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 Irish Math Olympiad, 5
Tags: number theory
For each integer $ n$ of the form $ n\equal{}p_1 p_2 p_3 p_4$, where $ p_1,p_2,p_3,p_4$ are distinct primes, let $ 1\equal{}d_1<d_2<...<d_{15}<d_{16}\equal{}n$ be the divisors of $ n$. Prove that if $ n<1995$, then $ d_9\minus{}d_8 \not\equal{} 22$.

1981 Swedish Mathematical Competition, 1
Tags: number theory , perfect square
Let $N = 11\cdots 122 \cdots 25$, where there are $n$ $1$s and $n+1$ $2$s. Show that $N$ is a perfect square.

2021 Estonia Team Selection Test, 3
Tags: number theory
For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$

2007 Turkey MO (2nd round), 1
Tags: modular arithmetic , number theory
Let $k>1$ be an integer, $p=6k+1$ be a prime number and $m=2^{p}-1$ . Prove that $\frac{2^{m-1}-1}{127m}$ is an integer.

2010 Hanoi Open Mathematics Competitions, 7
Tags: number theory , prime , quadratic , root
Determine all positive integer $a$ such that the equation $2x^2 - 30x + a = 0$ has two prime roots, i.e. both roots are prime numbers.

2022 Balkan MO Shortlist, N3
Tags: number theory
For every natural number $x{}$, let $P(x)$ be the product of the digits of the number $x{}$. Is there a natural number $n{}$ such that the numbers $P(n)$ and $P(n^2)$ are non-zero squares of natural numbers, where the number of digits of the number $n{}$ is equal to (a) 2021 and (b) 2022?

2016 Romania National Olympiad, 3
Tags: geometry , number theory , 3d geometry , sphere
We say that a rational number is [i]spheric[/i] if it is the sum of three squares of rational numbers (not necessarily distinct). Prove that: [b]a)[/b] $ 7 $ is not spheric. [b]b)[/b] a rational spheric number raised to the power of any natural number greater than $ 1 $ is spheric.

1987 Brazil National Olympiad, 1
Tags: number theory , algebra , coprime , prime number , polynomial
$p(x_1, x_2, ... , x_n)$ is a polynomial with integer coefficients. For each positive integer $r, k(r)$ is the number of $n$-tuples $(a_1, a_2,... , a_n)$ such that $0 \le a_i \le r-1 $ and $p(a_1, a_2, ... , a_n)$ is prime to $r$. Show that if $u$ and $v$ are coprime then $k(u\cdot v) = k(u)\cdot k(v)$, and if p is prime then $k(p^s) = p^{n(s-1)} k(p)$.

2014 China National Olympiad, 2
Tags: number theory
For the integer $n>1$, define $D(n)=\{ a-b\mid ab=n, a>b>0, a,b\in\mathbb{N} \}$. Prove that for any integer $k>1$, there exists pairwise distinct positive integers $n_1,n_2,\ldots,n_k$ such that $n_1,\ldots,n_k>1$ and $|D(n_1)\cap D(n_2)\cap\cdots\cap D(n_k)|\geq 2$.

2012 Indonesia TST, 4
Tags: function , number theory
Let $\mathbb{N}$ be the set of positive integers. For every $n \in \mathbb{N}$, define $d(n)$ as the number of positive divisors of $n$. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that: a) $d(f(x)) = x$ for all $x \in \mathbb{N}$ b) $f(xy)$ divides $(x-1)y^{xy-1}f(x)$ for all $x,y \in \mathbb{N}$

2017 IMO Shortlist, N7
Tags: equation , john berman more like jawnorz , polynomial , number theory
An ordered pair $(x, y)$ of integers is a primitive point if the greatest common divisor of $x$ and $y$ is $1$. Given a finite set $S$ of primitive points, prove that there exist a positive integer $n$ and integers $a_0, a_1, \ldots , a_n$ such that, for each $(x, y)$ in $S$, we have: $$a_0x^n + a_1x^{n-1} y + a_2x^{n-2}y^2 + \cdots + a_{n-1}xy^{n-1} + a_ny^n = 1.$$ [i]Proposed by John Berman, United States[/i]

2016 Austria Beginners' Competition, 1
Tags: number theory
Determine all nonnegative integers $n$ having two distinct positive divisors with the same distance from $\tfrac{n}{3}$. (Richard Henner)

2022 HMIC, 5
Tags: function , number theory , f_p
Let $\mathbb{F}_p$ be the set of integers modulo $p$. Call a function $f : \mathbb{F}_p^2 \to \mathbb{F}_p$ [i]quasiperiodic[/i] if there exist $a,b \in \mathbb{F}_p$, not both zero, so that $f(x + a, y + b) = f(x, y)$ for all $x,y \in \mathbb{F}_p$. Find the number of functions $\mathbb{F}_p^2 \to \mathbb{F}_p$ that can be written as the sum of some number of quasiperiodic functions.

