Olympiad problems

2006 Singapore Junior Math Olympiad, 1

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

1996 Poland - Second Round, 5

Tags: number theory , diophantine , diophantine equation

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

2016 Ecuador NMO (OMEC), 1

Tags: diophantine , diophantine equation

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

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: $$3x^2 = 9 + y^3.$$

1962 Swedish Mathematical Competition, 3

Tags: diophantine equation , diophantine , number theory

Find all pairs $(m, n)$ of integers such that $n^2 - 3mn + m - n = 0$.

2009 Belarus Team Selection Test, 4

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

Let $x,y,z$ be integer numbers satisfying the equality $yx^2+(y^2-z^2)x+y(y-z)^2=0$ a) Prove that number $xy$ is a perfect square. b) Prove that there are infinitely many triples $(x,y,z)$ satisfying the equality. I.Voronovich

2003 Dutch Mathematical Olympiad, 3

Tags: number theory , diophantine equation , diophantine , consecutive

Determine all positive integers$ n$ that can be written as the product of two consecutive integers and as well as the product of four consecutive integers numbers. In the formula: $n = a (a + 1) = b (b + 1) (b + 2) (b + 3)$.

1969 Dutch Mathematical Olympiad, 2

Tags: number theory , diophantine , diophantine equation

Prove that for all $n \in N$, $x^2 + y^2 = z^n$ has solutions with $x,y,z \in N$.

2016 Costa Rica - Final Round, A2

Tags: number theory , diophantine , diophantine equation

Find all integer solutions of the equation $p (x + y) = xy$, where $p$ is a prime number.

2016 Latvia Baltic Way TST, 19

Tags: number theory , diophantine equation , diophantine

Prove that for equation $$x^{2015} + y^{2015} = z^{2016}$$ there are infinitely many solutions where $x,y$ and $z$ are different natural numbers.

2017 Greece Junior Math Olympiad, 3

Tags: diophantine equation , diophantine , number theory

Find all triplets $(a,b,p)$ where $a,b$ are positive integers and $p$ is a prime number satisfying: $\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}$

2014 Hanoi Open Mathematics Competitions, 4

Tags: number theory , diophantine equation , diophantine , prime

If $p$ is a prime number such that there exist positive integers $a$ and $b$ such that $\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}$ then $p$ is (A): $3$, (B): $5$, (C): $11$, (D): $7$, (E) None of the above.

2022 Dutch BxMO TST, 3

Tags: diophantine equation , diophantine , number theory

Find all pairs $(p, q)$ of prime numbers such that $$p(p^2 -p - 1) = q(2q + 3).$$

2013 Junior Balkan Team Selection Tests - Moldova, 6

Tags: diophantine equation , diophantine , number theory

Determine all triplets of real numbers $(x, y, z)$ that satisfy the equation $4xyz = x^4 + y^4 + z^4 + 1$.

2007 QEDMO 4th, 1

Tags: diophantine equation , diophantine , number theory

Find all primes $p,$ $q,$ $r$ satisfying $p^{2}+2q^{2}=r^{2}.$

2021 Ukraine National Mathematical Olympiad, 5

Tags: diophantine equation , diophantine , number theory

Are there natural numbers $(m,n,k)$ that satisfy the equation $m^m+ n^n=k^k$ ?

2004 May Olympiad, 4

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

Find all the natural numbers $x, y, z$ that satisfy simultaneously $$\begin{cases} x y z=4104 \\ x+y+z=77 \end{cases}$$

2015 Saudi Arabia JBMO TST, 1

Tags: number theory , diophantine equation , diophantine

Find all the triples $(x,y,z)$ of positive integers such that $xy+yz+zx-xyz=2015$

VMEO III 2006, 10.2

Tags: number theory , diophantine equation , diophantine

Find all triples of integers $(x, y, z)$ such that $x^4 + 5y^4 = z^4$.

2012 Junior Balkan Team Selection Tests - Romania, 3

Tags: number theory , diophantine , diophantine equation

Let $m$ and $n$ be two positive integers, $m, n \ge 2$. Solve in the set of the positive integers the equation $x^n + y^n = 3^m$.

IV Soros Olympiad 1997 - 98 (Russia), 11.1

Tags: number theory , diophantine equation , diophantine

Solve the equation $xy =1997(x + y)$ in integers.

2013 Swedish Mathematical Competition, 1

Tags: number theory , diophantine , diophantine equation

For $r> 0$ denote by $B_r$ the set of points at distance at most $r$ length units from the origin. If $P_r$ is the set of the points in $B_r$ whit integer coordinates, show that the equation $$xy^3z + 2x^3z^3-3x^5y = 0$$ has an odd number of solutions $(x, y, z)$ in $P_r$.

2001 All-Russian Olympiad Regional Round, 11.1

Tags: number theory , diophantine , diophantine equation

Find all prime numbers $p$ and $q$ such that $p + q = (p -q)^3.$

2012 India Regional Mathematical Olympiad, 5

Tags: positive integer , number theory , diophantine equation , diophantine , prime

Determine with proof all triples $(a, b, c)$ of positive integers satisfying $\frac{1}{a}+ \frac{2}{b} +\frac{3}{c} = 1$, where $a$ is a prime number and $a \le b \le c$.