Olympiad problems

2013 Swedish Mathematical Competition, 3

Determine all primes $p$ and all non-negative integers $m$ and $n$, such that $$1 + p^n = m^3. $$

2010 Saudi Arabia Pre-TST, 4.1

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

Find all triples $(a, b, c)$ of positive integers for which $$\begin{cases} a + bc=2010 \\ b + ca = 250\end{cases}$$

2012 Dutch Mathematical Olympiad, 3

Tags: number theory , diophantine equation , diophantine

Determine all pairs $(p,m)$ consisting of a prime number $p$ and a positive integer $m$, for which $p^3 + m(p + 2) = m^2 + p + 1$ holds.

2019 Brazil Team Selection Test, 5

Tags: number theory , diophantine

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?

1966 Polish MO Finals, 1

Tags: number theory , diophantine , diophantine equation

Solve in integers the equation $$x^4 +4y^4 = 2(z^4 +4u^4)$$

2018 Puerto Rico Team Selection Test, 1

Tags: number theory , diophantine , diophantine equation

Find all pairs $(a, b)$ of positive integers that satisfy the equation $a^2 -3 \cdot 2^b = 1$.

1990 Austrian-Polish Competition, 2

Tags: number theory , diophantine , diophantine equation

Find all solutions in positive integers to $a^A = b^B = c^C = 1990^{1990}abc$, where $A = b^c, B = c^a, C = a^b$.

1963 Poland - Second Round, 3

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

Solve the system of equations in integers $$x + y + z = 3$$ $$x^3 + y^3 + z^3 = 3$$

2014 Junior Balkan Team Selection Tests - Romania, 2

Tags: number theory , diophantine , diophantine equation

Solve, in the positive integers, the equation $5^m + n^2 = 3^p$ .

2015 Dutch IMO TST, 2

Tags: number theory , diophantine , system of equations , 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 $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$

1991 All Soviet Union Mathematical Olympiad, 535

Tags: system of equations , number theory , diophantine

Find all integers $a, b, c, d$ such that $$\begin{cases} ab - 2cd = 3 \\ ac + bd = 1\end{cases}$$

2004 Singapore MO Open, 2

Tags: number theory , diophantine , diophantine equation

Find the number of ordered pairs $(a, b)$ of integers, where $1 \le a, b \le 2004$, such that $x^2 + ax + b = 167 y$ has integer solutions in $x$ and $y$. Justify your answer.

2009 QEDMO 6th, 1

Tags: number theory , diophantine , diophantine equation

Solve $y^5 - x^2 = 4$ in integers numbers $x,y$.

2019 Romania Team Selection Test, 4

Tags: number theory , diophantine

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?

2014 Hanoi Open Mathematics Competitions, 11

Tags: number theory , diophantine equation , diophantine

Find all pairs of integers $(x,y)$ satisfying the following equality $8x^2y^2 + x^2 + y^2 = 10xy$

2018 Hanoi Open Mathematics Competitions, 3

Tags: number theory , diophantine , inequalities

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. $12$

2004 May Olympiad, 4

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

Find all the natural numbers $x, y, z$ that satisfy simultaneously $$\begin{cases} x y z=4104 \\ x+y+z=77 \end{cases}$$

2013 Junior Balkan Team Selection Tests - Romania, 2

Tags: diophantine equation , diophantine , number theory

Find all positive integers $x,y,z$ such that $7^x + 13^y = 8^z$

2011 Junior Balkan Team Selection Tests - Romania, 4

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

Show that there is an infinite number of positive integers $t$ such that none of the equations $$ \begin{cases} x^2 + y^6 = t \\ x^2 + y^6 = t + 1 \\ x^2 - y^6 = t \\ x^2 - y^6 = t + 1 \end{cases}$$ has solutions $(x, y) \in Z \times Z$.

2017 Latvia Baltic Way TST, 14

Tags: diophantine , number theory , diophantine equation

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

2018 IMO Shortlist, N5

Tags: number theory , diophantine

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?

1948 Moscow Mathematical Olympiad, 154

Tags: combinatorics , number theory , inequalities , diophantine

How many different integer solutions to the inequality $|x| + |y| < 100$ are there?

1964 Swedish Mathematical Competition, 2

Tags: diophantine , number theory , diophantine equation

Find all positive integers $m, n$ such that $n + (n+1) + (n+2) + ...+ (n+m) = 1000$.

2002 Junior Balkan Team Selection Tests - Romania, 2

Tags: number theory , diophantine , diophantine equation , divisor

Let $k,n,p$ be positive integers such that $p$ is a prime number, $k < 1000$ and $\sqrt{k} = n\sqrt{p}$. a) Prove that if the equation $\sqrt{k + 100x} = (n + x)\sqrt{p}$ has a non-zero integer solution, then $p$ is a divisor of $10$. b) Find the number of all non-negative solutions of the above equation.

2013 Saudi Arabia BMO TST, 3

Tags: diophantine , number theory , diophantine equation

Find all positive integers $x, y, z$ such that $2^x + 21^y = z^2$