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

1995 IMO Shortlist, 4
Tags: algebra , number theory , equation , greatest common divisor
Find all $ x,y$ and $ z$ in positive integer: $ z \plus{} y^{2} \plus{} x^{3} \equal{} xyz$ and $ x \equal{} \gcd(y,z)$.

2017 May Olympiad, 1
Tags: number theory , digit
We shall call a positive integer [i]ascending [/i] if its digits read from left to right they are in strictly increasing order. For example, $458$ is ascending and $2339$ is not. Find the largest ascending number that is a multiple of $56$.

2016 Tournament Of Towns, 6
Tags: polynomial , number theory , algebra
$N $ different numbers are written on blackboard and one of these numbers is equal to $0$.One may take any polynomial such that each of its coefficients is equal to one of written numbers ( there may be some equal coefficients ) and write all its roots on blackboard.After some of these operations all integers between $-2016$ and $2016$ were written on blackboard(and some other numbers maybe). Find the smallest possible value of $N $.

2019 Hong Kong TST, 1
Tags: number theory , arithmetic function , number of divisors , divisor
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

1985 Tournament Of Towns, (091) T2
Tags: difference , prime , composite , number theory , combinatorics , maximum
From the set of numbers $1 , 2, 3, . . . , 1985$ choose the largest subset such that the difference between any two numbers in the subset is not a prime number (the prime numbers are $2, 3 , 5 , 7,... , 1$ is not a prime number) .

JOM 2015 Shortlist, A4
Tags: algebra , number theory
Suppose $ 2015= a_1 <a_2 < a_3<\cdots <a_k $ be a finite sequence of positive integers, and for all $ m, n \in \mathbb{N} $ and $1\le m,n \le k $, $$ a_m+a_n\ge a_{m+n}+|m-n| $$ Determine the largest possible value $ k $ can obtain.

2016 Greece JBMO TST, 3
Tags: number theory , greatest common divisor , divisor
Positive integer $n$ is such that number $n^2-9$ has exactly $6$ positive divisors. Prove that GCD $(n-3, n+3)=1$

1956 Putnam, A2
Tags: number theory , digit
Prove that every positive integer has a multiple whose decimal representation involves all ten digits.

2014 BMO TST, 5
Tags: modular arithmetic , number theory
Find all non-negative integers $k,n$ which satisfy $2^{2k+1} + 9\cdot 2^k+5=n^2$.

2025 Harvard-MIT Mathematics Tournament, 3
Tags: algebra , number theory
Given that $x, y,$ and $z$ are positive real numbers such that $$x^{\log_2(yz)}=2^8\cdot3^4, \quad y^{\log_2(zx)}=2^9\cdot3^6, \quad \text{and}\quad z^{\log_2(xy)}=2^5 \cdot 3^{10},$$ compute the smallest possible value of $xyz.$

2015 BMT Spring, 9
Tags: number theory , diophantine equation
There exists a unique pair of positive integers $k,n$ such that $k$ is divisible by $6$, and $\sum_{i=1}^ki^2=n^2$. Find $(k,n)$.

1999 Abels Math Contest (Norwegian MO), 2a
Tags: diophantine equation , diophantine , number theory
Find all integers $m$ and $n$ such that $2m^2 +n^2 = 2mn+3n$

2023/2024 Tournament of Towns, 3
Tags: number theory
3. Consider all 100-digit positive integers such that each decimal digit of these equals $2,3,4,5,6$, or 7 . How many of these integers are divisible by $2^{100}$ ? Pavel Kozhevnikov

2023 Princeton University Math Competition, A5 / B7
Tags: number theory
You play a game where you and an adversarial opponent take turns writing down positive integers on a chalkboard; the only condition is that, if $m$ and $n$ are written consecutively on the board, $\gcd(m,n)$ must be squarefree. If your objective is to make sure as many integers as possible that are strictly less than $404$ end up on the board (and your opponent is trying to minimize this quantity), how many more such integers can you guarantee will eventually be written on the board if you get to move first as opposed to when your opponent gets to move first?

2020 LIMIT Category 2, 12
Tags: gcd , number theory , limit
Let $A$ be the set $\{k^{19}-k: 1<k<20, k\in N\}$. Let $G$ be the GCD of all elements of $A$. Then the value of $G$ is?

2011 Ukraine Team Selection Test, 7
Tags: modular arithmetic , number theory , equation , lte lemma
Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

2017 Morocco TST-, 6
Tags: number theory , polynomial , functional equation , sum of digits
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]

2013 Flanders Math Olympiad, 1
Tags: number theory , digit , divisible , sum
A six-digit number is [i]balanced [/i] when all digits are different from zero and the sum of the first three digits is equal to the sum of the last three digits. Prove that the sum of all six-digit balanced numbers is divisible by $13$.

2023 Romanian Master of Mathematics Shortlist, N2
Tags: polynomial , number theory , decimal representation , algebra
For every non-negative integer $k$ let $S(k)$ denote the sum of decimal digits of $k$. Let $P(x)$ and $Q(x)$ be polynomials with non-negative integer coecients such that $S(P(n)) = S(Q(n))$ for all non-negative integers $n$. Prove that there exists an integer $t$ such that $P(x) - 10^tQ(x)$ is a constant polynomial.

2014 Contests, 2
Tags: floor function , number theory
Let $n$ be a natural number. Prove that, \[ \left\lfloor \frac{n}{1} \right\rfloor+ \left\lfloor \frac{n}{2} \right\rfloor + \cdots + \left\lfloor \frac{n}{n} \right\rfloor + \left\lfloor \sqrt{n} \right\rfloor \] is even.

2024 Indonesia TST, N
Tags: number theory
A natural number $n$ is called "good" if there exists natural numbers $a$ and $b$ such that $a+b=n$ and $ab \mid n^2+n+1$. Show that there are infinitely many "good" numbers

2007 Croatia Team Selection Test, 8
Tags: number theory
Positive integers $x>1$ and $y$ satisfy an equation $2x^2-1=y^{15}$. Prove that 5 divides $x$.

2020 South East Mathematical Olympiad, 4
Tags: number theory , maximum value , combinatorics
Let $a_1,a_2,\dots, a_{17}$ be a permutation of $1,2,\dots, 17$ such that $(a_1-a_2)(a_2-a_3)\dots(a_{17}-a_1)=n^{17}$ .Find the maximum possible value of $n$ .

KoMaL A Problems 2022/2023, A. 830
Tags: number theory
For $H\subset \mathbb Z$ and $n\in\mathbb Z$ let $h_n$ denote the number of finite subsets of $H$ in which the sum of the elements is $n$. Determine whether there exists $H\subset \mathbb Z$ for which $0\notin H$ and $h_n$ is a finite even number for every $n\in\mathbb{Z}$. (The sum of the elements of the empty set is $0$.) [i]Proposed by Csongor Beke, Cambridge[/i]

2015 Bundeswettbewerb Mathematik Germany, 2
Tags: number theory , decimal representation , fraction
In the decimal expansion of a fraction $\frac{m}{n}$ with positive integers $m$ and $n$ you can find a string of numbers $7143$ after the comma. Show $n>1250$. [i]Example:[/i] I mean something like $0.7143$.