Olympiad problems

2023 Junior Balkan Team Selection Tests - Moldova, 5

The positive integers $ a, b, c $ are the lengths of the sides of a right triangle. Prove that $abc$ is divisible by $60$.

2007 India IMO Training Camp, 2

Tags: modular arithmetic , diophantine equation , number theory

Find all integer solutions $(x,y)$ of the equation $y^2=x^3-p^2x,$ where $p$ is a prime such that $p\equiv 3 \mod 4.$

2020 Korea - Final Round, P4

Tags: number theory

Do there exist two positive reals $\alpha, \beta$ such that each positive integer appears exactly once in the following sequence? \[ 2020, [\alpha], [\beta], 4040, [2\alpha], [2\beta], 6060, [3\alpha], [3\beta], \cdots \] If so, determine all such pairs; if not, prove that it is impossible.

2013 Poland - Second Round, 1

Tags: trinomial , divisibility , number theory

Let $b$, $c$ be integers and $f(x) = x^2 + bx + c$ be a trinomial. Prove, that if for integers $k_1$, $k_2$ and $k_3$ values of $f(k_1)$, $f(k_2)$ and $f(k_3)$ are divisible by integer $n \neq 0$, then product $(k_1 - k_2)(k_2 - k_3)(k_3 - k_1)$ is divisible by $n$ too.

2011 Middle European Mathematical Olympiad, 8

Tags: modular arithmetic , arithmetic sequence , system of equations , number theory , greatest common divisor , relatively prime , algebra

We call a positive integer $n$ [i]amazing[/i] if there exist positive integers $a, b, c$ such that the equality \[n = (b, c)(a, bc) + (c, a)(b, ca) + (a, b)(c, ab)\] holds. Prove that there exist $2011$ consecutive positive integers which are [i]amazing[/i]. [b]Note.[/b] By $(m, n)$ we denote the greatest common divisor of positive integers $m$ and $n$.

2008 Baltic Way, 10

Tags: number theory

For a positive integer $ n$, let $ S(n)$ denote the sum of its digits. Find the largest possible value of the expression $ \frac {S(n)}{S(16n)}$.

2013 Danube Mathematical Competition, 2

Tags: multiple , sum , combinatorics , number theory

Consider $64$ distinct natural numbers, at most equal to $2012$. Show that it is possible to choose four of them, denoted as $a,b,c,d$ such that $ a+b-c-d$ to be a multiple of $2013$

2023 VN Math Olympiad For High School Students, Problem 3

Tags: number theory , algebra , prime number

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 $\mathbb{Z}[x].$

2023 India Regional Mathematical Olympiad, 3

Tags: number theory

For any natural number $n$, expressed in base 10 , let $s(n)$ denote the sum of all its digits. Find all natural numbers $m$ and $n$ such that $m<n$ and $$ (s(n))^2=m \text { and }(s(m))^2=n . $$

2008 Mongolia Team Selection Test, 1

Tags: number theory

Given an integer $ a$. Let $ p$ is prime number such that $ p|a$ and $ p \equiv \pm 3 (mod8)$. Define a sequence $ \{a_n\}_{n \equal{} 0}^\infty$ such that $ a_n \equal{} 2^n \plus{} a$. Prove that the sequence $ \{a_n\}_{n \equal{} 0}^\infty$ has finitely number of square of integer.

2018 Hanoi Open Mathematics Competitions, 15

Tags: diophantine equation , number theory

Find all pairs of prime numbers $(p,q)$ such that for each pair $(p,q)$, there is a positive integer m satisfying $\frac{pq}{p + q}=\frac{m^2 + 6}{m + 1}$.

2006 Argentina National Olympiad, 6

Tags: number theory

We will say that a natural number $n$ is [i]adequate[/i] if there exist $n$ integers $a_1,a_2,\ldots ,a_n$ (which are not necessarily positive and can be repeated) such that$$a_1+a_2+\cdots +a_n=a_1a_2 \cdots a_n=n.$$Determine all [i]adequate[/i] numbers.

