Olympiad problems

2014 Junior Balkan Team Selection Tests - Moldova, 2

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

2013 Greece JBMO TST, 3

Tags: diophantine equation , number theory , prime

If $p$ is a prime positive integer and $x,y$ are positive integers, find , in terms of $p$, all pairs $(x,y)$ that are solutions of the equation: $p(x-2)=x(y-1)$. (1) If it is also given that $x+y=21$, find all triplets $(x,y,p)$ that are solutions to equation (1).

2019 Teodor Topan, 1

Tags: equation , diophantine equation , number theory

Solve in the natural numbers the equation $ \log_{6n-19} (n!+1) =2. $ [i]Dragoș Crișan[/i]

2014 NIMO Problems, 15

Tags: diophantine equation

Let $A = (0,0)$, $B=(-1,-1)$, $C=(x,y)$, and $D=(x+1,y)$, where $x > y$ are positive integers. Suppose points $A$, $B$, $C$, $D$ lie on a circle with radius $r$. Denote by $r_1$ and $r_2$ the smallest and second smallest possible values of $r$. Compute $r_1^2 + r_2^2$. [i]Proposed by Lewis Chen[/i]

2019 India PRMO, 6

Tags: number theory , diophantine equation

Let $\overline{abc}$ be a three digit number with nonzero digits such that $a^2 + b^2 = c^2$. What is the largest possible prime factor of $\overline{abc}$

2012 International Zhautykov Olympiad, 3

Tags: modular arithmetic , greatest common divisor , diophantine equation , number theory

Find all integer solutions of the equation the equation $2x^2-y^{14}=1$.

2014 China Team Selection Test, 3

Tags: number theory , diophantine equation

Show that there are no 2-tuples $ (x,y)$ of positive integers satisfying the equation $ (x+1) (x+2)\cdots (x+2014)= (y+1) (y+2)\cdots (y+4028).$

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 $k + a^k = m + 2a^m$.

2015 USAJMO, 2

Tags: diophantine equation , algebra , number theory

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

2018 Spain Mathematical Olympiad, 5

Tags: number theory , diophantine equation

Let $a, b$ be coprime positive integers. A positive integer $n$ is said to be [i]weak[/i] if there do not exist any nonnegative integers $x, y$ such that $ax+by=n$. Prove that if $n$ is a [i]weak[/i] integer and $n < \frac{ab}{6}$, then there exists an integer $k \geq 2$ such that $kn$ is [i]weak[/i].

2021 Miklós Schweitzer, 2

Tags: factorial , diophantine equation , number theory

Prove that the equation \[ 2^x + 5^y - 31^z = n! \] has only a finite number of non-negative integer solutions $x,y,z,n$.

PEN P Problems, 15

Tags: quadratic , algebra , diophantine equation , additive number theory

Find all integers $m>1$ such that $m^3$ is a sum of $m$ squares of consecutive integers.

2023 Romanian Master of Mathematics, 1

Tags: number theory , prime number , diophantine equation

Determine all prime numbers $p$ and all positive integers $x$ and $y$ satisfying $$x^3+y^3=p(xy+p).$$

1999 Singapore Senior Math Olympiad, 1

Tags: diophantine , diophantine equation , number theory

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

1997 IMO Shortlist, 6

Tags: number theory , relatively prime , diophantine equation

(a) Let $ n$ be a positive integer. Prove that there exist distinct positive integers $ x, y, z$ such that \[ x^{n\minus{}1} \plus{} y^n \equal{} z^{n\plus{}1}.\] (b) Let $ a, b, c$ be positive integers such that $ a$ and $ b$ are relatively prime and $ c$ is relatively prime either to $ a$ or to $ b.$ Prove that there exist infinitely many triples $ (x, y, z)$ of distinct positive integers $ x, y, z$ such that \[ x^a \plus{} y^b \equal{} z^c.\]

2014 Junior Balkan Team Selection Tests - Moldova, 2

2013 Greece JBMO TST, 3

2019 Teodor Topan, 1

2014 NIMO Problems, 15

2019 India PRMO, 6

2012 International Zhautykov Olympiad, 3

2014 China Team Selection Test, 3

2021 Final Mathematical Cup, 1

2012 Czech-Polish-Slovak Junior Match, 5

2015 USAJMO, 2

2018 Spain Mathematical Olympiad, 5

2021 Miklós Schweitzer, 2

PEN P Problems, 15

2023 Romanian Master of Mathematics, 1

1999 Singapore Senior Math Olympiad, 1

1997 IMO Shortlist, 6

2020 China Team Selection Test, 4

2005 iTest, 16

2017 Junior Balkan Team Selection Tests - Moldova, Problem 1

2020 BMT Fall, 9

1956 Poland - Second Round, 4

PEN H Problems, 16

1993 Tournament Of Towns, (389) 1

2015 China Team Selection Test, 5

1981 IMO Shortlist, 12