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: 364

2022 239 Open Mathematical Olympiad, 4
Tags: prime , natural number , number theory
Vasya has a calculator that works with pairs of numbers. The calculator knows hoe to make a pair $(x+y,x)$ or a pair $(2x+y+1,x+y+1)$ from a pair $(x,y).$ At the beginning, the pair $(1,1)$ is presented on the calculator. Prove that for any natural $n$ there is exactly one pair $(n,k)$ that can be obtained using a calculator.

1985 Tournament Of Towns, (091) T2
Tags: combinatorics , number theory , difference , prime , composite , 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) .

2019 Peru EGMO TST, 1
Tags: prime , number theory , diophantine equation , diophantine
Find all the prime numbers $p, q$ and $r$ such that $p^2 + 1 = 74 (q^2 + r^2)$.

2013 Hanoi Open Mathematics Competitions, 1
Tags: prime , number theory , minimum
Write $2013$ as a sum of $m$ prime numbers. The smallest value of $m$ is: (A): $2$, (B): $3$, (C): $4$, (D): $1$, (E): None of the above.

2023 Indonesia TST, 1
Tags: prime , divisibility , number theory
Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

2019 Finnish National High School Mathematics Comp, 2
Tags: algebra , floor function , number theory , prime
Prove that the number $\lfloor (2+\sqrt5)^{2019} \rfloor$ is not prime.

2000 Estonia National Olympiad, 1
Tags: power of prime , prime , remainder , number theory
Find all prime numbers whose sixth power does not give remainder $1$ when dividing by $504$

2018 Saudi Arabia IMO TST, 1
Tags: number theory , prime , algebra , increasing , sequence
Consider the infinite, strictly increasing sequence of positive integer $(a_n)$ such that i. All terms of sequences are pairwise coprime. ii. The sum $\frac{1}{\sqrt{a_1a_2}} +\frac{1}{\sqrt{a_2a_3}}+ \frac{1}{\sqrt{a_3a_4}} + ..$ is unbounded. Prove that this sequence contains infinitely many primes.

2014 Estonia Team Selection Test, 1
Tags: prime , combinatorics
In Wonderland, the government of each country consists of exactly $a$ men and $b$ women, where $a$ and $b$ are fixed natural numbers and $b > 1$. For improving of relationships between countries, all possible working groups consisting of exactly one government member from each country, at least $n$ among whom are women, are formed (where $n$ is a fixed non-negative integer). The same person may belong to many working groups. Find all possibilities how many countries can be in Wonderland, given that the number of all working groups is prime.

1994 Poland - Second Round, 6
Tags: prime , divide , number theory
Let $p$ be a prime number. Prove that there exists $n \in Z$ such that $p | n^2 -n+3$ if and only if there exists $m \in Z$ such that $p | m^2 -m+25$.

2016 Mathematical Talent Reward Programme, MCQ: P 4
Tags: prime
There are 168 primes below 1000. Then sum of all primes below 1000 is [list=1] [*] 11555 [*] 76127 [*] 57298 [*] 81722 [/list]

2012 IMO Shortlist, N8
Tags: prime , gauss , divisibility , modular arithmetic , number theory
Prove that for every prime $p>100$ and every integer $r$, there exist two integers $a$ and $b$ such that $p$ divides $a^2+b^5-r$.

2009 Silk Road, 4
Tags: prime , perfect square , number theory
Prove that for any prime number $p$ there are infinitely many fours $(x, y, z, t)$ pairwise distinct natural numbers such that the number $(x^2+p t^2)(y^2+p t^2)(z^2+p t^2)$ is a perfect square.

1997 German National Olympiad, 6a
Tags: prime , algebra , divide , polynomial
Let us define $f$ and $g$ by $f(x) = x^5 +5x^4 +5x^3 +5x^2 +1$, $g(x) = x^5 +5x^4 +3x^3 -5x^2 -1$. Determine all prime numbers $p$ such that, for at least one integer $x, 0 \le x < p-1$, both $f(x)$ and $g(x)$ are divisible by $p$. For each such $p$, find all $x$ with this property.

2021 CHKMO, 2
Tags: prime , number theory , prime factorization
For each positive integer $n$ larger than $1$ with prime factorization $p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$, its [i]signature[/i] is defined as the sum $\alpha_1+\alpha_2+\cdots+\alpha_k$. Does there exist $2020$ consecutive positive integers such that among them, there are exactly $1812$ integers whose signatures are strictly smaller than $11$?

1994 Italy TST, 2
Tags: prime , perfect square , number theory
Find all prime numbers $p$ for which $\frac{2^{p-1} -1}{p}$ is a perfect square.

2009 Singapore Junior Math Olympiad, 3
Tags: prime , digit , number theory
Suppose $\overline{a_1a_2...a_{2009}}$ is a $2009$-digit integer such that for each $i = 1,2,...,2007$, the $2$-digit integer $\overline{a_ia_{i+1}}$ contains $3$ distinct prime factors. Find $a_{2008}$ (Note: $\overline{xyz...}$ denotes an integer whose digits are $x, y,z,...$.)

2015 Latvia Baltic Way TST, 13
Tags: floor function , prime , number theory
Are there positive real numbers $a$ and $b$ such that $[an+b]$ is prime for all natural values of $n$ ? $[x]$ denotes the integer part of the number $x$, the largest integer that does not exceed $x$.

2008 Postal Coaching, 2
Tags: prime number , divide , prime , number theory
Prove that an integer $n \ge 2$ is a prime if and only if $\phi (n)$ divides $(n - 1)$ and $(n + 1)$ divides $\sigma (n)$. [Here $\phi$ is the Totient function and $\sigma $ is the divisor - sum function.] [hide=Hint]$n$ is squarefree[/hide]

2002 Croatia Team Selection Test, 3
Tags: number theory , prime
Prove that if $n$ is a natural number such that $1 + 2^n + 4^n$ is prime then $n = 3^k$ for some $k \in N_0$.

2021 New Zealand MO, 3
Tags: prime , number theory
Let $\{x_1, x_2, x_3, ..., x_n\}$ be a set of $n$ distinct positive integers, such that the sum of any $3$ of them is a prime number. What is the maximum value of $n$?

2017 Israel Oral Olympiad, 3
Tags: prime , number theory , prime number
2017 prime numbers $p_1,...,p_{2017}$ are given. Prove that $\prod_{i<j} (p_i^{p_j}-p_j^{p_i})$ is divisible by 5777.

2018 Bosnia And Herzegovina - Regional Olympiad, 4
Tags: prime , sequence , composite , infinitely many solutions , number theory
We observe that number $10001=73\cdot137$ is not prime. Show that every member of infinite sequence $10001, 100010001, 1000100010001,...$ is not prime

2015 Regional Olympiad of Mexico Southeast, 1
Tags: prime , number theory
Find all integers $n>1$ such that every prime that divides $n^6-1$ also divides $n^5-n^3-n^2+1$.

2017 Romania Team Selection Test, P4
Tags: prime , arithmetic mean , fractional part , odd , quadratic residue , number theory
Given a positive odd integer $n$, show that the arithmetic mean of fractional parts $\{\frac{k^{2n}}{p}\}, k=1,..., \frac{p-1}{2}$ is the same for infinitely many primes $p$ .