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: 721

2023 IMC, 4
Tags: modular arithmetic , prime number , algebra , polynomial , number theory
Let $p$ be a prime number and let $k$ be a positive integer. Suppose that the numbers $a_i=i^k+i$ for $i=0,1, \ldots,p-1$ form a complete residue system modulo $p$. What is the set of possible remainders of $a_2$ upon division by $p$?

2014 German National Olympiad, 1
Tags: prime number , number theory
For which non-negative integers $n$ is \[K=5^{2n+3} + 3^{n+3} \cdot 2^n\] prime?

1985 IMO Shortlist, 7
Tags: prime number , sequence , inequality , number theory
The positive integers $x_1, \cdots , x_n$, $n \geq 3$, satisfy $x_1 < x_2 <\cdots< x_n < 2x_1$. Set $P = x_1x_2 \cdots x_n.$ Prove that if $p$ is a prime number, $k$ a positive integer, and $P$ is divisible by $pk$, then $\frac{P}{p^k} \geq n!.$

1995 Czech and Slovak Match, 6
Tags: prime number , algebra , number theory
Find all triples $(x; y; p)$ of two non-negative integers $x, y$ and a prime number p such that $ p^x-y^p=1 $

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.

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

1995 Moldova Team Selection Test, 2
Tags: prime number , number theory
Let $p{}$ be a prime number. Prove that the equation has $x^2-x+3-ps=0$ with $x,s\in\mathbb{Z}$ has solutions if and only if the equation $y^2-y+25-pt=0$ with $y,t\in\mathbb{Z}$ has solutions.

2016 Postal Coaching, 2
Tags: prime number , diophantine equation , number theory
Solve the equation for primes $p$ and $q$: $$p^3-q^3=pq^3-1.$$

1974 IMO Longlists, 35
Tags: prime number , number theory
If $p$ and $q$ are distinct prime numbers, then there are integers $x_0$ and $y_0$ such that $1 = px_0 + qy_0.$ Determine the maximum value of $b - a$, where $a$ and $b$ are positive integers with the following property: If $a \leq t \leq b$, and $t$ is an integer, then there are integers $x$ and $y$ with $0 \leq x \leq q - 1$ and $0 \leq y \leq p - 1$ such that $t = px + qy.$

1985 IMO Longlists, 22
Tags: prime number , sequence , inequality , number theory
The positive integers $x_1, \cdots , x_n$, $n \geq 3$, satisfy $x_1 < x_2 <\cdots< x_n < 2x_1$. Set $P = x_1x_2 \cdots x_n.$ Prove that if $p$ is a prime number, $k$ a positive integer, and $P$ is divisible by $pk$, then $\frac{P}{p^k} \geq n!.$

2022-23 IOQM India, 8
Tags: prime number , number theory
Suppose the prime numbers $p$ and $q$ satisfy $q^2+3p=197p^2+q$.Write $\frac{p}{q}$ as $l+\frac{m}{n}$, where $l,m,n$ are positive integers , $m<n$ and $GCD(m,n)=1$. Find the maximum value of $l+m+n$.

2022 VIASM Summer Challenge, Problem 1
Tags: prime number , number theory
Find all prime number pairs $(p,q)$ such that $p(p^2-p-1)=q(2q+3).$

2000 AMC 12/AHSME, 6
Tags: prime number , number theory
Two different prime numbers between $ 4$ and $ 18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $ \textbf{(A)}\ 21 \qquad \textbf{(B)}\ 60\qquad \textbf{(C)}\ 119 \qquad \textbf{(D)}\ 180\qquad \textbf{(E)}\ 231$

2020 Poland - Second Round, 5.
Tags: prime number , number theory
Let $p>$ be a prime number and $S$ be a set of $p+1$ integers. Prove that there exist pairwise distinct numbers $a_1,a_2,...,a_{p-1}\in S$ that $$ a_1+2a_2+3a_3+...+(p-1)a_{p-1}$$ is divisible by $p$.

2008 Irish Math Olympiad, 1
Tags: prime number , number theory
Let $ p_1, p_2, p_3$ and $ p_4$ be four different prime numbers satisying the equations $ 2p_1 \plus{} 3p_2 \plus{} 5p_3 \plus{} 7p_4 \equal{} 162$ $ 11p_1 \plus{} 7p_2 \plus{} 5p_3 \plus{} 4p_4 \equal{} 162$ Find all possible values of the product $ p_1p_2p_3p_4$

1996 National High School Mathematics League, 3
Tags: prime number , number theory
For a prime number $p$, there exists $n\in\mathbb{Z}_+$, $\sqrt{p+n}+\sqrt{n}$ is an integer, then $\text{(A)}$ there is no such $p$ $\text{(B)}$ there in only one such $p$ $\text{(C)}$ there is more than one such $p$, but finitely many $\text{(D)}$ there are infinitely many such $p$

1990 Irish Math Olympiad, 2
Tags: number theory , prime number , arithmetic sequence
Suppose that $p_1<p_2<\dots <p_{15}$ are prime numbers in arithmetic progression, with common difference $d$. Prove that $d$ is divisible by $2,3,5,7,11$ and $13$.

2020 Kosovo National Mathematical Olympiad, 4
Tags: prime number , compos , number theory
Let $p$ and $q$ be prime numbers. Show that $p^2+q^2+2020$ is composite.

2004 Korea Junior Math Olympiad, 3
Tags: multiple , prime number , number theory
For an arbitrary prime number $p$, show that there exists infinitely many multiples of $p$ that can be expressed as the form $$\frac{x^2+y+1}{x+y^2+1}$$ Where $x, y$ are some positive integers.

1999 China Team Selection Test, 2
Tags: prime number , number theory
Find all prime numbers $p$ which satisfy the following condition: For any prime $q < p$, if $p = kq + r, 0 \leq r < q$, there does not exist an integer $q > 1$ such that $a^{2} \mid r$.

2019 IFYM, Sozopol, 1
Tags: sequence , prime number , number theory
We define the sequence $a_n=(2n)^2+1$ for each natural number $n$. We will call one number [i]bad[/i], if there don’t exist natural numbers $a>1$ and $b>1$ such that $a_n=a^2+b^2$. Prove that the natural number $n$ is [i]bad[/i], if and only if $a_n$ is prime.

2010 Contests, 4
Tags: prime number , number theory
Determine the smallest positive integer $m$ with the property that $m^3-3m^2+2m$ is divisible by both $79$ and $83$.

2017 Thailand TSTST, 2
Tags: modular arithmetic , number theory , euler , quadratic , prime number , function
Suppose that for some $m,n\in\mathbb{N}$ we have $\varphi (5^m-1)=5^n-1$, where $\varphi$ denotes the Euler function. Show that $(m,n)>1$.

2019 BMT Spring, 6
Tags: prime number , number theory
Define $ f(n) = \dfrac{n^2 + n}{2} $. Compute the number of positive integers $ n $ such that $ f(n) \leq 1000 $ and $ f(n) $ is the product of two prime numbers.

2021 Argentina National Olympiad, 1
Tags: prime number , number theory
Determine all pairs of prime numbers $p$ and $q$ greater than $1$ and less than $100$, such that the following five numbers: $$p+6,p+10,q+4,q+10,p+q+1,$$ are all prime numbers.