Olympiad problems

1999 Estonia National Olympiad, 1

Prove that if $p$ is an odd prime, then $p^2(p^2 -1999)$ is divisible by $6$ but not by $12$.

1997 Israel National Olympiad, 5

Tags: coprime , number theory , primes

The natural numbers $a_1,a_2,...,a_n, n \ge 12$, are smaller than $9n^2$ and pairwise coprime. Show that at least one of these numbers is prime.

2023 USAMO, 5

Tags: AMC , USA(J)MO , USAMO , arithmetic sequence , modular arithmetic , primes

Let $n\geq3$ be an integer. We say that an arrangement of the numbers $1$, $2$, $\dots$, $n^2$ in a $n \times n$ table is [i]row-valid[/i] if the numbers in each row can be permuted to form an arithmetic progression, and [i]column-valid[/i] if the numbers in each column can be permuted to form an arithmetic progression. For what values of $n$ is it possible to transform any row-valid arrangement into a column-valid arrangement by permuting the numbers in each row?

2001 Estonia Team Selection Test, 5

Tags: primes , Product , Power , number theory , Digits

Find the exponent of $37$ in the representation of the number $111...... 11$ with $3\cdot 37^{2000}$ digits equals to $1$, as product of prime powers

2009 VJIMC, Problem 2

Tags: number theory , primes

Prove that the number $$2^{2^k-1}-2^k-1$$is composite (not prime) for all positive integers $k>2$.

2022 Ecuador NMO (OMEC), 6

Tags: number theory , primes , national olympiad

Prove that for all prime $p \ge 5$, there exist an odd prime $q \not= p$ such that $q$ divides $(p-1)^p + 1$

2012 Czech-Polish-Slovak Junior Match, 2

Tags: Diophantine equation , diophantine , primes , number theory

Determine all three primes $(a, b, c)$ that satisfied the equality $a^2+ab+b^2=c^2+3$.

2022 Mexico National Olympiad, 1

Tags: primes , reciprocals

A number $x$ is "Tlahuica" if there exist prime numbers $p_1,\ p_2,\ \dots,\ p_k$ such that \[x=\frac{1}{p_1}+\frac{1}{p_2}+\dots+\frac{1}{p_k}.\] Find the largest Tlahuica number $x$ such that $0<x<1$ and there exists a positive integer $m\leq 2022$ such that $mx$ is an integer.

2025 Bulgarian Winter Tournament, 11.4

Tags: Sequence , Divisibility , primes , infinite prime divisors , function , algebra , polynomial

Let $A$ be a set of $2025$ non-negative integers and $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ be a function with the following two properties: 1) For every two distinct positive integers $x,y$ there exists $a\in A$, such that $x-y$ divides $f(x+a) - f(y+a)$. 2) For every positive integer $N$ there exists a positive integer $t$ such that $f(x) \neq f(y)$ whenever $x,y \in [t, t+N]$ are distinct. Prove that there are infinitely many primes $p$ such that $p$ divides $f(x)$ for some positive integer $x$.

2016 Croatia Team Selection Test, Problem 4

Tags: number theory , Diophantine equation , prime numbers , primes

Find all pairs $(p,q)$ of prime numbers such that $$ p(p^2 - p - 1) = q(2q + 3) .$$

2017 Peru IMO TST, 10

Tags: squarefree , polynomial , Integer , primes , number theory

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.

2023 India IMO Training Camp, 3

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

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 $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

2012 Switzerland - Final Round, 4

Tags: number theory , prime , primes , recurrence relation

Show that there is no infinite sequence of primes $p_1, p_2, p_3, . . .$ there any for each $ k$: $p_{k+1} = 2p_k - 1$ or $p_{k+1} = 2p_k + 1$ is fulfilled. Note that not the same formula for every $k$.

2023 Germany Team Selection Test, 1

Tags: power of 2 , number theory , primes

Does there exist a positive odd integer $n$ so that there are primes $p_1$, $p_2$ dividing $2^n-1$ with $p_1-p_2=2$?

2004 Mexico National Olympiad, 1

Tags: primes , Perfect Square , number theory

Find all the prime number $p, q$ and r with $p < q < r$, such that $25pq + r = 2004$ and $pqr + 1 $ is a perfect square.

Mathley 2014-15, 7

Tags: number theory , primes , Diophantine equation , diophantine

Find all primes $p,q, r$ such that $\frac{p^{2q}+q^{2p}}{p^3-pq+q^3} = r$. Titu Andreescu, Mathematics Department, College of Texas, USA

2017 International Zhautykov Olympiad, 2

Tags: number theory , primes , Divisors

For each positive integer $k$ denote $C(k)$ to be sum of its distinct prime divisors. For example $C(1)=0,C(2)=2,C(45)=8$. Find all positive integers $n$ for which $C(2^n+1)=C(n)$.

2001 Estonia Team Selection Test, 5

Tags: primes , Product , Power , number theory , Digits

Find the exponent of $37$ in the representation of the number $111...... 11$ with $3\cdot 37^{2000}$ digits equals to $1$, as product of prime powers

2006 Junior Tuymaada Olympiad, 2

Tags: number theory , power of number , powers , primes

Ten different odd primes are given. Is it possible that for any two of them, the difference of their sixteenth powers to be divisible by all the remaining ones ?

2013 Ukraine Team Selection Test, 11

Tags: number theory , primes , Divisibility

Specified natural number $a$. Prove that there are an infinite number of prime numbers $p$ such that for some natural $n$ the number $2^{2^n} + a$ is divisible by $p$.

2023 Indonesia TST, 1

Tags: number theory , divsibility , primes

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

2021 Puerto Rico Team Selection Test, 6

Tags: number theory , primes

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 $2021$ .

2021 Romanian Master of Mathematics Shortlist, N2

Tags: number theory , primes , Combinatorial Number Theory , RMM Shortlist

We call a set of positive integers [i]suitable [/i] if none of its elements is coprime to the sum of all elements of that set. Given a real number $\varepsilon \in (0,1)$, prove that, for all large enough positive integers $N$, there exists a suitable set of size at least $\varepsilon N$, each element of which is at most $N$.

2020 Dutch BxMO TST, 5

Tags: number theory , primes , Product

A set S consisting of $2019$ (different) positive integers has the following property: [i]the product of every 100 elements of $S$ is a divisor of the product of the remaining $1919$ elements[/i]. What is the maximum number of prime numbers that $S$ can contain?

2015 Taiwan TST Round 2, 3

Tags: IMO Shortlist , number theory , primes , IMO Shortlist 2014 , N5

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]