Found problems: 916
Determine all triples of positive integers $(a, b, n)$ that satisfy the following equation: $a! + b! = 2^n$
Prove that the equation $\prod_{cyc} (x_1-x_2)= \prod_{cyc} (x_1-x_3)$ has a solution in natural numbers where all $x_i$ are different.
Find all natural numbers $x$ and $y$ such that $x^y-y^x=1$ .
Find all triples $(a,b,p)$ of positive integers with $p$ prime and \[ a^p=b!+p. \]
[b]a)[/b] Find all solutions of the equation $p^2+q^2+r^2=pqr$, where $p,q,r$ are positive primes.\\
[b]b)[/b] Show that for every positive integer $N$, there exist three integers $a,b,c\geq N$ with $a^2+b^2+c^2=abc$.
For each positive integer $k$ find the number of solutions in nonnegative integers $x,y,z$ with $x\le y \le z$ of the equation
$$8^k=x^3+y^3+z^3-3xyz$$
Find all pairs $(x, y)$ of rational numbers such that $$xy^2=x^2+2x-3$$
Provided the equation $xyz = p^n(x + y + z)$ where $p \geq 3$ is a prime and $n \in \mathbb{N}$. Prove that the equation has at least $3n + 3$ different solutions $(x,y,z)$ with natural numbers $x,y,z$ and $x < y < z$. Prove the same for $p > 3$ being an odd integer.
The sequence $ (a_n)$ is defined by: $ a_0\equal{}a_1\equal{}1$ and $ a_{n\plus{}1}\equal{}14a_n\minus{}a_{n\minus{}1}$ for all $ n\ge 1$.
Prove that $ 2a_n\minus{}1$ is a perfect square for any $ n\ge 0$.
Suppose that $p$ and $q$ are two different positive integers and $x$ is a real number. Form the product $(x+p)(x+q).$ Find the sum $S(x,n) = \sum (x+p)(x+q),$ where $p$ and $q$ take values from 1 to $n.$ Does there exist integer values of $x$ for which $S(x,n) = 0.$
Find all pairs $x,y$ of natural numbers that satisfy the equation
$$x^2-xy+2x-3y=1997$$
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!$.
Prove that for every positive integer $n$ there exist integers $a$ and $b,$ both greater than $1,$ such that $a ^ 2 + 1 = 2b ^ 2$ and $a - b$ is a multiple of $n.$
Let $p$ be a prime number in the form $p=4k+3$. Prove that if the numbers $x_0,y_0,z_0,t_0$ are solutions of the equation $x^{2p}+y^{2p}+z^{2p}=t^{2p}$, then at least one of them is divisible by $p$. [i](Plamen Koshlukov)[/i]
Solve in $\Bbb{N}^*$ the equation $$ 4^a \cdot 5^b - 3^c \cdot 11^d = 1.$$
Find all the triples $(x,y,z)$ of positive integers such that $xy+yz+zx-xyz=2015$
Find all $(x,y,z) \in {\mathbb{Z}}^3$ such that $x^{3}+y^{3}+z^{3}=x+y+z=3$.
Does there exist an integer such that its cube is equal to $3n^2 +3n+7$, where $n$ is integer?
Let $n$ be a positive integer. Suppose that the diophantine equation
\[z^n = 8 x^{2009} + 23 y^{2009} \]
uniquely has an integer solution $(x,y,z)=(0,0,0)$. Find the possible minimum value of $n$.
Let $A = (0,0)$, $B=(-1,-1)$, $C=(x,y)$, and $D=(x+1,y)$, where $x > y$ are positive integers. Suppose points $A$, $B$, $C$, $D$ lie on a circle with radius $r$. Denote by $r_1$ and $r_2$ the smallest and second smallest possible values of $r$. Compute $r_1^2 + r_2^2$.
[i]Proposed by Lewis Chen[/i]
Find non-negative integers $a, b, c, d$ such that $5^a + 6^b + 7^c + 11^d = 1999$.
Show that the equation ${n \choose k}=m^{l}$ has no integral solution with $l \ge 2$ and $4 \le k \le n-4$.
How many distinct integral solutions of the form $(x, y)$ exist to the equation$ 21x + 22y = 43$ such that $1 < x < 11$ and $y < 22$?
Find all pairs $(a, b)$ of positive integers that satisfy the equation $a^2 -3 \cdot 2^b = 1$.
Suppose that $A=1,2,$ or $3$. Let $a$ and $b$ be relatively prime integers such that $a^{2}+Ab^2 =s^3$ for some integer $s$. Then, there are integers $u$ and $v$ such that $s=u^2 +Av^2$, $a =u^3 - 3Avu^2$, and $b=3u^{2}v -Av^3$.