Olympiad problems

2023-24 IOQM India, 4

Let $x, y$ be positive integers such that $$ x^4=(x-1)\left(y^3-23\right)-1 . $$ Find the maximum possible value of $x+y$.

2007 Nicolae Coculescu, 2

Tags: equation , newton s binom , diophantine

[b]a)[/b] Prove that there exists two infinite sequences $ \left( a_n \right)_{n\ge 1} ,\left( b_n \right)_{n\ge 1} $ of nonnegative integers such that $ a_n>b_n $ and $ (2+\sqrt 3)^n =a_n (2+\sqrt 3) -b_n , $ for any natural numbers $ n. $ [b]b)[/b] Prove that the equation $ x^2-4xy+y^2=1 $ has infinitely many solutions in $ \mathbb{N}^2. $ [i]Florian Dumitrel[/i]

2008 Hanoi Open Mathematics Competitions, 4

Tags: number theory , diophantine equation , diophantine

Find all pairs $(m,n)$ of positive integers such that $m^2 + n^2 = 3(m + n)$.

1995 Israel Mathematical Olympiad, 4

Tags: diophantine , diophantine equation , number theory

Find all integers $m$ and $n$ satisfying $m^3 -n^3 - 9mn = 27$.

2017 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: number theory , diophantine , diophantine equation

Let $a$ be a fixed positive integer. Find the largest integer $b$ such that $(x+a)(x+b)=x+a+b$, for some integer $x$.

2017 QEDMO 15th, 1

Tags: diophantine , diophantine equation , number theory

Find all integers $x, y, z$ satisfy the $x^4-10y^4 + 3z^6 = 21$.

2014 Junior Balkan Team Selection Tests - Romania, 2

Tags: number theory , diophantine equation , diophantine

Determine all pairs $(a, b)$ of integers which satisfy the equality $\frac{a + 2}{b + 1} +\frac{a + 1}{b + 2} = 1 +\frac{6}{a + b + 1}$

2014 Saudi Arabia GMO TST, 1

Tags: number theory , diophantine equation , diophantine

Find all ordered triples $(a,b, c)$ of positive integers which satisfy $5^a + 3^b - 2^c = 32$

2004 Bosnia and Herzegovina Junior BMO TST, 1

Tags: diophantine equation , diophantine , number theory

In the set of integers solve the equation $\frac{1}{x}+\frac{1}{y}=\frac{1}{p}$, where $p$ is a prime number.

2016 Hanoi Open Mathematics Competitions, 5

Tags: diophantine equation , diophantine , number theory

There are positive integers $x, y$ such that $3x^2 + x = 4y^2 + y$, and $(x - y)$ is equal to (A): $2013$ (B): $2014$ (C): $2015$ (D): $2016$ (E): None of the above.

2017 Istmo Centroamericano MO, 4

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

Suppose that $a$ and $ b$ are distinct positive integers satisfying $20a + 17b = p$ and $17a + 20b = q$ for certain primes $p$ and $ q$. Determine the minimum value of $p + q$.

2014 Cuba MO, 7

Tags: diophantine equation , diophantine , number theory

Find all pairs of integers $(a, b)$ that satisfy the equation $$(a + 1)(b- 1) = a^2b^2.$$

2014 Junior Balkan Team Selection Tests - Moldova, 2

Tags: diophantine equation , diophantine , number theory

Determine all pairs of integers $(x, y)$ that satisfy equation $(y - 2) x^2 + (y^2 - 6y + 8) x = y^2 - 5y + 62$.

1966 Poland - Second Round, 1

Tags: diophantine , number theory

Solve the equation in natural numbers $$ x+y+z+t=xyzt. $$

2021 Final Mathematical Cup, 1

Tags: number theory , diophantine equation , diophantine

Find all integer $n$ such that the equation $2x^2 + 5xy + 2y^2 = n$ has integer solution for $x$ and $y$.

2012 Czech-Polish-Slovak Junior Match, 5

Tags: diophantine equation , diophantine , number theory

Find all triplets $(a, k, m)$ of positive integers that satisfy the equation $k + a^k = m + 2a^m$.

1999 Singapore Senior Math Olympiad, 1

Tags: diophantine , diophantine equation , number theory

Find all the integral solutions of the equation $\left( 1+\frac{1}{x}\right)^{x+1}=\left( 1+\frac{1}{1999}\right)^{1999}$

2005 iTest, 16

Tags: diophantine , number theory , diophantine equation

How many distinct integral solutions of the form $(x, y)$ exist to the equation$ 21x + 22y = 43$ such that $1 < x < 11$ and $y < 22$?

1956 Poland - Second Round, 4

Tags: diophantine , diophantine equation , number theory

Prove that the equation $ 2x^2 - 215y^2 = 1 $ has no integer solutions.

1993 Tournament Of Towns, (389) 1

Tags: diophantine , diophantine equation , number theory

Consider the set of solutions of the equation $$x^2+y^3=z^2.$$ in positive integers. Is it finite or infinite? (Folklore)

2011 Saudi Arabia Pre-TST, 3.2

Tags: diophantine equation , diophantine , number theory

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

2015 Costa Rica - Final Round, 4

Tags: number theory , diophantine equation , diophantine

Find all triples $(p,M, z)$ of integers, where $p$ is prime, $m$ is positive and $z$ is negative, that satisfy the equation $$p^3 + pm + 2zm = m^2 + pz + z^2$$

1998 Slovenia Team Selection Test, 4

Tags: gcd , number theory , diophantine equation , diophantine

Find all positive integers $x$ and $y$ such that $x+y^2+z^3 = xyz$, where $z$ is the greatest common divisor of $x$ and $y$

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.

2016 Junior Balkan Team Selection Tests - Moldova, 6

Tags: diophantine , diophantine equation , number theory

Determine all pairs $(x, y)$ of natural numbers satisfying the equation $5^x=y^4+4y+1$.