Found problems: 988
Determine whether there exist an infinite number of positive integers $x,y $ satisfying the condition: $x^2+y \mid x+y^2.$ Please prove it.
Show that the equation $2x^2-3x=3y^2$ has infinitely many solutions in positive integers.
Prove that $ \forall n > 1, n \in \mathbb{N}$ the equation \[ \sum^n_{k\equal{}1} \frac{x^k}{k!} \plus{} 1 \equal{} 0\] has no rational roots.
Find all pairs of integers $a, b$ such that the following system of equations has a unique integral solution $(x , y , z )$ :
$\begin{cases}x + y = a - 1 \\
x(y + 1) - z^2 = b \end{cases}$
Find all pairs $(p,q)$ of prime numbers such that
$$ p(p^2 - p - 1) = q(2q + 3) .$$
Prove that the equation $x^2 + x + 1 = py$ has solution $(x,y)$ for the infinite number of simple $p$.
Let $a, b, c $and $d$ be integers such that for all integers m and n, there exist integers $x$ and $y$ such that $ax + by = m$, and $cx + dy = n$. Prove that $ad - bc = \pm 1$.
Find all ordered triples of positive integers $(a,b,c)$ where $$\left(a+\frac{1}{a}\right)\left(b+\frac{1}{b}\right)=c+\frac{1}{c}.$$
[i]Proposed by vsamc[/i]
Find all triples $(x,y, z)$ of integers such that $$\begin{cases} x^2y + y^2z + z^2x= 2010^2 \\ xy^2 + yz^2 + zx^2= -2010 \end{cases}$$
The number of integer solutions $x$ of the equation below
$(12x -1)(6x - 1)(4x -1)(3x - 1) = 330$ is
(A): $0$, (B): $1$, (C): $2$, (D): $3$, (E): None of the above.
Find all integers $x, y, z$ satisfy the $x^4-10y^4 + 3z^6 = 21$.
For a positive integer $n$, denote $A_n=\{(x,y)\in\mathbb Z^2|x^2+xy+y^2=n\}$.
(a) Prove that the set $A_n$ is always finite.
(b) Prove that the number of elements of $A_n$ is divisible by $6$ for all $n$.
(c) For which $n$ is the number of elements of $A_n$ divisible by $12$?
Let $m, n, a, k$ be positive integers and $k>1$ such that the equality $$5^m+63n+49=a^k$$
holds. Find the minimum value of $k$.
Prove that the equation
$$3x(x-3y)=y^2+z^2$$doesn't have any integer solutions except $x=0,y=0,z=0$.
Let $p$ and $q$ be odd prime numbers. Assume that there exists a positive integer $n$ such that $pq-1= n^3$. Express $p+q$ in terms of $n$
Suppose that $2^{2n+1}+ 2^{n}+1=x^{k}$, where $k\geq2$ and $n$ are positive integers. Find all possible values of $n$.
[list=a]
[*] Find an example of three positive integers $a,b,c$ satisfying $31a+30b+28c=365$.
[*] Prove that any triplet $a,b,c$ satisfying the above condition, also satisfies $a+b+c=12$.
[/list]
Find all polynomials $f\in \mathbb{Z}[X]$ such that if $p$ is prime then $f(p)$ is also prime.
Find all integers $x,y$ such that $x^2(y-1)+y^2(x-1) = 1$.
Determine all pairs $(x, y)$ of integers such that \[1+2^{x}+2^{2x+1}= y^{2}.\]
Determine all solutions of the equation $4^x + 4^y + 4^z = u^2$ for integers $x,y,z$ and $u$.
Let $b$ be a positive integer such that $\gcd(b,6)=1$. Show that there are positive integers $x$ and $y$ such that $\frac1x+\frac1y=\frac3b$ if and only if $b$ is divisible by some prime number of form $6k-1$.
Find all integers $n > 2$ for which $(2n)! = (n-2)!n!(n+2)!$ .
Find all ordered pairs of positive integers $(m,n)$ such that :
$125*2^n-3^m=271$
Find all positive integers $n$ and prime numbers $p$ such that $$17^n \cdot 2^{n^2} - p =(2^{n^2+3}+2^{n^2}-1) \cdot n^2.$$
[i]Authored by Nikola Velov[/i]