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

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 $$ a|bc-1, b|ac+1, c|ab+1.$$

Russian TST 2020, P1
Tags: number theory , combinatorics
There are coins worth $1, 2, \ldots , b$ rubles, blue bills with worth $a{}$ rubles and red bills worth $a + b$ rubles. Ilya wants to exchange a certain amount into coins and blue bills, and use no more than $a-1$ coins. Pasha wants to exchange the same amount in coins and red bills, but use no more than $a{}$ coins. Prove that they have equally many ways of doing so.

1986 All Soviet Union Mathematical Olympiad, 426
Tags: number theory , square
Find all the natural numbers equal to the square of its divisors number.

2024 Bulgarian Spring Mathematical Competition, 12.3
Tags: number theory
For a positive integer $n$, denote with $b(n)$ the smallest positive integer $k$, such that there exist integers $a_1, a_2, \ldots, a_k$, satisfying $n=a_1^{33}+a_2^{33}+\ldots+a_k^{33}$. Determine whether the set of positive integers $n$ is finite or infinite, which satisfy: a) $b(n)=12;$ b) $b(n)=12^{12^{12}}.$

KoMaL A Problems 2022/2023, A.838
Tags: number theory , analytic number theory
Sets \(X\subset \mathbb{Z}^{+}\) and \(Y\subset \mathbb{Z}^{+}\) are called [i]comradely[/i], if every positive integer \(n\) can be written as \(n=xy\) for some \(x\in X\) and \(y\in Y\). Let \(X(n)\) and \(Y(n)\) denote the number of elements of \(X\) and \(Y\), respectively, among the first \(n\) positive integers. Let \(f\colon \mathbb{Z}^{+}\to \mathbb{R}^{+}\) be an arbitrary function that goes to infinity. Prove that one can find comradely sets \(X\) and \(Y\) such that \(\dfrac{X(n)}{n}\) and \(\dfrac{Y(n)}{n}\) goes to \(0\), and for all \(\varepsilon>0\) exists \(n \in \mathbb{Z}^+\) such that \[\frac{\min\big\{X(n), Y(n)\big\}}{f(n)}<\varepsilon. \]

2021 Honduras National Mathematical Olympiad, Problem 3
Tags: algebra , number theory
Let $a$ and $b$ be positive integers satisfying \[ \frac a{a-2} = \frac{b+2021}{b+2008} \] Find the maximum value $\dfrac ab$ can attain.

2025 Turkey Team Selection Test, 9
Tags: number theory with sequences , modular congruences , modular arithmetic , number theory
Let \(n\) be a positive integer. For every positive integer $1 \leq k \leq n$ the sequence ${\displaystyle {\{ a_{i}+ki\}}_{i=1}^{n }}$ is defined, where $a_1,a_2, \dots ,a_n$ are integers. Among these \(n\) sequences, for at most how many of them does all the elements of the sequence give different remainders when divided by \(n\)?

2020 Italy National Olympiad, #5
Tags: function , prime number , number theory
Le $S$ be the set of positive integers greater than or equal to $2$. A function $f: S\rightarrow S$ is italian if $f$ satifies all the following three conditions: 1) $f$ is surjective 2) $f$ is increasing in the prime numbers(that is, if $p_1<p_2$ are prime numbers, then $f(p_1)<f(p_2)$) 3) For every $n\in S$ the number $f(n)$ is the product of $f(p)$, where $p$ varies among all the primes which divide $n$ (For instance, $f(360)=f(2^3\cdot 3^2\cdot 5)=f(2)\cdot f(3)\cdot f(5)$). Determine the maximum and the minimum possible value of $f(2020)$, when $f$ varies among all italian functions.

2017 Switzerland - Final Round, 4
Tags: divide , number theory , divisible
Let $n$ be a natural number and $p, q$ be prime numbers such that the following statements hold: $$pq | n^p + 2$$ $$n + 2 | n^p + q^p.$$ Show that there is a natural number $m$ such that $q|4^mn + 2$ holds.

2022 Harvard-MIT Mathematics Tournament, 6
Tags: number theory
Let f be a function from $\{1, 2, . . . , 22\}$ to the positive integers such that $mn | f(m) + f(n)$ for all $m, n \in \{1, 2, . . . , 22\}$. If $d$ is the number of positive divisors of $f(20)$, compute the minimum possible value of $d$.

