Olympiad problems

2005 Austrian-Polish Competition, 9

Consider the equation $x^3 + y^3 + z^3 = 2$. a) Prove that it has infinitely many integer solutions $x,y,z$. b) Determine all integer solutions $x, y, z$ with $|x|, |y|, |z| \leq 28$.

2014 Ukraine Team Selection Test, 9

Tags: number theory , prime number , parameterization , diophantine , system of equations , diophantine equation

Let $m, n$ be odd prime numbers. Find all pairs of integers numbers $a, b$ for which the system of equations: $x^m+y^m+z^m=a$, $x^n+y^n+z^n=b$ has many solutions in integers $x, y, z$.

PEN H Problems, 14

Tags: diophantine equation

Show that the equation $x^2 +y^5 =z^3$ has infinitely many solutions in integers $x, y, z$ for which $xyz \neq 0$.

1978 Yugoslav Team Selection Test, Problem 1

Tags: diophantine equation , number theory

Find all integers $x,y,z$ such that $x^2(x^2+y)=y^{z+1}$.

2007 Puerto Rico Team Selection Test, 2

Tags: number theory , diophantine equation , diophantine , algebra , system of equations

Find the solutions of positive integers for the system $xy + x + y = 71$ and $x^2y + xy^2 = 880$.

PEN H Problems, 67

Tags: diophantine equation

Is there a positive integer $m$ such that the equation \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{abc}= \frac{m}{a+b+c}\] has infinitely many solutions in positive integers $a, b, c \;$?

2016 Ecuador Juniors, 2

Tags: diophantine , diophantine equation

Prove that there are no positive integers $x, y$ such that: $(x + 1)^2 + (x + 2)^2 +...+ (x + 9)^2 = y^2$

2021 Federal Competition For Advanced Students, P2, 3

Tags: number theory , diophantine equation , diophantine

Find all triples $(a, b, c)$ of natural numbers $a, b$ and $c$, for which $a^{b + 20} (c-1) = c^{b + 21} - 1$ is satisfied. (Walther Janous)

2005 Estonia Team Selection Test, 3

Tags: diophantine equation , diophantine , number theory

Find all pairs $(x, y)$ of positive integers satisfying the equation $(x + y)^x = x^y$.

1995 Irish Math Olympiad, 2

Tags: diophantine equation , number theory

Determine all integers $ a$ for which the equation $ x^2\plus{}axy\plus{}y^2\equal{}1$ has infinitely many distinct integer solutions $ x,y$.

2017 USA TSTST, 4

Tags: factorial , number theory , diophantine equation

Find all nonnegative integer solutions to $2^a + 3^b + 5^c = n!$. [i]Proposed by Mark Sellke[/i]

2010 Germany Team Selection Test, 3

Tags: induction , diophantine equation , number theory

Determine all $(m,n) \in \mathbb{Z}^+ \times \mathbb{Z}^+$ which satisfy $3^m-7^n=2.$

2021 Israel TST, 4

Tags: diophantine equation , number theory

Let $r$ be a positive integer and let $a_r$ be the number of solutions to the equation $3^x-2^y=r$ ,such that $0\leq x,y\leq 5781$ are integers. What is the maximal value of $a_r$?

1997 Tournament Of Towns, (558) 3

Tags: number theory , diophantine , diophantine equation

Prove that the equation $$xy(x -y) + yz(y-z) + zx(z-x) = 6$$ has infinitely many solutions in integers $x, y$ and $z$. (N Vassiliev)

2009 Irish Math Olympiad, 3

Tags: diophantine equation , number theory , modular arithmetic

Find all pairs $(a,b)$ of positive integers such that $(ab)^2 - 4(a+b)$ is the square of an integer.

2015 BAMO, 3

Tags: algebra , diophantine equation

Let $k$ be a positive integer. Prove that there exist integers $x$ and $y$, neither of which is divisible by $3$, such that $x^2+2y^2 = 3^k$.

1997 Slovenia National Olympiad, Problem 1

Tags: diophantine equation , number theory

Marko chose two prime numbers $a$ and $b$ with the same number of digits and wrote them down one after another, thus obtaining a number $c$. When he decreased $c$ by the product of $a$ and $b$, he got the result $154$. Determine the number $c$.

2020 OMpD, 1

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

Determine all pairs of positive integers $(x, y)$ such that: $$x^4 - 6x^2 + 1 = 7\cdot 2^y$$

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 $k$.

1966 Bulgaria National Olympiad, Problem 1

Tags: number theory , diophantine equation

Prove that the equation $$3x(x-3y)=y^2+z^2$$doesn't have any integer solutions except $x=0,y=0,z=0$.

2015 India PRMO, 14

Tags: diophantine equation , algebra , number theory

$14.$ If $3^x+2^y=985.$ and $3^x-2^y=473.$ What is the value of $xy ?$

2014 Contests, 1

Tags: modular arithmetic , diophantine equation

Find all triples of primes $(p,q,r)$ satisfying $3p^{4}-5q^{4}-4r^{2}=26$.

2013 Korea Junior Math Olympiad, 3

Tags: algebra , sequence , recurrence relation , sum , positive integer , diophantine equation

$\{a_n\}$ is a positive integer sequence such that $a_{i+2} = a_{i+1} +a_i$ (for all $i \ge 1$). For positive integer $n$, define as $$b_n=\frac{1}{a_{2n+1}}\Sigma_{i=1}^{4n-2}a_i$$ Prove that $b_n$ is positive integer.

1966 Polish MO Finals, 1

Tags: diophantine , diophantine equation , number theory

Solve in integers the equation $$x^4 +4y^4 = 2(z^4 +4u^4)$$

2016 Belarus Team Selection Test, 3

Tags: number theory , diophantine equation

Solve the equation $p^3-q^3=pq^3-1$ in primes $p,q$.