Olympiad problems

PEN H Problems, 25

What is the smallest positive integer $t$ such that there exist integers $x_{1},x_{2}, \cdots, x_{t}$ with \[{x_{1}}^{3}+{x_{2}}^{3}+\cdots+{x_{t}}^{3}=2002^{2002}\;\;?\]

PEN H Problems, 23

Tags: inequalities , Diophantine Equations

Find all $(x,y,z) \in {\mathbb{Z}}^3$ such that $x^{3}+y^{3}+z^{3}=x+y+z=3$.

2013 JBMO Shortlist, 6

Tags: number theory , Diophantine Equations , Integers

Solve in integers the system of equations: $$x^2-y^2=z$$ $$3xy+(x-y)z=z^2$$

PEN H Problems, 58

Tags: modular arithmetic , algorithm , Diophantine Equations

Solve in positive integers the equation $10^{a}+2^{b}-3^{c}=1997$.

PEN H Problems, 74

Tags: Diophantine Equations , pen

Find all pairs $(a,b)$ of positive integers that satisfy the equation \[a^{a^{a}}= b^{b}.\]

2017 Kyiv Mathematical Festival, 5

Tags: Kyiv mathematical festival , number theory , Diophantine Equations

Find all the pairs of integers $(x,y)$ for which $(x^2+y)(y^2+x)=(x+1)(y+1).$

PEN H Problems, 56

Tags: Diophantine Equations

Prove that the equation $\prod_{cyc} (x_1-x_2)= \prod_{cyc} (x_1-x_3)$ has a solution in natural numbers where all $x_i$ are different.

PEN H Problems, 38

Tags: abstract algebra , Diophantine Equations

Suppose that $p$ is an odd prime such that $2p+1$ is also prime. Show that the equation $x^{p}+2y^{p}+5z^{p}=0$ has no solutions in integers other than $(0,0,0)$.

2017 Istmo Centroamericano MO, 4

Tags: number theory , inequalities , Diophantine Equations , diophantine , system of equations

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 $p + q$.

2023 Brazil National Olympiad, 3

Tags: system of equations , Diophantine Equations , number theory

Let $n$ be a positive integer. Show that there are integers $x_1, x_2, \ldots , x_n$, [i]not all equal[/i], satisfying $$\begin{cases} x_1^2+x_2+x_3+\ldots+x_n=0 \\ x_1+x_2^2+x_3+\ldots+x_n=0 \\ x_1+x_2+x_3^2+\ldots+x_n=0 \\ \vdots \\ x_1+x_2+x_3+\ldots+x_n^2=0 \end{cases}$$ if, and only if, $2n-1$ is not prime.

PEN H Problems, 42

Tags: Diophantine Equations

Find all integers $a$ for which $x^3 -x+a$ has three integer roots.

PEN H Problems, 32

Tags: Diophantine Equations , pen

Let $n$ be a natural number. Solve in whole numbers the equation \[x^{n}+y^{n}=(x-y)^{n+1}.\]

2016 Latvia Baltic Way TST, 18

Tags: number theory , Diophantine Equations , system of equations

Solve the system of equations in integers: $$\begin{cases} a^3=abc+2a+2c \\ b^3=abc-c \\ c^3=abc-a+b \end{cases}$$

PEN H Problems, 31

Tags: Ring Theory , Diophantine Equations

Determine all integer solutions of the system \[2uv-xy=16,\] \[xv-yu=12.\]

PEN H Problems, 44

Tags: calculus , integration , Diophantine Equations

For all $n \in \mathbb{N}$, show that the number of integral solutions $(x, y)$ of \[x^{2}+xy+y^{2}=n\] is finite and a multiple of $6$.

2017 Hanoi Open Mathematics Competitions, 6

Tags: Diophantine Equations , diophantine , system of equations , number theory , parametric equation

Find all pairs of integers $a, b$ such that the following system of equations has a unique integral solution $(x , y , z )$ : $\begin{cases}x + y = a - 1 \\ x(y + 1) - z^2 = b \end{cases}$

2014 JBMO Shortlist, 2

Tags: modular arithmetic , Diophantine equation , Diophantine Equations

Find all triples of primes $(p,q,r)$ satisfying $3p^{4}-5q^{4}-4r^{2}=26$.

PEN H Problems, 7

Tags: Diophantine Equations

Determine all pairs $(x,y)$ of positive integers satisfying the equation \[(x+y)^{2}-2(xy)^{2}=1.\]

2014 Contests, 3

Tags: number theory , Diophantine equation , Diophantine Equations

a) Prove that the equation $2^x + 21^x = y^3$ has no solution in the set of natural numbers. b) Solve the equation $2^x + 21^y = z^2y$ in the set of non-negative integer numbers.

PEN H Problems, 6

Tags: Diophantine Equations , number theory , pen

Show that there are infinitely many pairs $(x, y)$ of rational numbers such that $x^3 +y^3 =9$.

PEN H Problems, 83

Tags: Diophantine Equations

Find all pairs $(a, b)$ of positive integers such that \[(\sqrt[3]{a}+\sqrt[3]{b}-1 )^{2}= 49+20 \sqrt[3]{6}.\]

PEN H Problems, 43

Tags: Diophantine Equations

Find all solutions in integers of $x^{3}+2y^{3}=4z^{3}$.

2014 Ukraine Team Selection Test, 9

Tags: number theory , prime numbers , parameterization , Diophantine Equations , 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 $x, y, z$.

PEN H Problems, 26

Tags: number theory , greatest common divisor , Diophantine Equations

Solve in integers the following equation \[n^{2002}=m(m+n)(m+2n)\cdots(m+2001n).\]

PEN H Problems, 33

Tags: geometry , 3D geometry , modular arithmetic , Diophantine Equations , pen

Does there exist an integer such that its cube is equal to $3n^2 +3n+7$, where $n$ is integer?