2013 Online Math Open Problems, 17
Tags: relatively prime , number theory
Determine the number of ordered pairs of positive integers $(x,y)$ with $y < x \le 100$ such that $x^2-y^2$ and $x^3 - y^3$ are relatively prime. (Two numbers are [i]relatively prime[/i] if they have no common factor other than $1$.) [i]Ray Li[/i]

2016 Ecuador NMO (OMEC), 6
Tags: number theory , divisible
A positive integer $n$ is "[i]olympic[/i]" if there are $n$ non-negative integers $x_1, x_2, ..., x_n$ that satisfy that: $\bullet$ There is at least one positive integer $j$: $1 \le j \le n$ such that $x_j \ne 0$. $\bullet$ For any way of choosing $n$ numbers $c_1, c_2, ..., c_n$ from the set $\{-1, 0, 1\}$, where not all $c_i$ are equal to zero, it is true that the sum $c_1x_1 + c_2x_2 +... + c_nx_n$ is not divisible by $n^3$. Find the largest positive "olympic" integer.

2015 India PRMO, 14
Tags: diophantine equation , algebra , number theory
$14.$ If $3^x+2^y=985.$ and $3^x-2^y=473.$ What is the value of $xy ?$

1994 Mexico National Olympiad, 2
Tags: integer , number theory , combinatorics
The $12$ numbers on a clock face are rearranged. Show that we can still find three adjacent numbers whose sum is $21$ or more.

2014 Tuymaada Olympiad, 1
Tags: number theory
Given are three different primes. What maximum number of these primes can divide their sum? [i](A. Golovanov)[/i]

2019 Belarus Team Selection Test, 5.1
Tags: polynomial , algebra , number theory , function
A function $f:\mathbb N\to\mathbb N$, where $\mathbb N$ is the set of positive integers, satisfies the following condition: for any positive integers $m$ and $n$ ($m>n$) the number $f(m)-f(n)$ is divisible by $m-n$. Is the function $f$ necessarily a polynomial? (In other words, is it true that for any such function there exists a polynomial $p(x)$ with real coefficients such that $f(n)=p(n)$ for all positive integers $n$?) [i](Folklore)[/i]

2015 China Northern MO, 4
Tags: number theory , combinatorics
It is known that $a_1, a_2,...a_{108}$ are $108$ different positive integers not exceeding $2015$. Prove that there is a positive integer $k$ such that there are at least four different pairs $(i, j) $satisfying $a_i-a_j =k$.

2023 Kazakhstan National Olympiad, 5
Tags: prime number , number theory
Solve the given equation in prime numbers $$p^3+q^3+r^3=p^2qr$$

2021 Moldova EGMO TST, 3
Tags: number theory
Prove that $9$ divides $A_n=16^n+4^n-2$ for every nonnegative integer $n$.

2021 BMT, 11
Tags: number theory
Compute the number of sequences of five positive integers $a_1,..., a_5$ where all $a_i \le 5$ and the greatest common divisor of all five integers is $1$.

2005 India IMO Training Camp, 2
Tags: number theory
Given real numbers $a,\alpha,\beta, \sigma \ and \ \varrho$ s.t. $\sigma, \varrho > 0$ and $\sigma \varrho = \frac{1}{16}$, prove that there exist integers $x$ and $y$ s.t. \[ - \sigma \leq (x+\alpha_(ax + y + \beta ) \leq \varrho \]

2019 Canadian Mathematical Olympiad Qualification, 8
Tags: inequalities , divide , number theory
For $t \ge 2$, defi ne $S(t)$ as the number of times $t$ divides into $t!$. We say that a positive integer $t$ is a [i]peak[/i] if $S(t) > S(u)$ for all values of $u < t$. Prove or disprove the following statement: For every prime $p$, there is an integer $k$ for which $p$ divides $k$ and $k$ is a peak.

1991 Spain Mathematical Olympiad, 5
Tags: number theory , digit sum , binary
For a positive integer $n$, let $s(n)$ denote the sum of the binary digits of $n$. Find the sum $s(1)+s(2)+s(3)+...+s(2^k)$ for each positive integer $k$.

2019 LIMIT Category A, Problem 12
Tags: number theory , diophantine equation , combinatorics
Compute the number of ordered quadruples of positive integers $(a,b,c,d)$ such that $$a!b!c!d!=24!$$$\textbf{(A)}~4$ $\textbf{(B)}~4!$ $\textbf{(C)}~4^4$ $\textbf{(D)}~\text{None of the above}$

2001 Junior Balkan Team Selection Tests - Romania, 2
Tags: number theory , calculus , integration
Find all $n\in\mathbb{Z}$ such that the number $\sqrt{\frac{4n-2}{n+5}}$ is rational.