Olympiad problems

1962 Swedish Mathematical Competition, 3

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

2016 Saudi Arabia IMO TST, 1

Tags: diophantine equation

Let $k$ be a positive integer. Prove that there exist integers $x$ and $y$, neither of which divisible by $7$, such that \begin{align*} x^2 + 6y^2 = 7^k. \end{align*}

2004 Regional Competition For Advanced Students, 1

Tags: number theory , diophantine equation

Determine all integers $ a$ and $ b$, so that $ (a^3\plus{}b)(a\plus{}b^3)\equal{}(a\plus{}b)^4$

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

1998 USAMTS Problems, 1

Tags: number theory , diophantine equation

Several pairs of positive integers $(m ,n )$ satisfy the condition $19m + 90 + 8n = 1998$. Of these, $(100, 1 )$ is the pair with the smallest value for $n$. Find the pair with the smallest value for $m$.

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.

2012 Turkey Junior National Olympiad, 1

Tags: quadratic , calculus , integration , diophantine equation , number theory

Let $x, y$ be integers and $p$ be a prime for which \[ x^2-3xy+p^2y^2=12p \] Find all triples $(x,y,p)$.

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: $\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}$

1989 IMO Shortlist, 5

Tags: algebra , polynomial , root , equation , diophantine equation

Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]

2022 JBMO Shortlist, N3

Tags: number theory , diophantine equation

Find all quadruples of positive integers $(p, q, a, b)$, where $p$ and $q$ are prime numbers and $a > 1$, such that $$p^a = 1 + 5q^b.$$

OMMC POTM, 2022 7

Tags: diophantine equation , number theory

Find all ordered triples of positive integers $(a,b,c)$ where $$\left(a+\frac{1}{a}\right)\left(b+\frac{1}{b}\right)=c+\frac{1}{c}.$$ [i]Proposed by vsamc[/i]

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.

PEN H Problems, 72

Tags: diophantine equation , induction , number theory , relatively prime , absolute value

Find all pairs $(x, y)$ of positive rational numbers such that $x^{y}=y^{x}$.

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 $p^{2}+2q^{2}=r^{2}.$

2012 Dutch Mathematical Olympiad, 1

Tags: diophantine equation , divisible , number theory

Let $a, b, c$, and $d$ be four distinct integers. Prove that $(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)$ is divisible by $12$.

2021 Ukraine National Mathematical Olympiad, 5

Tags: diophantine equation , diophantine , number theory

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

2015 Saudi Arabia JBMO TST, 1

Tags: number theory , diophantine equation , diophantine

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

2003 Greece JBMO TST, 5

Tags: number theory , diophantine equation

Find integer solutions of $x^3+y^3-2xy+x+y+2=0$

1990 India National Olympiad, 2

Tags: calculus , integration , quadratic , number theory , diophantine equation

Determine all non-negative integral pairs $ (x, y)$ for which \[ (xy \minus{} 7)^2 \equal{} x^2 \plus{} y^2.\]

VMEO III 2006, 10.2

Tags: number theory , diophantine equation , diophantine

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