Olympiad problems

2011 China Northern MO, 3

Find all positive integer solutions $(x, y, z)$ of the equation $1 + 2^x \cdot 7^y=z^2$.

2013 Balkan MO Shortlist, N5

Tags: number theory , diophantine equation , diophantine , prime

Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^2$

2015 Belarus Team Selection Test, 1

Tags: number theory , diophantine equation , diophantine

Solve the equation in nonnegative integers $a,b,c$: $3^a+2^b+2015=3c!$ I.Gorodnin

2004 Gheorghe Vranceanu, 4

Tags: equation , diophantine

Given a natural prime $ p, $ find the number of integer solutions of the equation $ p+xy=p(x+y). $

1979 Czech And Slovak Olympiad IIIA, 1

Tags: number theory , diophantine equation , diophantine

Let $n$ be a given natural number. Determine the number of all orderer triples $(x, y, z)$ of non-negative integers $x, y, z$ that satisfy the equation $$x + 2y + 5z=10n.$$

1967 All Soviet Union Mathematical Olympiad, 089

Tags: number theory , diophantine

Find all the integers $x,y$ satisfying equation $x^2+x=y^4+y^3+y^2+y$.

2009 Postal Coaching, 6

Tags: diophantine , inequalities , number theory

Find all pairs $(m, n)$ of positive integers $m$ and $n$ for which one has $$\sqrt{ m^2 - 4} < 2\sqrt{n} - m < \sqrt{ m^2 - 2}$$

2009 Hanoi Open Mathematics Competitions, 8

Tags: number theory , diophantine

Find all the pairs of the positive integers such that the product of the numbers of any pair plus the half of one of the numbers plus one third of the other number is three times less than $1004$.

2013 Czech And Slovak Olympiad IIIA, 1

Tags: number theory , diophantine equation , diophantine

Find all pairs of integers $a, b$ for which equality holds $\frac{a^2+1}{2b^2-3}=\frac{a-1}{2b-1}$

2015 QEDMO 14th, 3

Tags: number theory , diophantine , diophantine equation

Are there any rational numbers $x,y$ with $x^2 + y^2 = 2015$?

1992 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: diophantine , diophantine equation , number theory

Determine the integers $0 \le a \le b \le c \le d$ such that: $$2^n= a^2 + b^2 + c^2 + d^2.$$

1993 Tournament Of Towns, (360) 3

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

Positive integers $a$, $b$ and $c$ are positive integers with greatest common divisor equal to $1$ (i.e. they have no common divisors greater than $1$), and $$\frac{ab}{a-b}=c$$ Prove that $a -b$ is a perfect square. (SL Berlov)

2021 Mediterranean Mathematics Olympiad, 2

Tags: number theory , diophantine equation , diophantine

For every sequence $p_1<p_2<\cdots<p_8$ of eight prime numbers, determine the largest integer $N$ for which the following equation has no solution in positive integers $x_1,\ldots,x_8$: $$p_1\, p_2\, \cdots\, p_8 \left( \frac{x_1}{p_1}+ \frac{x_2}{p_2}+ ~\cdots~ +\frac{x_8}{p_8} \right) ~~=~~ N $$ [i]Proposed by Gerhard Woeginger, Austria[/i]

2020 Chile National Olympiad, 4

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

Determine all three integers $(x, y, z)$ that are solutions of the system $$x + y -z = 6$$ $$x^3 + y^3 -z^3 = 414$$

2016 India PRMO, 2

Tags: algebra , factorial , equation , diophantine , diophantine equation , wilson

Find the number of integer solutions of the equation $x^{2016} + (2016! + 1!) x^{2015} + (2015! + 2!) x^{2014} + ... + (1! + 2016!) = 0$

2013 Saudi Arabia BMO TST, 3

Tags: number theory , diophantine , diophantine equation

Find all positive integers $x, y, z$ such that $2^x + 21^y = z^2$

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)

2009 IMAC Arhimede, 5

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

Find all natural numbers $x$ and $y$ such that $x^y-y^x=1$ .

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 $x^2y + xy^2 = 880$.

1969 Swedish Mathematical Competition, 1

Tags: number theory , diophantine equation , diophantine

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

2005 All-Russian Olympiad Regional Round, 8.7

Tags: number theory , diophantine , system of equations

Find all pairs $(x, y)$ of natural numbers such that $$x + y = a^n, x^2 + y^2 = a^m$$ for some natural $a, n, m$.

1981 All Soviet Union Mathematical Olympiad, 316

Tags: diophantine equation , diophantine , number theory

Find the natural solutions of the equation $x^3 - y^3 = xy + 61$.

2016 Regional Olympiad of Mexico Northeast, 1

Tags: number theory , diophantine equation , diophantine

Determine if there is any triple of nonnegative integers, not necessarily different, $(a, b, c)$ such that: $$a^3 + b^3 + c^3 = 2016$$

2005 Austria Beginners' Competition, 1

Tags: number theory , diophantine , diophantine equation

Show that there are no positive integers $a$ und $b$ such that $4a(a + 1) = b(b + 3)$

1996 Israel National Olympiad, 7

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

Find all positive integers $a,b,c$ such that $$\begin{cases} a^2 = 4(b+c) \\ a^3 -2b^3 -4c^3 =\frac12 abc \end {cases}$$