Olympiad problems

2016 Hanoi Open Mathematics Competitions, 8

Find all positive integers $x,y,z$ such that $x^3 - (x + y + z)^2 = (y + z)^3 + 34$

1996 Singapore MO Open, 4

Tags: diophantine , diophantine equation , number theory

Determine all the solutions of the equation $x^3 + y^3 + z^3 = wx^2y^2z^2$ in natural numbers $x, y, z, w$. Justify your answer

2019 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , diophantine , diophantine equation

Determine all positive integers $k$ for which there exist positive integers $n$ and $m, m\ge 2$, such that $3^k + 5^k = n^m$

2018 Costa Rica - Final Round, N3

Tags: number theory , diophantine

Let $a$ and $b$ be positive integers such that $2a^2 + a = 3b^2 + b$. Prove that $a-b$ is a perfect square.

2020 Austrian Junior Regional Competition, 4

Tags: number theory , diophantine equation , diophantine

Find all positive integers $a$ for which the equation $7an -3n! = 2020$ has a positive integer solution $n$. (Richard Henner)

2005 Thailand Mathematical Olympiad, 13

Tags: number theory , diophantine , diophantine equation , odd

Find all odd integers $k$ for which there exists a positive integer $m$ satisfying the equation $k + (k + 5) + (k + 10) + ... + (k + 5(m - 1)) = 1372$.

2015 Hanoi Open Mathematics Competitions, 14

Tags: diophantine , number theory , diophantine equation

Determine all pairs of integers $(x, y)$ such that $2xy^2 + x + y + 1 = x^2 + 2y^2 + xy$.

Russian TST 2019, P3

Tags: number theory , diophantine

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

2006 Bosnia and Herzegovina Junior BMO TST, 1

Tags: number theory , diophantine , diophantine equation

. Find all triplets $(x, y, z)$, $x > y > z$ of positive integers such that $\frac{1}{x}+\frac{2}{y}+\frac{3}{z}= 1$

1989 Tournament Of Towns, (226) 4

Tags: number theory , diophantine , diophantine equation

Find the positive integer solutions of the equation $$ x+ \frac{1}{y+ \frac{1}{z}}= \frac{10}{7}$$ (G. Galperin)

2017 Hanoi Open Mathematics Competitions, 6

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

Find all pairs of integers $a, b$ such that the following system of equations has a unique integral solution $(x , y , z )$ : $\begin{cases}x + y = a - 1 \\ x(y + 1) - z^2 = b \end{cases}$

2010 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , diophantine , prime

Determine the prime numbers $p, q, r$ with the property $\frac {1} {p} + \frac {1} {q} + \frac {1} {r} \ge 1$

2008 Regional Olympiad of Mexico Center Zone, 1

Tags: number theory , diophantine equation , diophantine

Find all pairs of integers $ a, b $ that satisfy $a ^2-3a = b ^3-2$.

2013 Balkan MO Shortlist, N3

Tags: number theory , diophantine equation , diophantine

Determine all quadruplets ($x, y, z, t$) of positive integers, such that $12^x + 13^y - 14^z = 2013^t$.

2021 Austrian MO Beginners' Competition, 4

Tags: number theory , diophantine

Let $p$ be a prime number and let $m$ and $n$ be positive integers with $p^2 + m^2 = n^2$. Prove that $m> p$. (Karl Czakler)

1969 Swedish Mathematical Competition, 1

Tags: number theory , diophantine equation , diophantine

Find all integers m, n such that $m^3 = n^3 + n$.

2006 QEDMO 2nd, 1

Tags: number theory , diophantine , diophantine equation

Solve the equation $x^{2}+y^{2}=10xy$ for integers $x$ and $y$

1997 Singapore MO Open, 3

Tags: divisor , number theory , factor , diophantine equation , diophantine

Find all the natural numbers $N$ which satisfy the following properties: (i) $N$ has exactly $6$ distinct factors $1, d_1, d_2, d_3, d_4, N$ and (ii) $1 + N = 5(d_1 + d_2+d_3 + d_4)$. Justify your answers.

2021 Bosnia and Herzegovina Junior BMO TST, 2

Tags: number theory , diophantine , diophantine equation

Let $p, q, r$ be prime numbers and $t, n$ be natural numbers such that $p^2 +qt =(p + t)^n$ and $p^2 + qr = t^4$ . a) Show that $n < 3$. b) Determine all the numbers $p, q, r, t, n$ that satisfy the given conditions.

2014 Dutch Mathematical Olympiad, 1

Tags: number theory , diophantine equation , diophantine

Determine all triples $(a,b,c)$, where $a, b$, and $c$ are positive integers that satisfy $a \le b \le c$ and $abc = 2(a + b + c)$.

2017 Hanoi Open Mathematics Competitions, 2

Tags: diophantine equation , diophantine , number theory

How many pairs of positive integers $(x, y)$ are there, those satisfy the identity $2^x - y^2 = 4$? (A): $1$ (B): $2$ (C): $3$ (D): $4$ (E): None of the above.

2009 Postal Coaching, 2

Tags: diophantine , diophantine equation , number theory

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

2016 Latvia Baltic Way TST, 17

Tags: prime , diophantine equation , diophantine , number theory

Can you find five prime numbers $p, q, r, s, t$ such that $p^3+q^3+r^3+s^3 =t^3$?

1999 Singapore Senior Math Olympiad, 1

Tags: diophantine , number theory , diophantine equation

Find all the integral solutions of the equation $\left( 1+\frac{1}{x}\right)^{x+1}=\left( 1+\frac{1}{1999}\right)^{1999}$

2008 Hanoi Open Mathematics Competitions, 3

Tags: number theory , diophantine , diophantine equation

Show that the equation $x^2 + 8z = 3 + 2y^2$ has no solutions of positive integers $x, y$ and $z$.