Olympiad problems

2013 QEDMO 13th or 12th, 9

Are there infinitely many different natural numbers $a_1,a_2, a_3,...$ so that for every integer $k$ only finitely many of the numbers $a_1 + k$,$a_2 + k$,$a_3 + k$,$...$ are numbers prime?

2015 IFYM, Sozopol, 5

Tags: prime number , divisibility , prime divisor , number theory

Does there exist a natural number $n$ with exactly 3 different prime divisors $p$, $q$, and $r$, so that $p-1\mid n$, $qr-1\mid n$, $q-1\nmid n$, $r-1\nmid n$, and $3\nmid q+r$?

2023 Korea Junior Math Olympiad, 7

Tags: number theory

Find the smallest positive integer $N$ such that there are no different sets $A, B$ that satisfy the following conditions. (Here, $N$ is not a power of $2$. That is, $N \neq 1, 2^1, 2^2, \dots$.) [list] [*] $A, B \subseteq \{1, 2^1, 2^2, 2^3, \dots, 2^{2023}\} \cup \{ N \}$ [*] $|A| = |B| \geq 1$ [*] Sum of elements in $A$ and sum of elements in $B$ are equal. [/list]

2016 India Regional Mathematical Olympiad, 5

Tags: sum of squares , number theory

a.) A 7-tuple $(a_1,a_2,a_3,a_4,b_1,b_2,b_3)$ of pairwise distinct positive integers with no common factor is called a shy tuple if $$ a_1^2+a_2^2+a_3^2+a_4^2=b_1^2+b_2^2+b_3^2$$and for all $1 \le i<j \le 4$ and $1 \le k \le 3$, $a_i^2+a_j^2 \not= b_k^2$. Prove that there exists infinitely many shy tuples. b.) Show that $2016$ can be written as a sum of squares of four distinct natural numbers.

1992 AIME Problems, 1

Tags: totient function , number theory , prime number , search , relatively prime , function

Find the sum of all positive rational numbers that are less than $10$ and that have denominator $30$ when written in lowest terms.

2023 Olympic Revenge, 2

Tags: diophantine equation , number theory

Find all triples ($a$,$b$,$n$) of positive integers such that $$a^3=b^2+2^n$$

2025 Harvard-MIT Mathematics Tournament, 5

Tags: algebra , number theory

Let $\mathcal{S}$ be the set of all nonconstant polynomials $P$ with integer coefficients satisfying $P(\sqrt{3}+\sqrt{2})=P(\sqrt{3}-\sqrt{2}).$ If $Q$ is an element of $\mathcal{S}$ with minimal degree, compute the only possible value of $Q(10)-Q(0).$

1987 China Team Selection Test, 2

Tags: number theory

Find all positive integer $n$ such that the equation $x^3+y^3+z^3=n \cdot x^2 \cdot y^2 \cdot z^2$ has positive integer solutions.

2019 Switzerland Team Selection Test, 7

Tags: number theory

Prove that for all positive integers $n$ there are positive integers $a,b$ such that $$n\mid 4a^2+9b^2-1.$$

PEN E Problems, 31

Tags: number theory

Suppose $n$ and $r$ are nonnegative integers such that no number of the form $n^2+r-k(k+1) \text{ }(k\in\mathbb{N})$ equals to $-1$ or a positive composite number. Show that $4n^2+4r+1$ is $1$, $9$, or prime.

2007 Balkan MO Shortlist, N2

Tags: function , number theory

Prove that there are no distinct positive integers $x$ and $y$ such that $x^{2007} + y! = y^{2007} + x! $

MOAA Team Rounds, 2022.14

Tags: number theory

Find the greatest prime number $p$ for which there exists a prime number $q$ such that $p$ divides $4^q + 1$ and $q$ divides $4^p + 1$.

