Found problems: 916
Are there three integers $x,y,z$, such that $x^2 + y^3 = z^4$?
(a) Let $x$ and $y$ be two positive integers such that $\sqrt{x} +\sqrt{y}$ is an integer.
Show that $\sqrt{x}$ and $\sqrt{y}$ are both integers.
(b) Find all positive integers $x$ and $y$ such that $\sqrt{x} +\sqrt{y}=\sqrt{2007}$.
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 $p(n)=3$.
(a) Let $ n$ be a positive integer. Prove that there exist distinct positive integers $ x, y, z$ such that
\[ x^{n\minus{}1} \plus{} y^n \equal{} z^{n\plus{}1}.\]
(b) Let $ a, b, c$ be positive integers such that $ a$ and $ b$ are relatively prime and $ c$ is relatively prime either to $ a$ or to $ b.$ Prove that there exist infinitely many triples $ (x, y, z)$ of distinct positive integers $ x, y, z$ such that
\[ x^a \plus{} y^b \equal{} z^c.\]
For a positive integer $ n,$ $ a_{n}\equal{}n\sqrt{5}\minus{} \lfloor n\sqrt{5}\rfloor$. Compute the maximum value and the minimum value of $ a_{1},a_{2},\ldots ,a_{2009}.$
Find all solutions in positive integers to: $$\begin{cases} x_1^4 + 14x_1x_2 + 1 = y_1^4 \\ x_2^4 + 14x_2x_3 + 1 = y_2^4 \\ ... \\ x_n^4 + 14x_nx_1 + 1 = y_n^4 \end{cases}$$
Determine all triplet of non-negative integers $ (x,y,z) $ satisfy $$ 2^x3^y+1=7^z $$
Find all triples $(x, p, n)$ of non-negative integers such that $p$ is prime and $2x(x + 5) = p^n + 3(x - 1)$.
Find integers $m$ and $n$ such that $(5 + 3 \sqrt2)^m = (3 + 5 \sqrt2)^n$.
Find the positive integer solutions of the equation $$ x+ \frac{1}{y+ \frac{1}{z}}= \frac{10}{7}$$
(G. Galperin)
Solve in natural numbers the equation $x^2-y^2=105$.
Find all ordered triples $(a,b, c)$ of positive integers which satisfy $5^a + 3^b - 2^c = 32$
Let $k$ be a given positive integer. Find all triples of positive integers $a, b, c$, such that
$a + b + c = 3k + 1$,
$ab + bc + ca = 3k^2 + 2k$.
Slovakia
Find all pairs of natural numbers $(p, n)$ with $p$ prime such that $p^6+p^5+n^3+n=n^5+n^2$.
Solve in positive integers the equation $10^{a}+2^{b}-3^{c}=1997$.
Find all the prime numbers $p, q$ and $r$ such that $p^2 + 1 = 74 (q^2 + r^2)$.
Find all triples of positive integers $(x, y, z)$ that satisfy the equation
$$2020^x + 2^y = 2024^z.$$
[i]Proposed by Ognjen Tešić, Serbia[/i]
solve this diophantine equation y^2 = x^3 - 16
Suppose that $m$ and $n$ are integers, such that both the quadratic equations $x^2+mx-n=0$ and $x^2-mx+n=0$ have integer roots. Prove that $n$ is divisible by $6$.
Find all primes $p$ and all positive integers $a$ and $m$ such that $a\leq 5p^2$ and $(p-1)!+a=p^m$
Given $m\in\mathbb{N}$ and a prime number $p$, $p>m$, let
\[M=\{n\in\mathbb{N}\mid m^2+n^2+p^2-2mn-2mp-2np \,\,\, \text{is a perfect square} \} \]
Prove that $|M|$ does not depend on $p$.
[i]Proposed by Aleksandar Ivanov[/i]
Determine all triplets $(x, y, z)$ of positive integers with $x \le y \le z$ that satisfy $\left(1+\frac1x \right)\left(1+\frac1y \right)\left(1+\frac1z \right) = 3$
Prove that no integer $x$ and $y$ satisfy: $$3x^2 = 9 + y^3.$$
Show that the only solutions of the equation $x^{3}-3xy^2 -y^3 =1$ are given by $(x,y)=(1,0),(0,-1),(-1,1),(1,-3),(-3,2),(2,1)$.
Prove that the equation \[ x^n + 1 = y^{n+1}, \] where $n$ is a positive integer not smaller then 2, has no positive integer solutions in $x$ and $y$ for which $x$ and $n+1$ are relatively prime.