Olympiad problems

2011 Chile National Olympiad, 1

Find all the solutions $(a, b, c)$ in the natural numbers, verifying $1\le a \le b \le c$, of the equation$$\frac34=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$$

2015 India Regional MathematicaI Olympiad, 3

Tags: diophantine equation , system of equations , number theory

Find all integers $a,b,c$ such that $a^{2}=bc+4$ and $b^{2}=ca+4$.

2024 Junior Balkan MO, 3

Tags: number theory , diophantine equation , exponent

Find all triples of positive integers $(x, y, z)$ that satisfy the equation $$2020^x + 2^y = 2024^z.$$ [i]Proposed by Ognjen Tešić, Serbia[/i]

1984 IMO Shortlist, 11

Tags: equation , polynomial , diophantine equation , number theory

Let $n$ be a positive integer and $a_1, a_2, \dots , a_{2n}$ mutually distinct integers. Find all integers $x$ satisfying \[(x - a_1) \cdot (x - a_2) \cdots (x - a_{2n}) = (-1)^n(n!)^2.\]

PEN H Problems, 27

Tags: modular arithmetic , diophantine equation , inradius , perimeter , geometry

Prove that there exist infinitely many positive integers $n$ such that $p=nr$, where $p$ and $r$ are respectively the semi-perimeter and the inradius of a triangle with integer side lengths.

1999 Estonia National Olympiad, 1

Tags: number theory , diophantine , diophantine equation

Find all pairs of integers ($a, b$) such that $a^2 + b = b^{1999}$ .

1954 Moscow Mathematical Olympiad, 262

Tags: diophantine , number theory , diophantine equation

Are there integers $m$ and $n$ such that $m^2 + 1954 = n^2$?

2003 Irish Math Olympiad, 1

Tags: number theory , diophantine equation

find all solutions, not necessarily positive integers for $(m^2+ n)(m+ n^2)= (m+ n)^3$

2005 USAMO, 2

Tags: modular arithmetic , algebra , diophantine equation

Prove that the system \begin{align*} x^6+x^3+x^3y+y & = 147^{157} \\ x^3+x^3y+y^2+y+z^9 & = 157^{147} \end{align*} has no solutions in integers $x$, $y$, and $z$.

2002 Dutch Mathematical Olympiad, 2

Tags: diophantine , diophantine equation , number theory

Determine all triplets $(x, y, z)$ of positive integers with $x \le y \le z$ that satisfy $\left(1+\frac1x \right)\left(1+\frac1y \right)\left(1+\frac1z \right) = 3$

2010 China Northern MO, 3

Tags: diophantine , number theory , diophantine equation

Find all positive integer triples $(x, y, z)$ such that $1 + 2^x \cdot 3^y=5^z$ is true.

2015 Indonesia MO Shortlist, N4

Tags: greatest common divisor , exponential equation , exponential , diophantine equation , number theory

Suppose that the natural number $a, b, c, d$ satisfy the equation $a^ab^{a + b} = c^cd^{c + d}$. (a) If gcd $(a, b) = $ gcd $(c, d) = 1$, prove that $a = c$ and $b = d$. (b) Does the conclusion $a = c$ and $b = d$ apply, without the condition gcd $(a, b) = $ gcd $(c, d) = 1$?

PEN H Problems, 2

Tags: diophantine equation

The number $21982145917308330487013369$ is the thirteenth power of a positive integer. Which positive integer?

2016 Ecuador Juniors, 2

Tags: diophantine , diophantine equation

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

2012 Czech-Polish-Slovak Junior Match, 2

Tags: prime , diophantine , number theory , diophantine equation

Determine all three primes $(a, b, c)$ that satisfied the equality $a^2+ab+b^2=c^2+3$.

PEN H Problems, 71

Tags: modular arithmetic , diophantine equation

Let $n$ be a positive integer. Prove that the equation \[x+y+\frac{1}{x}+\frac{1}{y}=3n\] does not have solutions in positive rational numbers.

2022 Federal Competition For Advanced Students, P1, 4

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

Find all triples $(p, q, r)$ of prime numbers for which $4q - 1$ is a prime number and $$\frac{p + q}{p + r} = r - p$$ holds. [i](Walther Janous)[/i]

1998 Junior Balkan Team Selection Tests - Romania, 1

Tags: algebra , equation , diophantine equation

Solve in $ \mathbb{Z}^2 $ the following equation: $$ (x+1)(x+2)(x+3) +x(x+2)(x+3)+x(x+1)(x+3)+x(x+1)(x+2)=y^{2^x} . $$ [i]Adrian Zanoschi[/i]

2014 Contests, 3

Tags: number theory , diophantine equation

a) Prove that the equation $2^x + 21^x = y^3$ has no solution in the set of natural numbers. b) Solve the equation $2^x + 21^y = z^2y$ in the set of non-negative integer numbers.

2011 China Northern MO, 3

Tags: number theory , diophantine , diophantine equation

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

1967 IMO Longlists, 15

Tags: diophantine equation , trigonometric equation , trigonometry , number theory

Suppose $\tan \alpha = \dfrac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Prove that the number $\tan \beta$ for which $\tan {2 \beta} = \tan {3 \alpha}$ is rational only when $p^2 + q^2$ is the square of an integer.

1990 IMO Longlists, 26

Tags: search , diophantine equation , modular arithmetic , number theory

Prove that there exist infinitely many positive integers $n$ such that the number $\frac{1^2+2^2+\cdots+n^2}{n}$ is a perfect square. Obviously, $1$ is the least integer having this property. Find the next two least integers having this property.

2017 Istmo Centroamericano MO, 3

Tags: diophantine , number theory , diophantine equation

Find all ordered pairs of integers $(x, y)$ with $y \ge 0$ such that $x^2 + 2xy + y! = 131$.

2020 Korea National Olympiad, 4

Tags: diophantine equation , number theory

Find a pair of coprime positive integers $(m,n)$ other than $(41,12)$ such that $m^2-5n^2$ and $m^2+5n^2$ are both perfect squares.

PEN H Problems, 81

Tags: modular arithmetic , diophantine equation , relatively prime , number theory

Find a pair of relatively prime four digit natural numbers $A$ and $B$ such that for all natural numbers $m$ and $n$, $\vert A^m -B^n \vert \ge 400$.