Found problems: 916
Pairwise distinct prime numbers $p, q, r$ satisfy the equality $$rp^3 + p^2 + p = 2rq^2 +q^2 + q.$$
Determine all possible values of the product $pqr$.
Show that the equation $x^2 + 8z = 3 + 2y^2$ has no solutions of positive integers $x, y$ and $z$.
Find all 4-digit numbers $n$, such that $n=pqr$, where $p<q<r$ are distinct primes, such that $p+q=r-q$ and $p+q+r=s^2$, where $s$ is a prime number.
Find all integers $c$ such that the equation $(2a+b) (2b+a) =5^c$ has integer solutions.
Prove that there are infinite many positive integers $ n$ such that
$ n^2\plus{}1\mid n!$, and infinite many of those for which $ n^2\plus{}1 \nmid n!$.
Determine all nonnegative integer solutions of the equation $2^x-2^y = 1$
Can you find five prime numbers $p, q, r, s, t$ such that $p^3+q^3+r^3+s^3 =t^3$?
$\{a_n\}$ is a positive integer sequence such that $a_{i+2} = a_{i+1} +a_i$ (for all $i \ge 1$).
For positive integer $n$, de fine as $$b_n=\frac{1}{a_{2n+1}}\Sigma_{i=1}^{4n-2}a_i$$
Prove that $b_n$ is positive integer.
Find all triplets of positive integers $(x, y, z)$ such that $2^x+1=7^y+2^z$.
Find all solutions in positive integers to $(n+1)^k -1 = n!$
Let $n$ be a nonzero integer. Prove that $n^4-7n^2+1$ can never be a perfect square.
Find all triplets $(x,y,p)$ of positive integers such that $p$ be a prime number and $\frac{xy^3}{x+y}=p$
How many integer pairs $(x,y)$ satisfies $x^2+y^2=9999(x-y)$?
Find all prime numbers $p,q$, for which $p^{q+1}+q^{p+1}$ is a perfect square.
[i]Proposed by P. Boyvalenkov[/i]
Show that there are only finitely many triples $ (x,y,z)$ of positive integers satisfying the equation $ abc\equal{}2009(a\plus{}b\plus{}c).$
Prove that the product of five consecutive positive integers is never a perfect square.
Find the number of solutions to the equation$$x_1^4+x_2^4+\ldots+x_{10}^4=2011$$in the set of positive integers.
The non-negative integers $x,y$ satisfy $\sqrt{x}+\sqrt{x+60}=\sqrt{y}$. Find the largest possible value for $x$.
The number of real triples $(x , y , z )$ that satisfy the equation $x^4 + 4y^4 + z^4 + 4 = 8xyz$ is
(A): $0$, (B): $1$, (C): $2$, (D): $8$, (E): None of the above.
Find all triples $(x, y, z)$ of positive integers such that
\[x^2 + 4^y = 5^z. \]
[i]Proposed by Li4 and ltf0501[/i]
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.
Find all pairs of integers $ a, b $ that satisfy $a ^2-3a = b ^3-2$.
Show that the equation $y^{2}=x^{3}+2a^{3}-3b^2$ has no solution in integers if $ab \neq 0$, $a \not\equiv 1 \; \pmod{3}$, $3$ does not divide $b$, $a$ is odd if $b$ is even, and $p=t^2 +27u^2$ has a solution in integers $t,u$ if $p \vert a$ and $p \equiv 1 \; \pmod{3}$.
Given positive integers $a,b,c$ which are pairwise coprime. Let $f(n)$ denotes the number of the non-negative integer solution $(x,y,z)$ to the equation
$$ax+by+cz=n.$$
Prove that there exists constants $\alpha, \beta, \gamma \in \mathbb{R}$ such that for any non-negative integer $n$,
$$|f(n)- \left( \alpha n^2+ \beta n + \gamma \right) | < \frac{1}{12} \left( a+b+c \right).$$
Find all pairs of integers $(m, n)$ such that $$m+n = 3(mn+10).$$