2001 Balkan MO, 1

Tags: number theory , modular arithmetic

Let $a,b,n$ be positive integers such that $2^n - 1 =ab$. Let $k \in \mathbb N$ such that $ab+a-b-1 \equiv 0 \pmod {2^k}$ and $ab+a-b-1 \neq 0 \pmod {2^{k+1}}$. Prove that $k$ is even.

2011 Hanoi Open Mathematics Competitions, 7

Tags: number theory , divisible

How many positive integers a less than $100$ such that $4a^2 + 3a + 5$ is divisible by $6$.

2022 Canadian Mathematical Olympiad Qualification, 2

Tags: perfect square , number theory

Determine all pairs of integers $(m, n)$ such that $m^2 + n$ and $n^2 + m$ are both perfect squares.

2000 Tuymaada Olympiad, 5

Tags: number theory , prime , divisible

Are there prime $p$ and $q$ larger than $3$, such that $p^2-1$ is divisible by $q$ and $q^2-1$ divided by $p$?

2020 DMO Stage 1, 5.

Tags: number theory

Find the number of solutions to the given congruence$$x^{2}+y^{2}+z^{2} \equiv 2 a x y z \pmod p$$ where $p$ is an odd prime and $x,y,z \in \mathbb{Z}$. [i]Proposed by math_and_me[/i]

1983 All Soviet Union Mathematical Olympiad, 364

Tags: number theory , combinatorics

The kindergarten group is standing in the column of pairs. The number of boys equals the number of girls in each of the two columns. The number of mixed (boy and girl) pairs equals to the number of the rest pairs. Prove that the total number of children in the group is divisible by eight.

1991 Bundeswettbewerb Mathematik, 2

Tags: number theory , divide

Let $g$ be an even positive integer and $f(n) = g^n + 1$ , $(n \in N^* )$. Prove that for every positive integer $n$ we have: a) $f(n)$ divides each of the numbers $f(3n), f(5n), f(7n)$ b) $f(n)$ is relative prime to each of the numbers $f(2n), f(4n),f(6n),...$

2023 Bulgarian Autumn Math Competition, 9.4

Tags: number theory

Let $p, q$ be coprime integers, such that $|\frac{p} {q}| \leq 1$. For which $p, q$, there exist even integers $b_1, b_2, \ldots, b_n$, such that $$\frac{p} {q}=\frac{1}{b_1+\frac{1}{b_2+\frac{1}{b_3+\ldots}}}? $$

2004 Thailand Mathematical Olympiad, 13

Tags: remainder , number theory

Compute the remainder when $29^{30 }+ 31^{28} + 28! \cdot 30!$ is divided by $29 \cdot 31$.

2013 Saudi Arabia BMO TST, 8

Tags: number theory , sum , odd , algebra

Prove that the ratio $$\frac{1^1 + 3^3 + 5^5 + ...+ (2^{2013} - 1)^{(2^{2013} - 1)}}{2^{2013}}$$ is an odd integer.

2010 Tuymaada Olympiad, 4

Tags: floor function , polynomial , limit , number theory , algebra

Prove that for any positive real number $\alpha$, the number $\lfloor\alpha n^2\rfloor$ is even for infinitely many positive integers $n$.

2011 Switzerland - Final Round, 9

Tags: strong induction , ceiling function , induction , floor function , number theory

For any positive integer $n$ let $f(n)$ be the number of divisors of $n$ ending with $1$ or $9$ in base $10$ and let $g(n)$ be the number of divisors of $n$ ending with digit $3$ or $7$ in base $10$. Prove that $f(n)\geqslant g(n)$ for all nonnegative integers $n$. [i](Swiss Mathematical Olympiad 2011, Final round, problem 9)[/i]

2010 Saudi Arabia IMO TST, 1

Tags: algebra , number theory

Find all real numbers $x$ that can be written as $$x= \frac{a_0}{a_1a_2..a_n}+\frac{a_1}{a_2a_3..a_n}+\frac{a_2}{a_3a_4..a_n}+...+\frac{a_{n-2}}{a_{n-1}a_n}+\frac{a_{n-1}}{a_n}$$ where $n, a_1,a_2,...,a_n$ are positive integers and $1 = a_0 \le a_1 <... < a_n$