Olympiad problems

1994 Tuymaada Olympiad, 7

Prove that there are infinitely many natural numbers $a,b,c,u$ and $v$ with greatest common divisor $1$ satisfying the system of equations: $a+b+c=u+v$ and $a^2+b^2+c^2=u^2+v^2$

2022 Turkey Junior National Olympiad, 3

Tags: number theory , diophantine equation

Let $m, n, a, k$ be positive integers and $k>1$ such that the equality $$5^m+63n+49=a^k$$ holds. Find the minimum value of $k$.

2019 CMI B.Sc. Entrance Exam, 2

Tags: complex number , diophantine equation , algebra

$(a)$ Count the number of roots of $\omega$ of the equation $z^{2019} - 1 = 0 $ over complex numbers that satisfy \begin{align*} \vert \omega + 1 \vert \geq \sqrt{2 + \sqrt{2}} \end{align*} $(b)$ Find all real numbers $x$ that satisfy following equation $:$ \begin{align*} \frac{ 8^x + 27^x }{ 12^x + 18^x } = \frac{7}{6} \end{align*}

2019 Hanoi Open Mathematics Competitions, 7

Tags: number theory , diophantine , diophantine equation

Let $p$ and $q$ be odd prime numbers. Assume that there exists a positive integer $n$ such that $pq-1= n^3$. Express $p+q$ in terms of $n$

1986 IMO Shortlist, 4

Tags: number theory , diophantine equation , equation , algebra

Provided the equation $xyz = p^n(x + y + z)$ where $p \geq 3$ is a prime and $n \in \mathbb{N}$. Prove that the equation has at least $3n + 3$ different solutions $(x,y,z)$ with natural numbers $x,y,z$ and $x < y < z$. Prove the same for $p > 3$ being an odd integer.

1998 Korea - Final Round, 1

Tags: diophantine equation , ratio , number theory , relatively prime

Find all pairwise relatively prime positive integers $l, m, n$ such that \[(l+m+n)\left( \frac{1}{l}+\frac{1}{m}+\frac{1}{n}\right)\] is an integer.

2022 Korea Winter Program Practice Test, 1

Tags: number theory , diophantine equation

Prove that equation $y^2=x^3+7$ doesn't have any solution on integers.

2002 Federal Math Competition of S&M, Problem 3

Tags: number theory , diophantine equation

Find all pairs $(n,k)$ of positive integers such that $\binom nk=2002$.

2021 Olympic Revenge, 5

Tags: number theory , diophantine equation , arithmetic sequence , perfect square

Prove there aren't positive integers $a, b, c, d$ forming an arithmetic progression such that $ ab + 1, ac + 1, ad + 1, bc + 1, bd + 1, cd + 1 $ are all perfect squares.

2015 China Team Selection Test, 5

Tags: number theory , diophantine equation

FIx positive integer $n$. Prove: For any positive integers $a,b,c$ not exceeding $3n^2+4n$, there exist integers $x,y,z$ with absolute value not exceeding $2n$ and not all $0$, such that $ax+by+cz=0$

2016 Korea - Final Round, 3

Tags: number theory , diophantine equation

Prove that for all rationals $x,y$, $x-\frac{1}{x}+y-\frac{1}{y}=4$ is not true.

2010 Benelux, 4

Tags: number theory , diophantine equation

Find all quadruples $(a, b, p, n)$ of positive integers, such that $p$ is a prime and \[a^3 + b^3 = p^n\mbox{.}\] [i](2nd Benelux Mathematical Olympiad 2010, Problem 4)[/i]

PEN H Problems, 2

Tags: diophantine equation

The number $21982145917308330487013369$ is the thirteenth power of a positive integer. Which positive integer?

2023 SG Originals, Q2

Tags: number theory , diophantine equation

Find all positive integers $k$ such that there exists positive integers $a, b$ such that \[a^2+4=(k^2-4)b^2.\]

2011 Cuba MO, 2

Tags: diophantine , diophantine equation , number theory

Determine all the integer solutions of the equation $3x^4-2024y+1= 0$.

PEN H Problems, 59

Tags: diophantine equation , modular arithmetic

Solve the equation $28^x =19^y +87^z$, where $x, y, z$ are integers.

2013 District Olympiad, 1

Tags: number theory , diophantine equation , diophantine

Find all triples of integers $(x, y, z)$ such that $$x^2 + y^2 + z^2 = 16(x + y + z).$$

2007 Alexandru Myller, 1

Tags: perfect square , diophantine equation , number theory , divisibility , equation

Solve $ x^3-y^3=2xy+7 $ in integers.

2021 China Team Selection Test, 3

Tags: number theory , diophantine equation , algebra

Given positive integers $a,b,c$ which are pairwise coprime. Let $f(n)$ denotes the number of the non-negative integer solution $(x,y,z)$ to the equation $$ax+by+cz=n.$$ Prove that there exists constants $\alpha, \beta, \gamma \in \mathbb{R}$ such that for any non-negative integer $n$, $$|f(n)- \left( \alpha n^2+ \beta n + \gamma \right) | < \frac{1}{12} \left( a+b+c \right).$$

2024 CMI B.Sc. Entrance Exam, 5

Tags: diophantine equation , number theory , algebra

Find all solutions for positive integers $(x,y,k,m)$ such that \[ 20x^k+24y^m = 2024\] with $k, m > 1$

2014 Cuba MO, 1

Tags: number theory , diophantine , diophantine equation

Find all the integer solutions of the equation $ m^4 + 2n^2 = 9mn$.

1991 Tournament Of Towns, (291) 1

Tags: number theory , diophantine , diophantine equation

Find all natural numbers $n$, and all integers $x,y$ ($x\ne y$) for which the following equation is satisfied: $$x + x^2 + x^4 + ...+ x^{2^n} = y + y^2 + y^4 + ... + y^{2^n} .$$

2022 Czech-Polish-Slovak Junior Match, 4

Tags: diophantine equation , number theory , diophantine

Find all triples $(a, b, c)$ of integers that satisfy the equations $ a + b = c$ and $a^2 + b^3 = c^2$

1980 IMO Shortlist, 12

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

Find all pairs of solutions $(x,y)$: \[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]

2014 Indonesia MO Shortlist, N1

Tags: number theory , diophantine equation , prime

(a) Let $k$ be an natural number so that the equation $ab + (a + 1) (b + 1) = 2^k$ does not have a positive integer solution $(a, b)$. Show that $k + 1$ is a prime number. (b) Show that there are natural numbers $k$ so that $k + 1$ is prime numbers and equation $ab + (a + 1) (b + 1) = 2^k$ has a positive integer solution $(a, b)$.