2017 Saint Petersburg Mathematical Olympiad, 4
Tags: number theory
Each cell of a $3\times n$ table was filled by a number. In each of three rows, the number $1,2,…,n$ appear in some order. It is know that for each column, the sum of pairwise product of three numbers in it is a multiple of $n$. Find all possible value of $n$.

1997 All-Russian Olympiad, 1
Tags: modular arithmetic , number theory
Find all integer solutions of the equation $(x^2 - y^2)^2 = 1 + 16y$. [i]M. Sonkin[/i]

2008 Hanoi Open Mathematics Competitions, 2
Tags: combinatorics , algebra , number theory
How many integers belong to ($a,2008a$), where $a$ ($a > 0$) is given.

1990 IMO Shortlist, 25
Tags: algebra , number theory , functional equation , function
Let $ {\mathbb Q}^ \plus{}$ be the set of positive rational numbers. Construct a function $ f : {\mathbb Q}^ \plus{} \rightarrow {\mathbb Q}^ \plus{}$ such that \[ f(xf(y)) \equal{} \frac {f(x)}{y} \] for all $ x$, $ y$ in $ {\mathbb Q}^ \plus{}$.

Russian TST 2015, P3
Tags: number theory , operation
Fix positive integers $n$ and $k\ge 2$. A list of $n$ integers is written in a row on a blackboard. You can choose a contiguous block of integers, and I will either add $1$ to all of them or subtract $1$ from all of them. You can repeat this step as often as you like, possibly adapting your selections based on what I do. Prove that after a finite number of steps, you can reach a state where at least $n-k+2$ of the numbers on the blackboard are all simultaneously divisible by $k$.

2003 India IMO Training Camp, 7
Tags: polynomial , limit , algebra , number theory
$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$

2019 Nigerian Senior MO Round 4, 1
Tags: function , inequalities , functional equation , number theory
Let $f: N \to N$ be a function satisfying (a) $1\le f(x)-x \le 2019$ $\forall x \in N$ (b) $f(f(x))\equiv x$ (mod $2019$) $\forall x \in N$ Show that $\exists x \in N$ such that $f^k(x)=x+2019 k, \forall k \in N$

2002 India IMO Training Camp, 8
Tags: number theory
Let $\sigma(n)=\sum_{d|n} d$, the sum of positive divisors of an integer $n>0$. [list] [b](a)[/b] Show that $\sigma(mn)=\sigma(m)\sigma(n)$ for positive integers $m$ and $n$ with $gcd(m,n)=1$ [b](b)[/b] Find all positive integers $n$ such that $\sigma(n)$ is a power of $2$.[/list]

2002 USA Team Selection Test, 2
Tags: algebra , modular arithmetic , number theory , binomial theorem , summation
Let $p>5$ be a prime number. For any integer $x$, define \[{f_p}(x) = \sum_{k=1}^{p-1} \frac{1}{(px+k)^2}\] Prove that for any pair of positive integers $x$, $y$, the numerator of $f_p(x) - f_p(y)$, when written as a fraction in lowest terms, is divisible by $p^3$.

2019 German National Olympiad, 5
Tags: number theory , ratio , combinatorics
We are given two positive integers $p$ and $q$. Step by step, a rope of length $1$ is cut into smaller pieces as follows: In each step all the currently longest pieces are cut into two pieces with the ratio $p:q$ at the same time. After an unknown number of such operations, the currently longest pieces have the length $x$. Determine in terms of $x$ the number $a(x)$ of different lengths of pieces of rope existing at that time.

2024 239 Open Mathematical Olympiad, 3
Tags: number theory , combinatorics
There are $169$ non-zero digits written around a circle. Prove that they can be split into $14$ non-empty blocks of consecutive digits so that among the $14$ natural numbers formed by the digits in those blocks, at least $13$ of them are divisible by $13$ (the digits in each block are read in clockwise direction).

1986 IMO Shortlist, 3
Tags: geometry , number theory , algebra , diophantine equation , pythagorean theorem
Let $A, B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $ABC$ is an equilateral triangle whose side is $86$ meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y.$