Found problems: 916
Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.
a) Given that integers $a$ and $b$ satisfy the equality $$a^2 - (b^2 - 4b + 1) a - (b^4 - 2b^3) = 0 \,\,\, (*)$$, prove that $b^2 + a$ is a square of an integer.
b) Do there exist an infinitely many of pairs $(a,b)$ satisfying (*)?
Let $p$ be a fixed prime. Find all pairs $(x,y)$ of positive numbers satisfying $p(x-y) = xy$.
Solve the equation in nonnegative integers:\\
$2^x=5^y+3$
Find all pairs $(x, y)$ of positive rational numbers such that $x^{2}+3y^{2}=1$.
Is there a positive integer $m$ such that the equation \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{abc}= \frac{m}{a+b+c}\] has infinitely many solutions in positive integers $a, b, c \;$?
Find all positive integers $n$ for which the equation $$(x^2 + y^2)^n = (xy)^{2016}$$ has positive integer solutions.
Solve the equation $p^3-q^3=pq^3-1$ in primes $p,q$.
Find all the positive integers $a, b, c$ such that $$a! \cdot b! = a! + b! + c!$$
Solve the system of equations in integers:
$$\begin{cases} a^3=abc+2a+2c \\ b^3=abc-c \\ c^3=abc-a+b \end{cases}$$
Show that the equation $x^7 + y^7 = {1998}^z$ has no solution in positive integers.
Find all integer solutions $(p, q, r)$ of the equation $r + p ^ 4 = q ^ 4$ with the following conditions:
$\bullet$ $r$ is a positive integer with exactly $8$ positive divisors.
$\bullet$ $p$ and $q$ are prime numbers.
Find the smallest natural number n such that for all integers $m > n$ there are positive integers $x$ and $y$ for which the equality 1$7x + 23y = m$ holds
The cost of $1000$ grams of chocolate is $x$ dollars and the cost of $1000$ grams of potatoes is $y$ dollars, the numbers $x$ and $y$ are positive integers and have not more than $2$ digits. Mother said to Maria to buy $200$ grams of chocolate and $1000$ grams of potatoes that cost exactly $N$ dollars. Maria got confused and bought $1000$ grams of chocolate and $200$ grams of potatoes that cost exactly $M$ dollars ($M >N$). It turned out that the numbers $M$ and $N$ have no more than two digits and are formed of the same digits but in a different order. Find $x$ and $y$.
Determine all integers $ a$ for which the equation $ x^2\plus{}axy\plus{}y^2\equal{}1$ has infinitely many distinct integer solutions $ x,y$.
Find all positive integers $a,b,c$ and prime $p$ satisfying that
\[ 2^a p^b=(p+2)^c+1.\]
Given a prime number $n \ge 5$. Prove that for any natural number $a \le \frac{n}{2} $, we can search for natural number $b \le \frac{n}{2}$ so the number of non-negative integer solutions $(x, y)$ of the equation $ax+by=n$ to be odd*.
Clarification:
* For example when $n = 7, a = 3$, we can choose$ b = 1$ so that there number of solutions og $3x + y = 7$ to be $3$ (odd), namely: $(0, 7), (1, 4), (2, 1)$
Find all positive integers $x,y,z$ such that $x^3 - (x + y + z)^2 = (y + z)^3 + 34$
We say that a triple of integers $(x, y, z)$ is of [i]jenifer [/i] type if $x, y$, and $z$ are positive integers, with $y \ge 2$, and $$x^2 - 3y^2 = z^2 - 3.$$
a) Find a triple $(x, y, z)$ of the jenifer type with $x = 5$ and $x = 7$.
b) Show that for every $x \ge 5$ and odd there are at least two distinct triples $(x, y_1, z_1)$ and $(x, y_2, z_2)$ of jenifer type.
c) Find some triple $(x, y, z)$ of jenifer type with $x$ even.
Find the number of integer solutions of the equation
$x^{2016} + (2016! + 1!) x^{2015} + (2015! + 2!) x^{2014} + ... + (1! + 2016!) = 0$
For some integer n the equation $x^2 + y^2 + z^2 -xy -yz - zx = n$ has an integer solution $x, y, z$. Prove that the equation$ x^2 + y^2 - xy = n$ also has an integer solution $x, y$.
Alexandr Yuran
Find all pairs of coprime positive integers $(p,q)$ such that $p^2+2q^2+334=[p^2,q^2]$ where $[p^2,q^2]$ is the leact common multiple of $p^2,q^2$ .
Let $b$ be a positive integer. Determine all $2002$-tuples of non-negative integers $(a_{1}, a_{2}, \cdots, a_{2002})$ satisfying \[\sum^{2002}_{j=1}{a_{j}}^{a_{j}}=2002{b}^{b}.\]
Positive integer $k(\ge 8)$ is given. Prove that if there exists a pair of positive integers $(x,y)$ that satisfies the conditions below, then there exists infinitely many pairs $(x,y)$.
(1) $ $ $x\mid y^2-3, y\mid x^2-2$
(2) $ $ $gcd\left(3x+\frac{2(y^2-3)}{x},2y+\frac{3(x^2-2)}{y}\right)=k$ $ $
Find all integers $x,y$ satisfy the $x^3 + y^3 = 3xy$.