Found problems: 715
2016 Uzbekistan National Olympiad, 2
$n$ is natural number and $p$ is prime number. If $1+np$ is square of natural number then prove that $n+1$ is equal to some sum of $p$ square of natural numbers.
2019 Romania National Olympiad, 4
Let $p$ be a prime number. For any $\sigma \in S_p$ (the permutation group of $\{1,2,...,p \}),$ define the matrix $A_{\sigma}=(a_{ij}) \in \mathcal{M}_p(\mathbb{Z})$ as $a_{ij} = \sigma^{i-1}(j),$ where $\sigma^0$ is the identity permutation and $\sigma^k = \underbrace{\sigma \circ \sigma \circ ... \circ \sigma}_k.$
Prove that $D = \{ |\det A_{\sigma}| : \sigma \in S_p \}$ has at most $1+ (p-2)!$ elements.
2015 Nordic, 2
Find the primes ${p, q, r}$, given that one of the numbers ${pqr}$ and ${p + q + r}$ is ${101}$ times the other.
2008 Argentina Iberoamerican TST, 2
Set $S = \{1, 2, 3, ..., 2005\}$. If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$.
2016 Croatia Team Selection Test, Problem 4
Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime.
Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.
2005 China Western Mathematical Olympiad, 3
Set $S = \{1, 2, 3, ..., 2005\}$. If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$.
2023 VN Math Olympiad For High School Students, Problem 3
Given a polynomial with integer coefficents with degree $n>0:$$$P(x)=a_nx^n+...+a_1x+a_0.$$
Assume that there exists a prime number $p$ satisfying these conditions:
[i]i)[/i] $p|a_i$ for all $0\le i<n,$
[i]ii)[/i] $p\nmid a_n,$
[i]iii)[/i] $p^2\nmid a_0.$
Prove that $P(x)$ is irreducible in $\mathbb{Z}[x].$
1981 Czech and Slovak Olympiad III A, 4
Let $n$ be a positive integer. Show that there is a prime $p$ and a sequence $\left(a_k\right)_{k\ge1}$ of positive integers such that the sequence $\left(p+na_k\right)_{k\ge1}$ consists of distinct primes.
2008 Canada National Olympiad, 4
Determine all functions $ f$ defined on the natural numbers that take values among the natural numbers for which
\[ (f(n))^p \equiv n\quad {\rm mod}\; f(p)
\]
for all $ n \in {\bf N}$ and all prime numbers $ p$.
1990 IMO Longlists, 98
Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n \minus{} 1$ digits $ 1$ and one digit $ 7$ is prime.
2016 Bundeswettbewerb Mathematik, 1
A number with $2016$ zeros that is written as $101010 \dots 0101$ is given, in which the zeros and ones alternate. Prove that this number is not prime.
2020 Kosovo National Mathematical Olympiad, 4
Let $p$ and $q$ be prime numbers. Show that $p^2+q^2+2020$ is composite.
2018 Kürschák Competition, 2
Given a prime number $p$ and let $\overline{v_1},\overline{v_2},\dotsc ,\overline{v_n}$ be $n$ distinct vectors of length $p$ with integer coordinates in an $\mathbb{R}^3$ Cartesian coordinate system. Suppose that for any $1\leqslant j<k\leqslant n$, there exists an integer $0<\ell <p$ such that all three coordinates of $\overline{v_j} -\ell \cdot \overline{v_k} $ is divisible by $p$. Prove that $n\leqslant 6$.
2023 Bundeswettbewerb Mathematik, 1
Determine the greatest common divisor of the numbers $p^6-7p^2+6$ where $p$ runs through the prime numbers $p \ge 11$.
