Olympiad problems

2014 Cuba MO, 1

Find all the integer solutions of the equation $ m^4 + 2n^2 = 9mn$.

1991 Tournament Of Towns, (291) 1

Tags: number theory , diophantine , diophantine equation

Find all natural numbers $n$, and all integers $x,y$ ($x\ne y$) for which the following equation is satisfied: $$x + x^2 + x^4 + ...+ x^{2^n} = y + y^2 + y^4 + ... + y^{2^n} .$$

2018 Costa Rica - Final Round, N3

Tags: number theory , diophantine

Let $a$ and $b$ be positive integers such that $2a^2 + a = 3b^2 + b$. Prove that $a-b$ is a perfect square.

2022 Czech-Polish-Slovak Junior Match, 4

Tags: diophantine equation , number theory , diophantine

Find all triples $(a, b, c)$ of integers that satisfy the equations $ a + b = c$ and $a^2 + b^3 = c^2$

2018 Hanoi Open Mathematics Competitions, 3

Tags: inequalities , number theory , diophantine

How many integers $n$ are there those satisfy the following inequality $n^4 - n^3 - 3n^2 - 3n - 17 < 0$? A. $4$ B. $6$ C. $8$ D. $10$ E. $12$

1986 Poland - Second Round, 4

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

Natural numbers $ x, y, z $ whose greatest common divisor is equal to 1 satisfy the equation $$\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$$ Prove that $ x + y $ is the square of a natural number.

2005 Thailand Mathematical Olympiad, 6

Tags: number theory , diophantine equation , diophantine

Find the number of positive integer solutions to the equation $(x_1+x_2+x_3)^2(y_1+y_2) = 2548$.

2010 District Olympiad, 4

Tags: diophantine equation , number theory , diophantine

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

2010 Saudi Arabia BMO TST, 4

Tags: diophantine equation , diophantine , number theory

Find all primes $p, q$ satisfying the equation $2p^q - q^p = 7.$

1995 Austrian-Polish Competition, 7

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

Consider the equation $3y^4 + 4cy^3 + 2xy + 48 = 0$, where $c$ is an integer parameter. Determine all values of $c$ for which the number of integral solutions $(x,y)$ satisfying the conditions (i) and (ii) is maximal: (i) $|x|$ is a square of an integer; (ii) $y$ is a squarefree number.

2017 Latvia Baltic Way TST, 14

Tags: diophantine equation , number theory , diophantine

Can you find three natural numbers $a, b, c$ whose greatest common divisor is $1$ and which satisfy the equality $$ab + bc + ac = (a + b -c)(b + c - a)(c + a - b) ?$$

1941 Moscow Mathematical Olympiad, 088

Tags: number theory , diophantine

Solve in integers the equation $x + y = x^2 - xy + y^2$.

1983 Swedish Mathematical Competition, 3

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

The systems of equations \[\left\{ \begin{array}{l} 2x_1 - x_2 = 1 \\ -x_1 + 2x_2 - x_3 = 1 \\ -x_2 + 2x_3 - x_4 = 1 \\ -x_3 + 3x_4 - x_5 =1 \\ \cdots\cdots\cdots\cdots\\ -x_{n-2} + 2x_{n-1} - x_n = 1 \\ -x_{n-1} + 2x_n = 1 \\ \end{array} \right. \] has a solution in positive integers $x_i$. Show that $n$ must be even.

2022 Dutch IMO TST, 1

Tags: number theory , diophantine equation , diophantine

Determine all positive integers $n \ge 2$ which have a positive divisor $m | n$ satisfying $$n = d^3 + m^3.$$ where $d$ is the smallest divisor of $n$ which is greater than $1$.

2022 Argentina National Olympiad, 5

Tags: number theory , diophantine equation , diophantine

Find all pairs of positive integers $x,y$ such that $$x^3+y^3=4(x^2y+xy^2-5).$$

1964 Dutch Mathematical Olympiad, 3

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

Solve $ (n + 1)(n +10) = q^2$, for certain $q$ and maximum $n$.

2011 NZMOC Camp Selection Problems, 1

Tags: diophantine , diophantine equation , number theory

Find all pairs of positive integers $m$ and $n$ such that $$m! + n! = m^n.$$ .

2013 Singapore Senior Math Olympiad, 2

Tags: diophantine , diophantine equation , number theory

Find all pairs of integers $(m,n)$ such that $m^3-n^3=2mn +8$

1984 All Soviet Union Mathematical Olympiad, 379

Tags: diophantine , diophantine equation , number theory , algebra , exponential

Find integers $m$ and $n$ such that $(5 + 3 \sqrt2)^m = (3 + 5 \sqrt2)^n$.

2014 Ukraine Team Selection Test, 9

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

Let $m, n$ be odd prime numbers. Find all pairs of integers numbers $a, b$ for which the system of equations: $x^m+y^m+z^m=a$, $x^n+y^n+z^n=b$ has many solutions in integers $x, y, z$.

2016 NZMOC Camp Selection Problems, 7

Tags: diophantine , diophantine equation , number theory

Find all positive integers $n$ for which the equation $$(x^2 + y^2)^n = (xy)^{2016}$$ has positive integer solutions.

2017 Singapore Senior Math Olympiad, 1

Tags: diophantine , diophantine equation , number theory , factorial

Let $n$ be a positive integer and $a_1,a_2,...,a_{2n}$ be $2n$ distinct integers. Given that the equation $|x-a_1| |x-a_2| ... |x-a_{2n}| =(n!)^2$ has an integer solution $x = m$, find $m$ in terms of $a_1,a_2,...,a_{2n}$

2018 Finnish National High School Mathematics Comp, 5

Tags: number theory , diophantine equation , diophantine

Solve the diophantine equation $x^{2018}-y^{2018}=(xy)^{2017}$ when $x$ and $y$ are non-negative integers.

1984 Swedish Mathematical Competition, 5

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

Solve in natural numbers $a,b,c$ the system \[\left\{ \begin{array}{l}a^3 -b^3 -c^3 = 3abc \\ a^2 = 2(a+b+c)\\ \end{array} \right. \]

2014 Belarusian National Olympiad, 2

Tags: number theory , diophantine , diophantine equation

Pairwise distinct prime numbers $p, q, r$ satisfy the equality $$rp^3 + p^2 + p = 2rq^2 +q^2 + q.$$ Determine all possible values of the product $pqr$.