Olympiad problems

2017 JBMO Shortlist, NT5

Find all positive integers $n$ such that there exists a prime number $p$, such that $p^n-(p-1)^n$ is a power of $3$. Note. A power of $3$ is a number of the form $3^a$ where $a$ is a positive integer.

2021 Indonesia TST, C

Tags: combinatorics , subset , counting , prime , summation

Let $p$ be an odd prime. Determine the number of nonempty subsets from $\{1, 2, \dots, p - 1\}$ for which the sum of its elements is divisible by $p$.

2023 Belarus Team Selection Test, 2.1

Tags: number theory , divisibility , prime

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

1999 IMO Shortlist, 1

Tags: number theory , divisibility , prime

Find all the pairs of positive integers $(x,p)$ such that p is a prime, $x \leq 2p$ and $x^{p-1}$ is a divisor of $ (p-1)^{x}+1$.

2016 Latvia Baltic Way TST, 17

Tags: number theory , diophantine equation , diophantine , prime

Can you find five prime numbers $p, q, r, s, t$ such that $p^3+q^3+r^3+s^3 =t^3$?

2013 Balkan MO Shortlist, N6

Tags: number theory , diophantine equation , diophantine , prime

Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^3$

1958 Poland - Second Round, 1

Tags: prime , number theory , prime number

Prove that if $ a $ is an integer different from $ 1 $ and $ - 1 $, then $ a^4 + 4 $ is not a prime number.

2025 Israel TST, P2

Tags: number theory , prime

Prove that for all primes $ p $ such that $ p \equiv 3 \pmod{4} $ or $ p \equiv 5 \pmod{8} $, there exist integers \[ 1 \leq a_1 < a_2 < \cdots < a_{(p-1)/2} < p \] such that \[ \prod_{\substack{1 \leq i < j \leq (p-1)/2}} (a_i + a_j)^2 \equiv 1 \pmod{p}. \]

2011 Indonesia TST, 4

Tags: number theory , prime , sum of divisors , divide

Given $N = 2^ap_1p_2...p_m$, $m \ge 1$, $a \in N$ with $p_1, p_2,..., p_m$ are different primes. It is known that $\sigma (N) = 3N $ where $\sigma (N)$ is the sum of all positive integers which are factors of $N$. Show that there exists a prime number $p$ such that $2^p- 1$ is also a prime, and $2^p - 1|N$.

2022 IMO Shortlist, N2

Tags: divisibility , number theory , prime

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

2016 JBMO Shortlist, 2

Tags: combinatorics , sum , prime

The natural numbers from $1$ to $50$ are written down on the blackboard. At least how many of them should be deleted, in order that the sum of any two of the remaining numbers is not a prime?

2001 Estonia National Olympiad, 4

Tags: composite , sum of powers , prime , number theory

Prove that for any integer $a > 1$ there is a prime $p$ for which $1+a+a^2+...+ a^{p-1}$ is composite.

2005 iTest, 17

Tags: number theory , sum of digits , prime

On the $2004$ iTest, we defined an [i]optimus [/i] prime to be any prime number whose digits sum to a prime number. (For example, $83$ is an optimus prime, because it is a prime number and its digits sum to $11$, which is also a prime number.) Given that you select a prime number under $100$, find the probability that is it not an optimus prime.

2014 Danube Mathematical Competition, 1

Tags: number theory , sum , natural , prime

Determine the natural number $a =\frac{p+q}{r}+\frac{q+r}{p}+\frac{r+p}{q}$ where $p, q$ and $r$ are prime positive numbers.

2017 Peru IMO TST, 10

Tags: squarefree , polynomial , integer , number theory , prime

Let $P (n)$ and $Q (n)$ be two polynomials (not constant) whose coefficients are integers not negative. For each positive integer $n$, define $x_n = 2016^{P (n)} + Q (n)$. Prove that there exist infinite primes $p$ for which there is a positive integer $m$, squarefree, such that $p | x_m$. Clarification: A positive integer is squarefree if it is not divisible by the square of any prime number.

1941 Moscow Mathematical Olympiad, 085

Tags: prime , remainder , number theory

Prove that the remainder after division of the square of any prime $p > 3$ by $12$ is equal to $1$.

2017 Brazil National Olympiad, 6.

Tags: number theory , divisibility , prime , group theory

[b]6.[/b] Let $a$ be a positive integer and $p$ a prime divisor of $a^3-3a+1$, with $p \neq 3$. Prove that $p$ is of the form $9k+1$ or $9k-1$, where $k$ is integer.

2017 Romania Team Selection Test, P4

Tags: quadratic residue , number theory , arithmetic mean , fractional part , odd , prime

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 $p$ .

2016 Croatia Team Selection Test, Problem 4

Tags: number theory , prime number , prime , primitive root , decimal representation

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 $0,1, \ldots, 9$.

2023 Brazil Team Selection Test, 3

Tags: number theory , prime factorization , function , prime number , prime

Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$. (For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

2021 Puerto Rico Team Selection Test, 6

Tags: number theory , prime

Two positive integers $n,m\ge 2$ are called [i]allies[/i] if when written as a product of primes (not necessarily different): $n=p_1p_2...p_s$ and $m=q_1q_2...q_t$, turns out that: $$p_1 + p_2 + ... + p_s = q_1 + q_2 + ... + q_t$$ (a) Show that the biggest ally of any positive integer has to have only $2$ and $3$ in its prime factorization. (b) Find the biggest number which is allied of $2021$ .

2023 Switzerland - Final Round, 6

Tags: number theory , divisibility , prime

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

2019 Tournament Of Towns, 1

Tags: inequalities , number theory , prime , number of divisors , prime factorization , factor , divisor

Let us call the number of factors in the prime decomposition of an integer $n > 1$ the complexity of $n$. For example, [i]complexity [/i] of numbers $4$ and $6$ is equal to $2$. Find all $n$ such that all integers between $n$ and $2n$ have complexity a) not greater than the complexity of $n$. b) less than the complexity of $n$. (Boris Frenkin)

2019 Cono Sur Olympiad, 4

Tags: number theory , diophantine equation , prime

Find all positive prime numbers $p,q,r,s$ so that $p^2+2019=26(q^2+r^2+s^2)$.

1978 IMO Longlists, 17

Tags: quadratic , number theory , prime , sum of squares

Prove that for any positive integers $x, y, z$ with $xy-z^2 = 1$ one can find non-negative integers $a, b, c, d$ such that $x = a^2 + b^2, y = c^2 + d^2, z = ac + bd$. Set $z = (2q)!$ to deduce that for any prime number $p = 4q + 1$, $p$ can be represented as the sum of squares of two integers.