Found problems: 436
For a given natural number $ n $, find the number of solutions to the equation $ \sqrt{x} + \sqrt{y} = n $ in natural numbers $ x, y $.
For prime numbers $p$ and $q$, natural numbers $n$, $k$, $r$, the equality $p^{2k}+q^{2n}=r^2$ holds. Prove that the number $r$ is prime.
Determine all pairs $(x, y)$ of integers for which satisfy the equality $\sqrt{x-\sqrt{y}}+ \sqrt{x+\sqrt{y}}= \sqrt{xy}$
Find all integers $x,y$ that satisfy the equation $x+y=x^2-xy+y^2$
Are there any (not necessarily positive) integers $m$ and $n$ such that
a) $\frac{1}{m}-\frac{1}{n}=\frac{1}{m-n}$ ?
b) $\frac{1}{m}-\frac{1}{n}=\frac{1}{n-m}$
Assume that $a$ and $b$ are integers. Prove that the equation $a^2 +b^2 +x^2 = y^2$ has an integer solution $x,y$ if and only if the product $ab$ is even.
For $r> 0$ denote by $B_r$ the set of points at distance at most $r$ length units from the origin.
If $P_r$ is the set of the points in $B_r$ whit integer coordinates, show that the equation $$xy^3z + 2x^3z^3-3x^5y = 0$$
has an odd number of solutions $(x, y, z)$ in $P_r$.
Find all quadruples $(p, q, m, n)$ of natural numbers such that $p$ and $q$ are prime and the the following equation is fulfilled: $$p^m - q^3 = n^3$$
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 $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$
Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^2$
Find all the natural numbers $N$ which satisfy the following properties:
(i) $N$ has exactly $6$ distinct factors $1, d_1, d_2, d_3, d_4, N$ and
(ii) $1 + N = 5(d_1 + d_2+d_3 + d_4)$.
Justify your answers.
Find all integers $m, n$ such that $2n^3 - m^3 = mn^2 + 11$.
Find all positive integers $x$ and $y$ such that $x+y^2+z^3 = xyz$, where $z$ is the greatest common divisor of $x$ and $y$
Find all quadruples $(p, q, r, n)$ of prime numbers $p, q, r$ and positive integer numbers $n$, such that
$$p^2 = q^2 + r^n$$
(Walther Janous)
There are positive integers $x, y$ such that $3x^2 + x = 4y^2 + y$, and $(x - y)$ is equal to
(A): $2013$ (B): $2014$ (C): $2015$ (D): $2016$ (E): None of the above.
(a) Let $a$ and $b$ be rational numbers such that line $y = ax + b$ intersects the circle $x^2 + y^2 = 5$ at two different points. Show that if one of the intersections has two rational coordinates, so does the other intersection.
(b) Show that there are infinitely many triples ($k, n, m$) that are such that $k^2 + n^2 = 5m^2$, where $k, n$ and $m$ are integers, and not all three have any in common prime factor.
Prove that $(0,1)$, $(0, -1)$,$( -1,1)$ and $(-1,-1)$ are the only integer solutions of $$x^2 + x +1 = y^2.$$
Find all the integral solutions of the equation $\left( 1+\frac{1}{x}\right)^{x+1}=\left( 1+\frac{1}{1999}\right)^{1999}$
Let $n$ be a positive integer. Determine all positive integers $p$ for which there exist positive integers $x_1 < x_2 <...< x_n$ such that $\frac{1}{x_1}+\frac{2}{x_2}+ ... +\frac{n}{x_n}= p$
Irish Mathematical Olympiad
Find all pairs of integers $(x, y)$ such that $y^3 = 8x^6 + 2x^3 y -y^2$.
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.
Find all pairs of positive integers $m$ and $n$ such that $$m! + n! = m^n.$$
.
Prove that the number of solutions in $ \left( \mathbb{N}\cup\{ 0 \} \right)\times \left( \mathbb{N}\cup\{ 0 \} \right)\times \left( \mathbb{N}\cup\{ 0 \} \right) $ of the parametric equation
$$ \sqrt{x^2+y+n}+\sqrt{y^2+x+n} = z, $$
is greater than zero and finite, for nay natural number $ n. $
A triplet of positive integers $(x, y, z)$ satisfying $x, y, z > 1$ and $x^3 - yz^3 = 2021$ is called [i]primary [/i] if at least two of the integers $x, y, z$ are prime numbers.
a) Find at least one primary triplet.
b) Show that there are infinitely many primary triplets.
Find all integer solutions of the equation $14^x - 3^y = 2015$.
(Malik Talbi)