Olympiad problems

1982 Tournament Of Towns, (025) 3

Prove that the equation $m!n! = k!$ has infinitely many solutions in which $m, n$ and $k$ are natural numbers greater than unity .

2011 NZMOC Camp Selection Problems, 4

Tags: diophantine , diophantine equation , number theory

Find all pairs of positive integers $m$ and $n$ such that $$(m + 1)! + (n + 1)! = m^2n.$$

1998 Switzerland Team Selection Test, 2

Tags: diophantine , diophantine equation , number theory

Find all nonnegative integer solutions $(x,y,z)$ of the equation $\frac{1}{x+2}+\frac{1}{y+2}=\frac{1}{2} +\frac{1}{z+2}$

2016 Finnish National High School Mathematics Comp, 4

Tags: number theory , diophantine equation , diophantine

How many pairs $(a, b)$ of positive integers $a,b$ solutions of the equation $(4a-b)(4b-a )=1770^n$ exist , if $n$ is a positive integer?

2003 All-Russian Olympiad Regional Round, 11.1

Tags: number theory , diophantine , diophantine equation

Find all prime $p$, for each of which there are such natural $ x$ and $y$ such that $p^x = y^3 + 1$.

2014 Austria Beginners' Competition, 1

Tags: number theory , diophantine equation , diophantine

Determine all solutions of the diophantine equation $a^2 = b \cdot (b + 7)$ in integers $a\ge 0$ and $b \ge 0$. (W. Janous, Innsbruck)

2015 Cuba MO, 7

Tags: number theory , diophantine , diophantine equation

If $p$ is a prime number and $x, y$ are positive integers, find in terms of $p$, all pairs $(x, y)$ that satisfy the equation: $$p(x -2) = x(y -1).$$ If $x+y = 21$, find all triples $(x, y, p)$ that satisfy this equation.

1999 Estonia National Olympiad, 1

Tags: diophantine , diophantine equation , number theory

Find all pairs of integers $(m, n)$ such that $(m - n)^2 =\frac{4mn}{m + n - 1}$

Mathley 2014-15, 2

Tags: diophantine equation , number theory , diophantine

Let $n$ be a positive integer and $p$ a prime number $p > n + 1$. Prove that the following equation does not have integer solution $$1 + \frac{x}{n + 1} + \frac{x^2}{2n + 1} + ...+ \frac{x^p}{pn + 1} = 0$$ Luu Ba Thang, Department of Mathematics, College of Education

2011 Saudi Arabia Pre-TST, 3.4

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

Find all quadruples $(x,y,z,w)$ of integers satisfying the system of equations $$x + y + z + w = xy + yz + zx + w^2 - w = xyz - w^3 = - 1$$

2016 Abels Math Contest (Norwegian MO) Final, 2a

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

Find all positive integers $a, b, c, d$ with $a \le b$ and $c \le d$ such that $\begin{cases} a + b = cd \\ c + d = ab \end{cases}$ .

1945 Moscow Mathematical Olympiad, 103

Tags: diophantine , number theory

Solve in integers the equation $xy + 3x - 5y = - 3$.

2002 Junior Balkan Team Selection Tests - Romania, 2

Tags: number theory , diophantine , diophantine equation

Find all positive integers $a, b,c,d$ such that $a + b + c + d - 3 = ab + cd$.

1971 Polish MO Finals, 4

Tags: number theory , diophantine

Prove that if positive integers $x,y,z$ satisfy the equation $$x^n + y^n = z^n,$$ then $\min\, (x,y) \ge n$.

2011 Dutch Mathematical Olympiad, 1

Tags: number theory , factorial , diophantine , diophantine equation

Determine all triples of positive integers $(a, b, n)$ that satisfy the following equation: $a! + b! = 2^n$

2020 Switzerland - Final Round, 5

Tags: diophantine equation , factorial , number theory , diophantine

Find all the positive integers $a, b, c$ such that $$a! \cdot b! = a! + b! + c!$$

2012 NZMOC Camp Selection Problems, 3

Tags: number theory , diophantine equation , diophantine

Find all triples of positive integers $(x, y, z)$ with $$\frac{xy}{z}+ \frac{yz}{x}+\frac{zx}{y}= 3$$

2008 Mathcenter Contest, 6

Tags: number theory , diophantine equation , diophantine

Find the total number of integer solutions of the equation $$x^5-y^2=4$$ [i](Erken)[/i]

2015 HMIC, 5

Tags: cyclotomic field , roots of unity , diophantine

Let $\omega = e^{2\pi i /5}$ be a primitive fifth root of unity. Prove that there do not exist integers $a, b, c, d, k$ with $k > 1$ such that \[(a + b \omega + c \omega^2 + d \omega^3)^{k}=1+\omega.\] [i]Carl Lian[/i]

2009 Federal Competition For Advanced Students, P1, 1

Tags: factorial , exponential , exponential inequality , inequalities , algebra , diophantine , induction

Show that for all positive integer $n$ the following inequality holds $3^{n^2} > (n!)^4$ .

2018 Puerto Rico Team Selection Test, 1

Tags: number theory , diophantine , diophantine equation

Find all pairs $(a, b)$ of positive integers that satisfy the equation $a^2 -3 \cdot 2^b = 1$.

2012 QEDMO 11th, 1

Tags: diophantine equation , diophantine , number theory

Find all $x, y, z \in N_0$ with $(2^x + 1) (2^y-1) = 2^z-1$.

1905 Eotvos Mathematical Competition, 1

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

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 $7^a= 4^b + 5^c + 6^d$.

1986 Greece Junior Math Olympiad, 1

Tags: number theory , diophantine equation , diophantine

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