Found problems: 916
Find all positive integers $m, n$ such that $\frac{1}{m} + \frac{1}{n} - \frac{1}{mn} =\frac{2}{5}$.
Show that for every $n\in\mathbb{N}$, $n\sqrt{2}-\lfloor n\sqrt{2}\rfloor>\frac{1}{2n \sqrt{2}}$ and that for every $\epsilon >0$, there exists an $n\in\mathbb{N}$ such that $ n\sqrt{2}-\lfloor n\sqrt{2}\rfloor < \frac{1}{2n \sqrt{2}}+\epsilon$.
Find all prime numbers $p, q$ such that$$p(p+1)(p^2+1) = q^2(q^2+q+1) + 2025.$$
[i]Proposed by Md. Fuad Al Alam[/i]
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 $a_1,a_2,...,a_{2n}$
Find all $(x,y,z) \in {\mathbb{N}}^3$ such that $x^{4}-y^{4}=z^{2}$.
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)
Find all integer solutions of the equation $p (x + y) = xy$, where $p$ is a prime number.
Find all quadruples of positive integers $(p, q, a, b)$, where $p$ and $q$ are prime numbers and $a > 1$, such that $$p^a = 1 + 5q^b.$$
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 $m$?
Prove that the number $99999+111111\sqrt{3}$ cannot be written in the form $(A+B\sqrt{3})^2$, where $A$ and $B$ are integers.
Marko chose two prime numbers $a$ and $b$ with the same number of digits and wrote them down one after another, thus obtaining a number $c$. When he decreased $c$ by the product of $a$ and $b$, he got the result $154$. Determine the number $c$.
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}.$
Find all triples ($a$,$b$,$n$) of positive integers such that $$a^3=b^2+2^n$$
Find all pairs of natural numbers $(\alpha,\beta)$ for which, if $\delta$ is the greatest common divisor of $\alpha,\beta$, and $\varDelta$ is the least common multiple of $\alpha,\beta$, then
\[ \delta + \Delta = 4(\alpha + \beta) + 2021\]
Find all pairs of natural numbers $(a, b)$ that fulfill $a^b=(a+b)^a$.
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
$$7 \cdot 4^n = a^2 + b^2 + c^2 + d^2.$$
Does there exist positive integers $a, b, c$ such that $4(ab - a - c^2) = b$?
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.
Find all natural numbers $m$ and $n$ such that $3^m+n!-1$ is the square of a natural number.