Olympiad problems

2011 Kyiv Mathematical Festival, 1

Solve the equation $mn =$ (gcd($m,n$))$^2$ + lcm($m, n$) in positive integers, where gcd($m, n$) – greatest common divisor of $m,n$, and lcm($m, n$) – least common multiple of $m,n$.

1999 AMC 12/AHSME, 4

Tags: number theory , prime number

Find the sum of all prime numbers between $ 1$ and $ 100$ that are simultaneously $ 1$ greater than a multiple of $ 4$ and $ 1$ less than a multiple of $ 5$. $ \textbf{(A)}\ 118\qquad \textbf{(B)}\ 137\qquad \textbf{(C)}\ 158\qquad \textbf{(D)}\ 187 \qquad \textbf{(E)}\ 245$

2004 Federal Math Competition of S&M, 1

Tags: number theory , diophantine equation

Find all pairs of positive integers $(a,b)$ such that $5a^b - b = 2004$.

2014 Contests, 4

Tags: algebra , functional equation , number theory

Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that \[ m^2 + f(n) \mid mf(m) +n \] for all positive integers $m$ and $n$.

2010 Slovenia National Olympiad, 1

Tags: prime number , number theory

Find all prime numbers $p, q$ and $r$ such that $p>q>r$ and the numbers $p-q, p-r$ and $q-r$ are also prime.

2014 Junior Regional Olympiad - FBH, 4

Tags: number theory , remainder

Positive integer $n$ when divided with number $3$ gives remainder $a$, when divided with $5$ has remainder $b$ and when divided with $7$ gives remainder $c$. Find remainder when dividing number $n$ with $105$ if $4a+3b+2c=30$

2023 Balkan MO, 3

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)

2023 Indonesia TST, 1

Tags: number theory

A number is called [i]Norwegian[/i] if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number. (Note: The total number of positive divisors of a Norwegian number is allowed to be larger than $3$.)

1991 IberoAmerican, 5

Tags: number theory , algebra , polynomial , quadratic

Let $P(x,\, y)=2x^{2}-6xy+5y^{2}$. Let us say an integer number $a$ is a value of $P$ if there exist integer numbers $b$, $c$ such that $P(b,\, c)=a$. a) Find all values of $P$ lying between 1 and 100. b) Show that if $r$ and $s$ are values of $P$, then so is $rs$.

2019 Czech and Slovak Olympiad III A, 5

Tags: number theory

Prove that there are infinitely many integers which cannot be expressed as $2^a+3^b-5^c$ for non-negative integers $a,b,c$.

2017 Moldova Team Selection Test, 10

Tags: number theory

Let $p$ be an odd prime. Prove that the number $$\left\lfloor \left(\sqrt{5}+2\right)^{p}-2^{p+1}\right\rfloor$$ is divisible by $20p$.

2014 Romania National Olympiad, 1

Tags: number theory

Find x, y, z $\in Z$\\$x^2+y^2+z^2=2^n(x+y+z)$\\$n\in N$

2021 Bolivian Cono Sur TST, 2

Tags: number theory , least common multiple , lcm

Find all posible pairs of positive integers $x,y$ such that $$\text{lcm}(x,y+3001)=\text{lcm}(y,x+3001)$$

2007 Romania National Olympiad, 3

Tags: number theory

For which integers $n\geq 2$, the number $(n-1)^{n^{n+1}}+(n+1)^{n^{n-1}}$ is divisible by $n^{n}$ ?

1971 Dutch Mathematical Olympiad, 4

Tags: number theory , sum

For every positive integer $n$ there exist unambiguously determined non-negative integers $a(n)$ and $b(n)$ such that $$n = 2^{a(n)}(2b(n)+1),$$ For positive integer $k$ we define $S(k)$ by: $$a(1) + a(2) + ... + a(2^k) = S(k)$$ Express $S(k)$ in terms of $k$.

2015 Bundeswettbewerb Mathematik Germany, 2

Tags: number theory , fraction , decimal representation

In the decimal expansion of a fraction $\frac{m}{n}$ with positive integers $m$ and $n$ you can find a string of numbers $7143$ after the comma. Show $n>1250$. [i]Example:[/i] I mean something like $0.7143$.

1969 IMO Longlists, 25

Tags: number theory , divisibility , frobenius , additive number theory

$(GBR 2)$ Let $a, b, x, y$ be positive integers such that $a$ and $b$ have no common divisor greater than $1$. Prove that the largest number not expressible in the form $ax + by$ is $ab - a - b$. If $N(k)$ is the largest number not expressible in the form $ax + by$ in only $k$ ways, find $N(k).$

2023 Ukraine National Mathematical Olympiad, 11.1

Tags: number theory , divisibility

Set $M$ contains $n \ge 2$ positive integers. It's known that for any two different $a, b \in M$, $a^2+1$ is divisible by $b$. What is the largest possible value of $n$? [i]Proposed by Oleksiy Masalitin[/i]

1964 Bulgaria National Olympiad, Problem 1

Tags: number theory , digit

A $6n$-digit number is divisible by $7$. Prove that if its last digit is moved to the beginning of the number then the new number is also divisible by $7$.

2021 SG Originals, Q5

Tags: algebra , number theory , diophantine equation

Find all $a,b \in \mathbb{N}$ such that $$2049^ba^{2048}-2048^ab^{2049}=1.$$ [i]Proposed by fattypiggy123 and 61plus[/i]

1936 Eotvos Mathematical Competition, 3

Tags: number theory

Let $a$ be any positive integer. Prove that there exists a unique pair of positive integers $x$ and $y$ such that $$x +\frac12 (x + y - 1)(x + y- 2) = a.$$

2023 Belarusian National Olympiad, 10.6

Tags: number theory

Prove that for any positive integer $n$ there exists a positive integer $k$ such that $3^k+4^k-1 \vdots 12^n$

2015 EGMO, 4

Tags: number theory , sequence , integer sequence

Determine whether there exists an infinite sequence $a_1, a_2, a_3, \dots$ of positive integers which satisfies the equality \[a_{n+2}=a_{n+1}+\sqrt{a_{n+1}+a_{n}} \] for every positive integer $n$.

1981 IMO, 3

Tags: number theory , algebra , polynomial , vieta , diophantine equation

Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.

2011 Romania Team Selection Test, 4

Tags: number theory

Show that: a) There are infinitely many positive integers $n$ such that there exists a square equal to the sum of the squares of $n$ consecutive positive integers (for instance, $2$ is one such number as $5^2=3^2+4^2$). b) If $n$ is a positive integer which is not a perfect square, and if $x$ is an integer number such that $x^2+(x+1)^2+...+(x+n-1)^2$ is a perfect square, then there are infinitely many positive integers $y$ such that $y^2+(y+1)^2+...+(y+n-1)^2$ is a perfect square.