2007 India Regional Mathematical Olympiad, 2

Tags: number theory , search , greatest common divisor

Let $ a, b, c$ be three natural numbers such that $ a < b < c$ and $ gcd (c \minus{} a, c \minus{} b) \equal{} 1$. Suppose there exists an integer $ d$ such that $ a \plus{} d, b \plus{} d, c \plus{} d$ form the sides of a right-angled triangle. Prove that there exist integers, $ l,m$ such that $ c \plus{} d \equal{} l^{2} \plus{} m^{2} .$ [b][Weightage 17/100][/b]

2015 Saudi Arabia IMO TST, 3

Tags: sum of digits , combinatorics , number theory

Find the number of binary sequences $S$ of length $2015$ such that for any two segments $I_1, I_2$ of $S$ of the same length, we have • The sum of digits of $I_1$ differs from the sum of digits of $I_2$ by at most $1$, • If $I_1$ begins on the left end of S then the sum of digits of $I_1$ is not greater than the sum of digits of $I_2$, • If $I_2$ ends on the right end of S then the sum of digits of $I_2$ is not less than the sum of digits of $I_1$. Lê Anh Vinh

1997 Estonia Team Selection Test, 3

Tags: number theory

It is known that for every integer $n > 1$ there is a prime number among the numbers $n+1,n+2,...,2n-1.$ Determine all positive integers $n$ with the following property: Every integer $m > 1$ less than $n$ and coprime to $n$ is prime.

2016 Japan Mathematical Olympiad Preliminary, 2

Tags: number theory

For $1\leq n\leq 2016$, how many integers $n$ satisfying the condition: the reminder divided by $20$ is smaller than the one divided by $16$.

2017 Olympic Revenge, 1

Tags: number theory

Prove that does not exist positive integers $a$, $b$ and $k$ such that $4abk-a-b$ is a perfect square.

2022 Czech-Polish-Slovak Junior Match, 2

Tags: diophantine , diophantine equation , number theory

Solve the following system of equations in integer numbers: $$\begin{cases} x^2 = yz + 1 \\ y^2 = zx + 1 \\ z^2 = xy + 1 \end{cases}$$

2024 China Team Selection Test, 7

Tags: number theory

For coprime positive integers $a,b$,denote $(a^{-1}\bmod{b})$ by the only integer $0\leq m<b$ such that $am\equiv 1\pmod{b}$ (1)Prove that for pairwise coprime integers $a,b,c$, $1<a<b<c$,we have\[(a^{-1}\bmod{b})+(b^{-1}\bmod{c})+(c^{-1}\bmod{a})>\sqrt a.\] (2)Prove that for any positive integer $M$,there exists pairwise coprime integers $a,b,c$, $M<a<b<c$ such that \[(a^{-1}\bmod{b})+(b^{-1}\bmod{c})+(c^{-1}\bmod{a})< 100\sqrt a.\]

1999 Moldova Team Selection Test, 12

Tags: number theory

Solve the equation in postive integers $$x^2+y^2+1998=1997x-1999y.$$

2019 Bosnia and Herzegovina EGMO TST, 2

Tags: divisor , number theory

Let $1 = d_1 < d_2 < ...< d_k = n$ be all natural divisors of the natural number $n$. Find all possible values of the number $k$ if $n=d_2d_3 + d_2d_5+d_3d_5$.

2020 Bulgaria National Olympiad, P4

Tags: number theory

Are there positive integers $m>4$ and $n$, such that a) ${m \choose 3}=n^2;$ b) ${m \choose 4}=n^2+9?$

2011 JBMO Shortlist, 3

Tags: number theory

Find all positive integers $n$ such that the equation $y^2 + xy + 3x = n(x^2 + xy + 3y)$ has at least a solution $(x, y)$ in positive integers.

1973 Poland - Second Round, 6

Tags: number theory , greatest common divisor

Prove that for every non-negative integer $m$ there exists a polynomial w with integer coefficients such that $2^m$ is the greatest common divisor of the numbers $$ a_n = 3^n + w(n), n = 0, 1, 2, ....$$

2023 Balkan MO Shortlist, N3

Tags: number theory

For each positive integer $n$, denote by $\omega(n)$ the number of distinct prime divisors of $n$ (for example, $\omega(1)=0$ and $\omega(12)=2$). Find all polynomials $P(x)$ with integer coefficients, such that whenever $n$ is a positive integer satisfying $\omega(n)>2023^{2023}$, then $P(n)$ is also a positive integer with \[\omega(n)\ge\omega(P(n)).\] Greece (Minos Margaritis - Iasonas Prodromidis)