Found problems: 364
1999 Romania National Olympiad, 2
Let $a, b, c$ be non zero integers,$ a\ne c$ such that $$\frac{a}{c}=\frac{a^2+b^2}{c^2+b^2}$$
Prove that $a^2 +b^2 +c^2$ cannot be a prime number.
2013 VJIMC, Problem 1
Let $S_n$ denote the sum of the first $n$ prime numbers. Prove that for any $n$ there exists the square of an integer between $S_n$ and $S_{n+1}$.
2002 Croatia Team Selection Test, 3
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 Baltic Way, 17
Distinct positive integers $a, b, c, d$ satisfy
$$\begin{cases} a \mid b^2 + c^2 + d^2,\\
b\mid a^2 + c^2 + d^2,\\
c \mid a^2 + b^2 + d^2,\\
d \mid a^2 + b^2 + c^2,\end{cases}$$
and none of them is larger than the product of the three others. What is the largest possible number of primes among them?
2015 ELMO Problems, 4
Let $a > 1$ be a positive integer. Prove that for some nonnegative integer $n$, the number $2^{2^n}+a$ is not prime.
[i]Proposed by Jack Gurev[/i]
2025 Kosovo National Mathematical Olympiad`, P4
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ for which these two conditions hold simultaneously
(i) For all $m,n \in \mathbb{N}$ we have:
$$ \frac{f(mn)}{\gcd(m,n)} = \frac{f(m)f(n)}{f(\gcd(m,n))};$$
(ii) For all prime numbers $p$, there exists a prime number $q$ such that $f(p^{2025})=q^{2025}$.
2011 Brazil Team Selection Test, 2
Let $n\ge 3$ be an integer such that for every prime factor $q$ of $n-1$ exists an integer $a > 1$ such that $a^{n-1} \equiv 1 \,(\mod n \, )$ and $a^{\frac{n-1} {q}}\not\equiv 1 \,(\mod n \, )$. Prove that $n$ is not prime.
2024-IMOC, N1
Proof that for every primes $p$, $q$
\[p^{q^2-q+1}+q^{p^2-p+1}-p-q\]
is never a perfect square.
[i]Proposed by chengbilly[/i]
2012 Korea Junior Math Olympiad, 6
$p > 3$ is a prime number such that $p|2^{p-1} - 1$ and $p \nmid 2^x - 1$ for $x = 1, 2,...,p-2$. Let $p = 2k + 3$. Now we define sequence $\{a_n\}$ as $$a_i = a_{i+k} = 2^i \,\, (1 \le i \le k ), \,\,\,\, a_{j+2k} = a_ja_{j+k} \,\, (j \le 1)$$
Prove that there exist $2k$ consecutive terms of sequence $a_{x+1},a_{x+2},..., a_{x+2k}$ such that $a_{x+i } \not\equiv a_{x+j}$ (mod $p$) for all $1 \le i < j \le 2k$ .
2015 Gulf Math Olympiad, 1
a) Suppose that $n$ is an odd integer. Prove that $k(n-k)$ is divisible by $2$ for all positive integers $k$.
b) Find an integer $k$ such that $k(100-k)$ is not divisible by $11$.
c) Suppose that $p$ is an odd prime, and $n$ is an integer.
Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by $p$.
d) Suppose that $p,q$ are two different odd primes, and $n$ is an integer.
Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by any of $p,q$.
2010 Junior Balkan Team Selection Tests - Romania, 1
Determine the prime numbers $p, q, r$ with the property $\frac {1} {p} + \frac {1} {q} + \frac {1} {r} \ge 1$
2015 Bosnia And Herzegovina - Regional Olympiad, 2
Find all triplets $(p,a,b)$ of positive integers such that $$p=b\sqrt{\frac{a-8b}{a+8b}}$$ is prime
2021 Dutch Mathematical Olympiad, 5
We consider an integer $n > 1$ with the following property: for every positive divisor $d$ of $n$ we have that $d + 1$ is a divisor of$ n + 1$. Prove that $n$ is a prime number.
2016 Hanoi Open Mathematics Competitions, 15
Find all polynomials of degree $3$ with integer coeffcients such that $f(2014) = 2015, f(2015) = 2016$ and $f(2013) - f(2016)$ is a prime number.
2010 NZMOC Camp Selection Problems, 3
Find all positive integers n such that $n^5 + n + 1$ is prime.
2009 Bundeswettbewerb Mathematik, 2
Let $n$ be an integer that is greater than $1$. Prove that the following two statements are equivalent:
(A) There are positive integers $a, b$ and $c$ that are not greater than $n$ and for which that polynomial $ax^2 + bx + c$ has two different real roots $x_1$ and $x_2$ with $| x_2- x_1 | \le \frac{1}{n}$
(B) The number $n$ has at least two different prime divisors.
2012 Mathcenter Contest + Longlist, 2
Let $p=2^n+1$ and $3^{(p-1)/2}+1\equiv 0 \pmod p$. Show that $p$ is a prime.
[i](Zhuge Liang) [/i]
2025 Macedonian Balkan MO TST, 4
Let $n$ be a positive integer. Prove that for every odd prime $p$ dividing $n^2 + n + 2$, there exist integers $a, b$ such that $p = a^2 + 7b^2$.
2013 Dutch BxMO/EGMO TST, 3
Find all triples $(x,n,p)$ of positive integers $x$ and $n$ and primes $p$ for which the following holds $x^3 + 3x + 14 = 2 p^n$
2009 Singapore Junior Math Olympiad, 3
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 $x, y,z,...$.)
2016 JBMO Shortlist, 1
Determine the largest positive integer $n$ that divides $p^6 - 1$ for all primes $p > 7$.
1998 Switzerland Team Selection Test, 6
Find all prime numbers $p$ for which $p^2 +11$ has exactly six positive divisors.
2021 CHKMO, 2
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 $11$?
2013 Cuba MO, 4
We say that a positive integer is [i]decomposed [/i] if it is prime and also If a line is drawn separating it into two numbers, those two numbers are never composite. For example 1997 is [i]decomposed [/i] since it is prime, it is divided into: $1$, $997$; $19$, $97$; $199$, $7$ and none of those numbers are compound. How many [i]decomposed [/i] numbers are there between $2000$ and $3000$?
2009 Tournament Of Towns, 7
Initially a number $6$ is written on a blackboard. At $n$-th step an integer $k$ on the blackboard is replaced by $k+gcd(k,n)$. Prove that at each step the number on the blackboard increases either by $1$ or by a prime number.