Olympiad problems

2009 Federal Competition For Advanced Students, P1, 1

Tags: inequalities , factorial , exponential , exponential inequality , diophantine , induction , algebra

Show that for all positive integer $n$ the following inequality holds $3^{n^2} > (n!)^4$ .

2011 Dutch Mathematical Olympiad, 1

Tags: diophantine , number theory , factorial , diophantine equation

Determine all triples of positive integers $(a, b, n)$ that satisfy the following equation: $a! + b! = 2^n$

1990 Austrian-Polish Competition, 4

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

Find all solutions in positive integers to: $$\begin{cases} x_1^4 + 14x_1x_2 + 1 = y_1^4 \\ x_2^4 + 14x_2x_3 + 1 = y_2^4 \\ ... \\ x_n^4 + 14x_nx_1 + 1 = y_n^4 \end{cases}$$

2021 Mediterranean Mathematics Olympiad, 2

Tags: number theory , diophantine equation , diophantine

For every sequence $p_1<p_2<\cdots<p_8$ of eight prime numbers, determine the largest integer $N$ for which the following equation has no solution in positive integers $x_1,\ldots,x_8$: $$p_1\, p_2\, \cdots\, p_8 \left( \frac{x_1}{p_1}+ \frac{x_2}{p_2}+ ~\cdots~ +\frac{x_8}{p_8} \right) ~~=~~ N $$ [i]Proposed by Gerhard Woeginger, Austria[/i]

2020 Francophone Mathematical Olympiad, 4

Tags: number theory , diophantine equation , diophantine

Find all the integers $x, y$ and $z$ greater than or equal to $0$ such that $2^x + 9 \cdot 7^y = z^3$

2002 Argentina National Olympiad, 2

Tags: number theory , diophantine , diophantine equation

Determine the smallest positive integer $k$ so that the equation $$2002x+273y=200201+k$$ has integer solutions, and for that value of $k$, find the number of solutions $\left (x,y\right )$ with $x$, $y$ positive integers that have the equation.

2014 Swedish Mathematical Competition, 6

Tags: diophantine , number theory , diophantine equation

Determine all odd primes $p$ and $q$ such that the equation $x^p + y^q = pq$ at least one solution $(x, y)$ where $x$ and $y$ are positive integers.

1992 All Soviet Union Mathematical Olympiad, 580

Tags: inequalities , diophantine , algebra

If $a > b > c > d > 0$ are integers such that $ad = bc$, show that $$(a - d)^2 \ge 4d + 8$$

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 $n$.

2015 Swedish Mathematical Competition, 2

Tags: diophantine , diophantine equation , number theory

Determine all integer solutions to the equation $x^3 + y^3 + 2015 = 0$.

2005 iTest, 34

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

If $x$ is the number of solutions to the equation $a^2 + b^2 + c^2 = d^2$ of the form $(a,b,c,d)$ such that $\{a,b,c\}$ are three consecutive square numbers and $d$ is also a square number, find $x$.

2012 Cuba MO, 5

Tags: diophantine , diophantine equation , number theory

Find all pairs $(m, n)$ of positive integers such that $m^2 + n^2 =(m + 1)(n + 1).$

I Soros Olympiad 1994-95 (Rus + Ukr), 9.5

Tags: diophantine equation , diophantine , number theory

Find the triplets of natural numbers $(p,q,r)$ that satisfy the equality $$\frac{1}{p}+\frac{q}{q^r -1}=1.$$

2019 Philippine TST, 2

Tags: diophantine , number theory

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

2015 Canadian Mathematical Olympiad Qualification, 1

Tags: number theory , diophantine

Find all integer solutions to the equation $7x^2y^2 + 4x^2 = 77y^2 + 1260$.

2001 Croatia Team Selection Test, 3

Tags: number theory , diophantine equation , diophantine

Find all solutions of the equation $(a^a)^5 = b^b$ in positive integers.

2003 Dutch Mathematical Olympiad, 3

Tags: diophantine , consecutive , diophantine equation , number theory

Determine all positive integers$ n$ that can be written as the product of two consecutive integers and as well as the product of four consecutive integers numbers. In the formula: $n = a (a + 1) = b (b + 1) (b + 2) (b + 3)$.

2017 Singapore Junior Math Olympiad, 2

Tags: factorial , diophantine , diophantine equation , number theory

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 $a_1,a_2,...,a_{2n}$

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.

1979 Chisinau City MO, 179

Tags: diophantine , number theory , diophantine equation

Prove that the equation $x^2 + y^2 = 1979$ has no integer solutions.

1999 Estonia National Olympiad, 1

Tags: diophantine , number theory , diophantine equation

Find all pairs of integers $(m, n)$ such that $(m - n)^2 =\frac{4mn}{m + n - 1}$

2000 Singapore Senior Math Olympiad, 2

Tags: number theory , factorial , diophantine equation , diophantine

Prove that there exist no positive integers $m$ and $n$ such that $m > 5$ and $(m - 1)! + 1 = m^n$.

2011 Denmark MO - Mohr Contest, 3

Tags: number theory , diophantine

Determine all the ways in which the fraction $\frac{1}{11}$ can be written as $\frac{1}{n}+\frac{1}{m}$ , where $n$ and $m$ are two different positive integers.

2019 Thailand TST, 2

Tags: diophantine , number theory

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

1995 Argentina National Olympiad, 2

Tags: diophantine equation , diophantine , number theory

For each positive integer $n$ let $p(n)$ be the number of ordered pairs $(x,y)$ of positive integers such that$$\dfrac{1}{x}+\dfrac{1}{y} =\dfrac{1}{n}.$$For example, for $n=2$ the pairs are $(3,6),(4,4),(6,3)$. Therefore $p(2)=3$. a) Determine $p(n)$ for all $n$ and calculate $p(1995)$. b) Determine all pairs $n$ such that $p(n)=3$.