Olympiad problems

1987 Poland - Second Round, 5

Determine all prime numbers $ p $ and natural numbers $ x, y $ for which $ p^x-y^3 = 1 $.

2021 New Zealand MO, 4

Tags: diophantine equation , diophantine , number theory

Find all triples $(x, p, n)$ of non-negative integers such that $p$ is prime and $2x(x + 5) = p^n + 3(x - 1)$.

II Soros Olympiad 1995 - 96 (Russia), 9.2

Tags: diophantine equation , diophantine , number theory

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

2002 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , diophantine equation , diophantine

Let $m,n > 1$ be integer numbers. Solve in positive integers $x^n+y^n = 2^m$.

2016 Hanoi Open Mathematics Competitions, 8

Tags: number theory , diophantine equation , diophantine

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

2016 Czech-Polish-Slovak Junior Match, 6

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

Let $k$ be a given positive integer. Find all triples of positive integers $a, b, c$, such that $a + b + c = 3k + 1$, $ab + bc + ca = 3k^2 + 2k$. Slovakia

1981 Polish MO Finals, 5

Tags: number theory , diophantine

Determine all pairs of integers $(x,y)$ satisfying the equation $$x^3 +x^2y+xy^2 +y^3 = 8(x^2 +xy+y^2 +1).$$

1997 All-Russian Olympiad Regional Round, 8.7

Tags: number theory , diophantine , diophantine equation

Find all pairs of prime numbers $p$ and $q$ such that $p^3-q^5 = (p+q)^2$.

2021 New Zealand MO, 5

Tags: diophantine equation , number theory , diophantine

Find all pairs of integers $x, y$ such that $y^5 + 2xy = x^2 + 2y^4.$ .

2017 Switzerland - Final Round, 7

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

Let $n$ be a natural number such that there are exactly$ 2017$ distinct pairs of natural numbers $(a, b)$, which the equation $$\frac{1}{a}+\frac{1}{b}=\frac{1}{n}$$ fulfilld. Show that $n$ is a perfect square . Remark: $(7, 4) \ne (4, 7)$

2019 Romania National Olympiad, 4

Tags: number theory , diophantine equation , diophantine

Find the natural numbers $x, y, z$ that verify the equation: $$2^x + 3 \cdot 11^y =7^z$$

2007 BAMO, 4

Tags: number theory , divisible , diophantine

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 $3$.

1917 Eotvos Mathematical Competition, 1

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

If $a$ and $b$ are integers and if the solutions of the system of equations $$y - 2x - a = 0$$ $$y^2 - xy + x^2 - b = 0$$ are rational, prove that the solutions are integers.

2012 NZMOC Camp Selection Problems, 6

Tags: diophantine equation , diophantine , number theory

Let $a, b$ and $c$ be positive integers such that $a^{b+c} = b^{c} c$. Prove that b is a divisor of $c$, and that $c$ is of the form $d^b$ for some positive integer $d$.

1963 Poland - Second Round, 3

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

Solve the system of equations in integers $$x + y + z = 3$$ $$x^3 + y^3 + z^3 = 3$$

1955 Moscow Mathematical Olympiad, 302

Tags: diophantine , diophantine equation , number theory

Find integer solutions of the equation $x^3 - 2y^3 - 4z^3 = 0$.

VMEO IV 2015, 10.3

Tags: number theory , diophantine equation , diophantine

Find all triples of integers $(a, b, c)$ satisfying $a^2 + b^2 + c^2 =3(ab + bc + ca).$

2017 Regional Olympiad of Mexico Northeast, 3

Tags: number theory , diophantine equation , diophantine

Prove that there is no pair of relatively prime positive integers $(a, b)$ that satisfy the equation $$a^3 + 2017a = b^3 -2017b.$$

2001 Grosman Memorial Mathematical Olympiad, 6

Tags: diophantine equation , diophantine , number theory

(a) Find a pair of integers (x,y) such that $15x^2 +y^2 = 2^{2000}$ (b) Does there exist a pair of integers $(x,y)$ such that $15x^2 + y^2 = 2^{2000}$ and $x$ is odd?

1990 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: floor function , algebra , diophantine

How many positive integer solutions does the equation have $$\left\lfloor\frac{x}{10}\right\rfloor= \left\lfloor\frac{x}{11}\right\rfloor + 1?$$ ($\lfloor x \rfloor$ denotes the integer part of $x$, for example $\lfloor 2\rfloor = 2$, $\lfloor \pi\rfloor = 3$, $\lfloor \sqrt2 \rfloor =1$)

1987 Poland - Second Round, 5

2021 New Zealand MO, 4

II Soros Olympiad 1995 - 96 (Russia), 9.2

2002 Junior Balkan Team Selection Tests - Romania, 1

2016 Hanoi Open Mathematics Competitions, 8

2016 Czech-Polish-Slovak Junior Match, 6

1981 Polish MO Finals, 5

1997 All-Russian Olympiad Regional Round, 8.7

2021 New Zealand MO, 5

2017 Switzerland - Final Round, 7

2019 Romania National Olympiad, 4

2007 BAMO, 4

1917 Eotvos Mathematical Competition, 1

2012 NZMOC Camp Selection Problems, 6

1963 Poland - Second Round, 3

1955 Moscow Mathematical Olympiad, 302

VMEO IV 2015, 10.3

2017 Regional Olympiad of Mexico Northeast, 3

2001 Grosman Memorial Mathematical Olympiad, 6

1990 Rioplatense Mathematical Olympiad, Level 3, 1

1950 Poland - Second Round, 6

1964 Swedish Mathematical Competition, 2

VMEO III 2006, 12.2

2022 Austrian Junior Regional Competition, 4

1979 Chisinau City MO, 179