Olympiad problems

1997 Brazil Team Selection Test, Problem 3

Let $b$ be a positive integer such that $\gcd(b,6)=1$. Show that there are positive integers $x$ and $y$ such that $\frac1x+\frac1y=\frac3b$ if and only if $b$ is divisible by some prime number of form $6k-1$.

1984 Bulgaria National Olympiad, Problem 1

Tags: diophantine equation , number theory

Solve the equation $5^x7^y+4=3^z$ in nonnegative integers.

2005 Swedish Mathematical Competition, 1

Tags: number theory , diophantine equation

Find all integer solutions $x$,$y$ of the equation $(x+y^2)(x^2+y)=(x+y)^3$.

2025 Kosovo EGMO Team Selection Test, P2

Tags: nt , diophantine equation , number theory

Find all natural numbers $m$ and $n$ such that $3^m+n!-1$ is the square of a natural number.

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.

2018 Pan-African Shortlist, N5

Tags: diophantine equation , number theory

Find all quadruplets $(a, b, c, d)$ of positive integers such that \[ \left( 1 + \frac{1}{a} \right) \left( 1 + \frac{1}{b} \right) \left( 1 + \frac{1}{c} \right) \left( 1 + \frac{1}{d} \right) = 4. \]

PEN H Problems, 30

Tags: diophantine equation

Let $a$, $b$, $c$ be given integers, $a>0$, $ac-b^2=p$ a squarefree positive integer. Let $M(n)$ denote the number of pairs of integers $(x, y)$ for which $ax^2 +bxy+cy^2=n$. Prove that $M(n)$ is finite and $M(n)=M(p^{k} \cdot n)$ for every integer $k \ge 0$.

PEN H Problems, 49

Tags: abstract algebra , diophantine equation , search , group theory

Show that the only solutions of the equation $x^{3}-3xy^2 -y^3 =1$ are given by $(x,y)=(1,0),(0,-1),(-1,1),(1,-3),(-3,2),(2,1)$.

1997 Tuymaada Olympiad, 2

Tags: system of equations , natural number , diophantine equation , algebra

Solve in natural numbers the system of equations $3x^2+6y^2+5z^2=1997$ and $3x+6y+5z=161$ .

2016 Tuymaada Olympiad, 4

Tags: number theory , diophantine equation

For each positive integer $k$ find the number of solutions in nonnegative integers $x,y,z$ with $x\le y \le z$ of the equation $$8^k=x^3+y^3+z^3-3xyz$$

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]

2022 Saudi Arabia JBMO TST, 1

Tags: diophantine equation , number theory

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

II Soros Olympiad 1995 - 96 (Russia), 9.2

Tags: diophantine , diophantine equation , number theory

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

2010 Czech And Slovak Olympiad III A, 1

Tags: number theory , diophantine equation

Determine all pairs of integers $a, b$ for which they apply $4^a + 4a^2 + 4 = b^2$ .

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 $n^2 + n + p = 1996$.

2015 USAJMO, 2

Tags: diophantine equation , algebra , number theory

Solve in integers the equation \[ x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3. \]

2012 Greece JBMO TST, 2

Tags: coprime , diophantine equation , least common multiple , number theory

Find all pairs of coprime positive integers $(p,q)$ such that $p^2+2q^2+334=[p^2,q^2]$ where $[p^2,q^2]$ is the leact common multiple of $p^2,q^2$ .

PEN H Problems, 73

Tags: diophantine equation , greatest common divisor

Find all pairs $(a,b)$ of positive integers that satisfy the equation \[a^{b^{2}}= b^{a}.\]

2010 District Olympiad, 4

Tags: diophantine , number theory , diophantine equation

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

PEN H Problems, 28

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

Let $a, b, c$ be positive integers such that $a$ and $b$ are relatively prime and $c$ is relatively prime either to $a$ or $b$. Prove that there exist infinitely many triples $(x, y, z)$ of distinct positive integers such that \[x^{a}+y^{b}= z^{c}.\]

2022 Junior Balkan Mathematical Olympiad, 3

Tags: diophantine equation , number theory

Find all quadruples of positive integers $(p, q, a, b)$, where $p$ and $q$ are prime numbers and $a > 1$, such that $$p^a = 1 + 5q^b.$$