Olympiad problems

For given positive integers $n$ and $p$, find neaessary and sufficient conditions for the system of equations $$x + py = n , \\ x + y = p^2$$ to have a solution $(x, y, z)$ of positive integers. Prove also that there is at most one such solution.

2009 Junior Balkan Team Selection Tests - Romania, 1

Tags: diophantine equation , diophantine , number theory

Find all non-negative integers $a,b,c,d$ such that $7^a= 4^b + 5^c + 6^d$.

2008 Costa Rica - Final Round, 5

Tags: diophantine equation , other problems

Let $ p$ be a prime number such that $ p\minus{}1$ is a perfect square. Prove that the equation $ a^{2}\plus{}(p\minus{}1)b^{2}\equal{}pc^{2}$ has infinite many integer solutions $ a$, $ b$ and $ c$ with $ (a,b,c)\equal{}1$

PEN H Problems, 86

Tags: diophantine equation , geometry , function , euler , least common multiple

A triangle with integer sides is called Heronian if its area is an integer. Does there exist a Heronian triangle whose sides are the arithmetic, geometric and harmonic means of two positive integers?

1986 Greece Junior Math Olympiad, 1

Tags: number theory , diophantine equation , diophantine

Find all pairs of integers $(x,y)$ such that $$(x+1)(y+1)(x+y)(x^2+y^2)=16x^2y^2$$

2006 Singapore Junior Math Olympiad, 1

Tags: diophantine , diophantine equation , number theory

Find all integers $x,y$ that satisfy the equation $x+y=x^2-xy+y^2$

2014 Indonesia MO Shortlist, N2

Tags: number theory , composite number , composite , diophantine equation

Suppose that $a, b, c, k$ are natural numbers with $a, b, c \ge 3$ which fulfill the equation $abc = k^2 + 1$. Show that at least one between $a - 1, b - 1, c -1$ is composite number.

1996 Poland - Second Round, 5

Tags: number theory , diophantine , diophantine equation

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

2015 Junior Balkan Team Selection Tests - Romania, 2

Tags: number theory , diophantine equation , romania jbmo tst

Solve in $\Bbb{N}^*$ the equation $$ 4^a \cdot 5^b - 3^c \cdot 11^d = 1.$$

2018 Lusophon Mathematical Olympiad, 4

Tags: positive integer , diophantine equation , number theory

Determine the pairs of positive integer numbers $m$ and $n$ that satisfy the equation $m^2=n^2 +m+n+2018$.

2010 Contests, 3

Tags: induction , diophantine equation , number theory

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

2016 Ecuador NMO (OMEC), 1

Tags: diophantine , diophantine equation

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

2014 Stars Of Mathematics, 1

Tags: number theory , diophantine equation

Prove that for any integer $n>1$ there exist infinitely many pairs $(x,y)$ of integers $1<x<y$, such that $x^n+y \mid x+y^n$. ([i]Dan Schwarz[/i])

2024 Bangladesh Mathematical Olympiad, P1

Tags: diophantine equation , number theory , divisibility , factorization

Find all non-negative integers $x, y$ such that\[x^3y+x+y=xy+2xy^2\]

2022 Kosovo & Albania Mathematical Olympiad, 4

Tags: diophantine equation , algebra

Let $A$ be the set of natural numbers $n$ such that the distance of the real number $n\sqrt{2022} - \frac13$ from the nearest integer is at most $\frac1{2022}$. Show that the equation $$20x + 21y = 22z$$ has no solutions over the set $A$.

2016 India PRMO, 1

Tags: algebra , diophantine equation , diophantine

Consider all possible integers $n \ge 0$ such that $(5 \cdot 3^m) + 4 = n^2$ holds for some corresponding integer $m \ge 0$. Find the sum of all such $n$.

1978 Dutch Mathematical Olympiad, 1

Tags: diophantine , diophantine equation , number theory

Prove that no integer $x$ and $y$ satisfy: $$3x^2 = 9 + y^3.$$