Olympiad problems

2016 Belarus Team Selection Test, 3

Solve the equation $2^a-5^b=3$ in positive integers $a,b$.

2015 Indonesia MO Shortlist, N4

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

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 $(c, d) = 1$?

2022 Korea Winter Program Practice Test, 1

Tags: number theory , diophantine equation

Prove that equation $y^2=x^3+7$ doesn't have any solution on integers.

2021 Cyprus JBMO TST, 2

Tags: number theory , diophantine equation , gcd , lcm

Find all pairs of natural numbers $(\alpha,\beta)$ for which, if $\delta$ is the greatest common divisor of $\alpha,\beta$, and $\varDelta$ is the least common multiple of $\alpha,\beta$, then \[ \delta + \Delta = 4(\alpha + \beta) + 2021\]

2014 Dutch Mathematical Olympiad, 1

Tags: diophantine equation , diophantine , number theory

Determine all triples $(a,b,c)$, where $a, b$, and $c$ are positive integers that satisfy $a \le b \le c$ and $abc = 2(a + b + c)$.

2015 BMT Spring, 9

Tags: number theory , diophantine equation

There exists a unique pair of positive integers $k,n$ such that $k$ is divisible by $6$, and $\sum_{i=1}^ki^2=n^2$. Find $(k,n)$.

2022 3rd Memorial "Aleksandar Blazhevski-Cane", P3

Tags: number theory , diophantine equation , 3 variables

Find all triplets of positive integers $(x, y, z)$ such that $x^2 + y^2 + x + y + z = xyz + 1$. [i]Proposed by Viktor Simjanoski[/i]

2018 Pan-African Shortlist, N1

Tags: number theory , diophantine equation

Does there exist positive integers $a, b, c$ such that $4(ab - a - c^2) = b$?

2006 Petru Moroșan-Trident, 2

Tags: diophantine equation , algebra , equation

Solve the following Diophantines. [b]a)[/b] $ x^2+y^2=6z^2 $ [b]b)[/b] $ x^2+y^2-2x+4y-1=0 $ [i]Dan Negulescu[/i]

2019 Belarus Team Selection Test, 1.3

Tags: number theory , diophantine equation , perfect power

Given the equation $$ a^b\cdot b^c=c^a $$ in positive integers $a$, $b$, and $c$. [i](i)[/i] Prove that any prime divisor of $a$ divides $b$ as well. [i](ii)[/i] Solve the equation under the assumption $b\ge a$. [i](iii)[/i] Prove that the equation has infinitely many solutions. [i](I. Voronovich)[/i]

2013 Bulgaria National Olympiad, 6

Tags: diophantine equation , number theory

Given $m\in\mathbb{N}$ and a prime number $p$, $p>m$, let \[M=\{n\in\mathbb{N}\mid m^2+n^2+p^2-2mn-2mp-2np \,\,\, \text{is a perfect square} \} \] Prove that $|M|$ does not depend on $p$. [i]Proposed by Aleksandar Ivanov[/i]

2004 Thailand Mathematical Olympiad, 11

Tags: diophantine , diophantine equation , number theory

Find the number of positive integer solutions to $(x_1 + x_2 + x_3)(y_1 + y_2 + y_3 + y_4) = 91$

2015 USAMO, 1

Tags: diophantine equation , algebra , number theory

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

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)

2006 Federal Math Competition of S&M, Problem 2

Tags: diophantine equation , number theory

Given prime numbers $p$ and $q$ with $p<q$, determine all pairs $(x,y)$ of positive integers such that $$\frac1x+\frac1y=\frac1p-\frac1q.$$

2024 Ecuador NMO (OMEC), 5

Tags: number theory , diophantine equation

Find all triples of non-negative integer numbers $(E, C, U)$ such that $EC \ge 1$ and: $$2^{3^E}+3^{2^C}=593 \cdot 5^U$$

1980 IMO Shortlist, 3

Tags: number theory , equation , algebra , diophantine equation

Prove that the equation \[ x^n + 1 = y^{n+1}, \] where $n$ is a positive integer not smaller then 2, has no positive integer solutions in $x$ and $y$ for which $x$ and $n+1$ are relatively prime.

2015 Dutch IMO TST, 2

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

Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$ with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$

2017 German National Olympiad, 6

Tags: number theory , diophantine equation

Prove that there exist infinitely many positive integers $m$ such that there exist $m$ consecutive perfect squares with sum $m^3$. Specify one solution with $m>1$.