2022 Pan-American Girls' Math Olympiad, 6
Ana and Bety play a game alternating turns. Initially, Ana chooses an odd possitive integer and composite $n$ such that $2^j<n<2^{j+1}$ with $2<j$. In her first turn Bety chooses an odd composite integer $n_1$ such that
\[n_1\leq \frac{1^n+2^n+\dots+(n-1)^n}{2(n-1)^{n-1}}.\]
Then, on her other turn, Ana chooses a prime number $p_1$ that divides $n_1$. If the prime that Ana chooses is $3$, $5$ or $7$, the Ana wins; otherwise Bety chooses an odd composite positive integer $n_2$ such that \[n_2\leq \frac{1^{p_1}+2^{p_1}+\dots+(p_1-1)^{p_1}}{2(p_1-1)^{p_1-1}}.\]
After that, on her turn, Ana chooses a prime $p_2$ that divides $n_2,$, if $p_2$ is $3$, $5$, or $7$, Ana wins, otherwise the process repeats. Also, Ana wins if at any time Bety cannot choose an odd composite positive integer in the corresponding range. Bety wins if she manages to play at least $j-1$ turns. Find which of the two players has a winning strategy.
2019 Dutch IMO TST, 4
Find all functions $f : Z \to Z$ satisfying
$\bullet$ $ f(p) > 0$ for all prime numbers $p$,
$\bullet$ $p| (f(x) + f(p))^{f(p)}- x$ for all $x \in Z$ and all prime numbers $p$.
2020 Kazakhstan National Olympiad, 3
Let $p$ be a prime number and $k,r$ are positive integers such that $p>r$. If $pk+r$ divides $p^p+1$ then prove that $r$ divides $k$.
2019 India PRMO, 20
Consider the set $E$ of all natural numbers $n$ such that whenn divided by $11, 12, 13$, respectively, the remainders, int that order, are distinct prime numbers in an arithmetic progression. If $N$ is the largest number in $E$, find the sum of digits of $N$.
2015 AMC 10, 11
Among the positive integers less than $100$, each of whose digits is a prime number, one is selected at random. What is the probablility that the selected number is prime?
$\textbf{(A) } \dfrac{8}{99}
\qquad\textbf{(B) } \dfrac{2}{5}
\qquad\textbf{(C) } \dfrac{9}{20}
\qquad\textbf{(D) } \dfrac{1}{2}
\qquad\textbf{(E) } \dfrac{9}{16}
$
2024 Assara - South Russian Girl's MO, 2
Let $p$ be a prime number. Positive integers numbers $a$ and $b$ are such $\frac{p}{a}+\frac{p}{b}=1$ and $a+b$ is divisible by $p$. What values can an expression $\frac{a+b}{p}$ take?
[i]Yu.A.Karpenko[/i]
2008 Bulgarian Autumn Math Competition, Problem 8.3
Prove that there exists a prime number $p$, such that the sum of digits of $p$ is a composite odd integer. Find the smallest such $p$.
2009 China Team Selection Test, 2
Find all integers $ n\ge 2$ having the following property: for any $ k$ integers $ a_{1},a_{2},\cdots,a_{k}$ which aren't congruent to each other (modulo $ n$), there exists an integer polynomial $ f(x)$ such that congruence equation $ f(x)\equiv 0 (mod n)$ exactly has $ k$ roots $ x\equiv a_{1},a_{2},\cdots,a_{k} (mod n).$
2010 Contests, 1
Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that
\[f(a+1), f(a+2), \dots, f(a+n)\]
form an arithmetic progression.
2018 IFYM, Sozopol, 3
Let $p$ be some prime number.
a) Prove that there exist positive integers $a$ and $b$ such that $a^2 + b^2 + 2018$ is multiple of $p$.
b) Find all $p$ for which the $a$ and $b$ from a) can be chosen in such way that both these numbers aren’t multiples of $p$.
2002 Manhattan Mathematical Olympiad, 1
Famous French mathematician Pierre Fermat believed that all numbers of the form $F_n = 2^{2^n} + 1$ are prime for all non-negative integers $n$. Indeed, one can check that $F_0 = 3$, $F_1 = 5$, $F_2 = 17$, $F_3 = 257$ are all prime.
a) Prove that $F_5$ is divisible by $641$. (Hence Fermat was wrong.)
b) Prove that if $k \ne n$ then $F_k$ and $F_n$ are relatively prime (i.e. they do not have any common divisor except $1$)
(Notice: using b) one can prove that there are infinitely many prime numbers)