Found problems: 436
Determine the prime numbers $p, q, r$ with the property that: $p(p-7) + q (q-7) = r (r-7)$.
Find all positive integers $m, n$ such that $\frac{1}{m} + \frac{1}{n} - \frac{1}{mn} =\frac{2}{5}$.
Four positive integers $x,y,z$ and $t$ satisfy the relations
\[ xy - zt = x + y = z + t. \]
Is it possible that both $xy$ and $zt$ are perfect squares?
Let $n$ be a positive integer and $a_1,a_2,...,a_{2n}$ be $2n$ distinct integers. Given that the equation $|x-a_1| |x-a_2| ... |x-a_{2n}| =(n!)^2$ has an integer solution $x = m$, find $m$ in terms of $a_1,a_2,...,a_{2n}$
Two positive integers $m, n$ satisfy the two equations $m^2 + n^2 = 3789$ and $gcd (m, n) + lcm (m, n) = 633$. Compute $m + n$.
Let $a$ be a prime number and $n > 2$ an integer.
Find all integer solutions of the equation $x^n +ay^n = a^2z^n$
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Find all positive integers $(m,n)$ such that $m^2 + n^2 + 3 = 4(m + n)$
Does there exist an infinite set of triples of integers $x, y, z$ (not necessarily positive) such that
$$x^2 + y^2 + z^2 = x^3 + y^3+z^3?$$
(NB Vassiliev)
How many integers $n$ are there those satisfy the following inequality $n^4 - n^3 - 3n^2 - 3n - 17 < 0$?
A. $4$ B. $6$ C. $8$ D. $10$ E. $12$
Find all integer solutions of the equation $p (x + y) = xy$, where $p$ is a prime number.
The positive integer $m$ and non-negative integers $x_0, x_1,..., x_{1001}$ satisfy the following equation: $$m^{x_0} =\sum_{i=1}^{1001}m^{x_i}.$$ How many possibilities are there for the value of $m$?
Find all prime $p$, for each of which there are such natural $ x$ and $y$ such that $p^x = y^3 + 1$.
Find all pairs of integers $(x, y)$ satisfying the condition $12x^2 + 6xy + 3y^2 = 28(x + y)$.
Let $n\ge 1$ be a natural number. Determine all positive integer solutions of the equation
$$7 \cdot 4^n = a^2 + b^2 + c^2 + d^2.$$
Let $k,n,p$ be positive integers such that $p$ is a prime number, $k < 1000$ and $\sqrt{k} = n\sqrt{p}$.
a) Prove that if the equation $\sqrt{k + 100x} = (n + x)\sqrt{p}$ has a non-zero integer solution, then $p$ is a divisor of $10$.
b) Find the number of all non-negative solutions of the above equation.
Find all triples of positive integers $(m,p,q)$ such that $2^mp^2 + 27 = q^3$ and $p$ is a prime.
Determine all triples of positive integers $(a, b, n)$ that satisfy the following equation: $a! + b! = 2^n$
Find all natural numbers $x$ and $y$ such that $x^y-y^x=1$ .
Find all triples $(a, b, c)$ of positive integers for which $$\begin{cases} a + bc=2010 \\ b + ca = 250\end{cases}$$
Find all pairs $x,y$ of natural numbers that satisfy the equation
$$x^2-xy+2x-3y=1997$$
Solve in integers the equation $x + y = x^2 - xy + y^2$.
Prove that there are no integers $x, y, z$ satisfying the equation $$x^2+y^2-z^2=xyz-2.$$
[i]Proposed by Navid Safaei[/i]
Find all the triples $(x,y,z)$ of positive integers such that $xy+yz+zx-xyz=2015$
Find non-negative integers $a, b, c, d$ such that $5^a + 6^b + 7^c + 11^d = 1999$.
How many distinct integral solutions of the form $(x, y)$ exist to the equation$ 21x + 22y = 43$ such that $1 < x < 11$ and $y < 22$?