \cdot 7^{11}$. For $k$ a positive integer, let $f(k)$ be the number of integers $0 \le x < n$ such that $x^2 \equiv k^2$ (mod $n$). Compute the number of positive integers k such that $k | f(k)$.

2019 Estonia Team Selection Test, 10
Tags: combinatorics , number theory , algebra , projective geometry , geometry
Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

2019 Austrian Junior Regional Competition, 4
Tags: number theory , divisible , divide , prime
Let $p, q, r$ and $s$ be four prime numbers such that $$5 <p <q <r <s <p + 10.$$ Prove that the sum of the four prime numbers is divisible by $60$. (Walther Janous)

2014 France Team Selection Test, 4
Tags: number theory , algebra , functional equation
Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that \[ m^2 + f(n) \mid mf(m) +n \] for all positive integers $m$ and $n$.

1997 Dutch Mathematical Olympiad, 1
Tags: number theory , sum of digits , product
For each positive integer $n$ we define $f (n)$ as the product of the sum of the digits of $n$ with $n$ itself. Examples: $f (19) = (1 + 9) \times 19 = 190$, $f (97) = (9 + 7) \times 97 = 1552$. Show that there is no number $n$ with $f (n) = 19091997$.

2000 Junior Balkan Team Selection Tests - Romania, 2
Tags: number theory , modular arithmetic , algebra
Let be a natural power of two. Find the number of numbers equivalent with $ 1 $ modulo $ 3 $ that divide it. [i]Dan Brânzei[/i]

2014 India Regional Mathematical Olympiad, 6
Tags: number theory , grid , combinatorics
Suppose $n$ is odd and each square of an $n \times n$ grid is arbitrarily filled with either by $1$ or by $-1$. Let $r_j$ and $c_k$ denote the product of all numbers in $j$-th row and $k$-th column respectively, $1 \le j, k \le n$. Prove that $$\sum_{j=1}^{n} r_j+ \sum_{k=1}^{n} c_k\ne 0$$

2021 Vietnam TST, 1
Tags: number theory , inequalities
Define the sequence $(a_n)$ as $a_1 = 1$, $a_{2n} = a_n$ and $a_{2n+1} = a_n + 1$ for all $n\geq 1$. a) Find all positive integers $n$ such that $a_{kn} = a_n$ for all integers $1 \leq k \leq n$. b) Prove that there exist infinitely many positive integers $m$ such that $a_{km} \geq a_m$ for all positive integers $k$.

2012 Dutch IMO TST, 3
Tags: number theory , integer
Determine all positive integers that cannot be written as $\frac{a}{b} + \frac{a+1}{b+1}$ where $a$ and $b$ are positive integers.

2017-IMOC, N8
Tags: number theory
Find all pairs $(p,n)$ of integers so that $p$ is a prime and there exists $x,y\not\equiv0\pmod p$ with $$x^2+y^2\equiv n\pmod p.$$

2001 Italy TST, 3
Tags: number theory
Find all pairs $ (p, q)$ of prime numbers such that $ p$ divides $ 5^q \plus{} 1$ and $ q$ divides $ 5^p \plus{} 1$.

2001 JBMO ShortLists, 2
Tags: modular arithmetic , number theory
Let $P_n \ (n=3,4,5,6,7)$ be the set of positive integers $n^k+n^l+n^m$, where $k,l,m$ are positive integers. Find $n$ such that: i) In the set $P_n$ there are infinitely many squares. ii) In the set $P_n$ there are no squares.

2015 Regional Competition For Advanced Students, 1
Tags: number theory , gcd
Determine all triples $(a,b,c)$ of positive integers satisfying the conditions $$\gcd(a,20) = b$$ $$\gcd(b,15) = c$$ $$\gcd(a,c) = 5$$ (Richard Henner)

2020 Princeton University Math Competition, B2
Tags: number theory
Last year, the U.S. House of Representatives passed a bill which would make Washington, D.C. into the $51$st state. Naturally, the mathematicians are upset that Congress won’t prioritize mathematical interest of flag design in choosing how many U.S. states there should be. Suppose the U.S. flag must contain, as it does now, stars arranged in rows alternating between $n$ and $n - 1$ stars, starting and ending with rows of n stars, where $n \ge 2$ is some integer and the flag has more than one row. What is the minimum number of states that the U.S. would need to contain so that there are at least three different ways, excluding rotations, to arrange the stars on the flag?

2013 Macedonia National Olympiad, 1
Tags: modular arithmetic , number theory
Let $ p,q,r $ be prime numbers. Solve the equation $ p^{2q}+q^{2p}=r $

2006 Poland - Second Round, 1
Tags: pigeonhole principle , number theory
Let $c$ be fixed natural number. Sequence $(a_n)$ is defined by: $a_1=1$, $a_{n+1}=d(a_n)+c$ for $n=1,2,...$. where $d(m)$ is number of divisors of $m$. Prove that there exist $k$ natural such that sequence $a_k,a_{k+1},...$ is periodic.

2012 Kazakhstan National Olympiad, 1
Tags: algebra , difference of squares , special factorizations , number theory
Solve the equation $p+\sqrt{q^{2}+r}=\sqrt{s^{2}+t}$ in prime numbers.

2025 Macedonian Mathematical Olympiad, Problem 4
Tags: number theory
Let $P(x)=a x^{75}+b$ be a polynomial where \(a\) and \(b\) are coprime integers in the set \(\{1,2,\dots,151\}\), and suppose it satisfies the following condition: there exists at most one prime \(p\) such that for every positive integer \(k\), \(p\mid P(k)\). Prove that for every prime \(q \neq p\) there exists a positive integer \(k\) for which $q^2 \mid P(k).$

2009 Estonia Team Selection Test, 2
Tags: number theory , mean , coprime
Call a finite set of positive integers [i]independent [/i] if its elements are pairwise coprime, and [i]nice [/i] if the arithmetic mean of the elements of every non-empty subset of it is an integer. a) Prove that for any positive integer $n$ there is an $n$-element set of positive integers which is both independent and nice. b) Is there an infinite set of positive integers whose every independent subset is nice and which has an $n$-element independent subset for every positive integer $n$?

1989 China Team Selection Test, 2
Tags: number theory
Let $v_0 = 0, v_1 = 1$ and $v_{n+1} = 8 \cdot v_n - v_{n-1},$ $n = 1,2, ...$. Prove that in the sequence $\{v_n\}$ there aren't terms of the form $3^{\alpha} \cdot 5^{\beta}$ with $\alpha, \beta \in \mathbb{N}.$

2014 Contests, Problem 2
Tags: number theory
Clau writes all four-digit natural numbers where $3$ and $7$ are always together. How many digits does she write in total?

2015 Iran Team Selection Test, 3
Tags: number theory
Let $ b_1<b_2<b_3<\dots $ be the sequence of all natural numbers which are sum of squares of two natural numbers. Prove that there exists infinite natural numbers like $m$ which $b_{m+1}-b_m=2015$ .