Olympiad problems

2010 Federal Competition For Advanced Students, P2, 2

Determine all triples $(x, y, z)$ of positive integers $x > y > z > 0$, such that $x^2 = y \cdot 2^z + 1$

2022 Austrian Junior Regional Competition, 4

Tags: diophantine equation , diophantine , number theory

Determine all prime numbers $p, q$ and $r$ with $p + q^2 = r^4$. [i](Karl Czakler)[/i]

2009 Postal Coaching, 6

Tags: inequalities , diophantine , number theory

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

1996 Poland - Second Round, 5

Tags: diophantine equation , diophantine , number theory

Find all integers $x,y$ such that $x^2(y-1)+y^2(x-1) = 1$.

1981 Tournament Of Towns, (007) 1

Tags: diophantine equation , diophantine , number theory

Find all integer solutions to the equation $y^k = x^2 + x$, where $k$ is a natural number greater than $1$.

2019 Ecuador NMO (OMEC), 5

Tags: diophantine , number theory

Let $a, b, c$ be integers not all the same with $a, b, c\ge 4$ that satisfy $$4abc = (a + 3) (b + 3) (c + 3).$$ Find the numerical value of $a + b + c$.

2003 Chile National Olympiad, 2

Tags: diophantine , diophantine equation , number theory

Find all primes $p, q$ such that $p + q = (p-q)^3$.

2006 Alexandru Myller, 1

Tags: diophantine , number theory

Find a countable family of natural solutions to $ \frac{1}{a} +\frac{1}{b} +\frac{1}{ab}=\frac{1}{c} . $

2005 Estonia Team Selection Test, 3

Tags: diophantine , diophantine equation , number theory

Find all pairs $(x, y)$ of positive integers satisfying the equation $(x + y)^x = x^y$.

2017 Swedish Mathematical Competition, 2

Tags: diophantine , diophantine equation , number theory

Let $p$ be a prime number. Find all pairs of coprime positive integers $(m,n)$ such that $$ \frac{p+m}{p+n}=\frac{m}{n}+\frac{1}{p^2}.$$

1941 Moscow Mathematical Olympiad, 088

Tags: diophantine , number theory

Solve in integers the equation $x + y = x^2 - xy + y^2$.

2017 District Olympiad, 2

Tags: equation , parameterization , diophantine

Let $ E(x,y)=\frac{x}{y} +\frac{x+1}{y+1} +\frac{x+2}{y+2} . $ [b]a)[/b] Solve in $ \mathbb{N}^2 $ the equation $ E(x,y)=3. $ [b]b)[/b] Show that there are infinitely many natural numbers $ n $ such that the equation $ E(x,y)=n $ has at least one solution in $ \mathbb{N}^2. $

VI Soros Olympiad 1999 - 2000 (Russia), 9.4

Tags: diophantine , diophantine equation , number theory

Are there integers $k$ and $m$ for which $$\frac{(k-3)(k-2)(k-1)k+1}{(k+1)(k+2)(k+3)(k+4)+1}=m(m+1)+(m+1)(m+2)+(m+2)m \,\, ?$$

1967 German National Olympiad, 5

Tags: diophantine equation , diophantine , number theory

For each natural number $n$, determine the number $A(n)$ of all integer nonnegative solutions the equation $$5x + 2y + z = 10n.$$

1996 Estonia National Olympiad, 1

Tags: sum , product , diophantine equation , diophantine , number theory

Find all pairs of integers $(x, y)$ such that ths sum of the fractions $\frac{19}{x}$ and $\frac{96}{y}$ would be equal to their product.

2017 Hanoi Open Mathematics Competitions, 3

Tags: diophantine , diophantine equation , number theory

The number of real triples $(x , y , z )$ that satisfy the equation $x^4 + 4y^4 + z^4 + 4 = 8xyz$ is (A): $0$, (B): $1$, (C): $2$, (D): $8$, (E): None of the above.

1949 Moscow Mathematical Olympiad, 158

Tags: diophantine , number theory

a) Prove that $x^2 + y^2 + z^2 = 2xyz$ for integer $x, y, z$ only if $x = y = z = 0$. b) Find integers $x, y, z, u$ such that $x^2 + y^2 + z^2 + u^2 = 2xyzu$.

1984 Poland - Second Round, 1

Tags: diophantine , diophantine equation , number theory

For a given natural number $ n $, find the number of solutions to the equation $ \sqrt{x} + \sqrt{y} = n $ in natural numbers $ x, y $.

2008 Mathcenter Contest, 6

Tags: diophantine , diophantine equation , number theory

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

II Soros Olympiad 1995 - 96 (Russia), 9.2

Tags: diophantine , diophantine equation , number theory

Find the integers $x, y, z$ for which $$\dfrac{1}{x+\dfrac{1}{y+\dfrac{1}{z}}}=\dfrac{7}{17}$$

2007 BAMO, 4

Tags: divisible , diophantine , number theory

Let $N$ be the number of ordered pairs $(x,y)$ of integers such that $x^2+xy+y^2 \le 2007$. Remember, integers may be positive, negative, or zero! (a) Prove that $N$ is odd. (b) Prove that $N$ is not divisible by $3$.

1996 Greece Junior Math Olympiad, 4b

Tags: diophantine , diophantine equation , prime , number theory

Determine whether exist a prime number $p$ and natural number $n$ such that $n^2 + n + p = 1996$.

2010 District Olympiad, 4

Tags: diophantine , number theory , diophantine equation

Find all non negative integers $(a, b)$ such that $$a + 2b - b^2 =\sqrt{2a + a^2 + |2a + 1 - 2b|}.$$

2023 Puerto Rico Team Selection Test, 1

Tags: diophantine , diophantine equation , number theory

Determine all triples $(a, b, c)$ of positive integers such that $$a! +b! = 2^{c!} .$$

2014 Ukraine Team Selection Test, 9

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

Let $m, n$ be odd prime numbers. Find all pairs of integers numbers $a, b$ for which the system of equations: $x^m+y^m+z^m=a$, $x^n+y^n+z^n=b$ has many solutions in integers $x, y, z$.