Olympiad problems

1997 Tuymaada Olympiad, 2

Solve in natural numbers the system of equations $3x^2+6y^2+5z^2=1997$ and $3x+6y+5z=161$ .

PEN H Problems, 39

Tags: quadratics , Diophantine Equations

Let $A, B, C, D, E$ be integers, $B \neq 0$ and $F=AD^{2}-BCD+B^{2}E \neq 0$. Prove that the number $N$ of pairs of integers $(x, y)$ such that \[Ax^{2}+Bxy+Cx+Dy+E=0,\] satisfies $N \le 2 d( \vert F \vert )$, where $d(n)$ denotes the number of positive divisors of positive integer $n$.

PEN H Problems, 69

Tags: number theory , relatively prime , Diophantine Equations

Determine all positive rational numbers $r \neq 1$ such that $\sqrt[r-1]{r}$ is rational.

PEN H Problems, 65

Tags: modular arithmetic , number theory , Diophantine Equations , pen

Determine all pairs $(x, y)$ of integers such that \[(19a+b)^{18}+(a+b)^{18}+(19b+a)^{18}\] is a nonzero perfect square.

PEN H Problems, 4

Tags: conics , ellipse , parameterization , greatest common divisor , Diophantine Equations

Find all pairs $(x, y)$ of positive rational numbers such that $x^{2}+3y^{2}=1$.

2022 Czech-Polish-Slovak Junior Match, 2

Tags: number theory , diophantine , Diophantine Equations

Solve the following system of equations in integer numbers: $$\begin{cases} x^2 = yz + 1 \\ y^2 = zx + 1 \\ z^2 = xy + 1 \end{cases}$$

PEN H Problems, 88

Tags: induction , number theory , Diophantine equation , Diophantine Equations

(Leo Moser) Show that the Diophantine equation \[\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}+\frac{1}{x_{1}x_{2}\cdots x_{n}}= 1\] has at least one solution for every positive integers $n$.

2014 IMAC Arhimede, 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.

1979 IMO, 2

Tags: algebra , system of equations , Diophantine Equations , IMO , Cauchy-Schwarz inequality

Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$

2015 India Regional MathematicaI Olympiad, 3

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

Find all integers $a,b,c$ such that $a^{2}=bc+4$ and $b^{2}=ca+4$.

PEN H Problems, 51

Tags: modular arithmetic , Diophantine Equations

Prove that the product of five consecutive positive integers is never a perfect square.

PEN H Problems, 45

Tags: Diophantine Equations

Show that there cannot be four squares in arithmetical progression.

PEN H Problems, 13

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

Find all pairs $(x,y)$ of positive integers that satisfy the equation \[y^{2}=x^{3}+16.\]

PEN H Problems, 27

Tags: geometry , perimeter , inradius , modular arithmetic , Diophantine Equations

Prove that there exist infinitely many positive integers $n$ such that $p=nr$, where $p$ and $r$ are respectively the semi-perimeter and the inradius of a triangle with integer side lengths.

PEN H Problems, 34

Tags: quadratics , Diophantine Equations

Are there integers $m$ and $n$ such that $5m^2 -6mn+7n^2 =1985$?

2022 Czech-Polish-Slovak Junior Match, 4

Tags: number theory , Diophantine equation , Diophantine Equations , diophantine

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

PEN H Problems, 78

Tags: Diophantine Equations , pen

Let $x, y$, and $z$ be integers with $z>1$. Show that \[(x+1)^{2}+(x+2)^{2}+\cdots+(x+99)^{2}\neq y^{z}.\]

PEN H Problems, 73

Tags: greatest common divisor , Diophantine Equations

Find all pairs $(a,b)$ of positive integers that satisfy the equation \[a^{b^{2}}= b^{a}.\]

PEN H Problems, 35

Tags: algebra , polynomial , Diophantine Equations

Find all cubic polynomials $x^3 +ax^2 +bx+c$ admitting the rational numbers $a$, $b$ and $c$ as roots.

PEN H Problems, 11

Tags: Diophantine Equations

Find all $(x,y,n) \in {\mathbb{N}}^3$ such that $\gcd(x, n+1)=1$ and $x^{n}+1=y^{n+1}$.

PEN H Problems, 10

Tags: modular arithmetic , Diophantine Equations , pen

Prove that there are unique positive integers $a$ and $n$ such that \[a^{n+1}-(a+1)^{n}= 2001.\]

PEN H Problems, 84

Tags: Diophantine Equations

For what positive numbers $a$ is \[\sqrt[3]{2+\sqrt{a}}+\sqrt[3]{2-\sqrt{a}}\] an integer?

PEN H Problems, 14

Tags: Diophantine Equations , pen

Show that the equation $x^2 +y^5 =z^3$ has infinitely many solutions in integers $x, y, z$ for which $xyz \neq 0$.

2022 German National Olympiad, 4

Tags: number theory , number theory proposed , Diophantine equation , Diophantine Equations , cubic equation

Determine all $6$-tuples $(x,y,z,u,v,w)$ of integers satisfying the equation \[x^3+7y^3+49z^3=2u^3+14v^3+98w^3.\]

PEN H Problems, 29

Tags: quadratics , algebra , quadratic formula , Diophantine Equations

Find all pairs of integers $(x, y)$ satisfying the equality \[y(x^{2}+36)+x(y^{2}-36)+y^{2}(y-12)=0.\]