Found problems: 988
Solve in integers the equation
\[ x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3. \]
Find all triples $(p,M, z)$ of integers, where $p$ is prime, $m$ is positive and $z$ is negative, that satisfy the equation
$$p^3 + pm + 2zm = m^2 + pz + z^2$$
Solve the equation $x^2 +7=2^n$ in integers.
Find all pairs $(n,k)$ of positive integers such that $\binom nk=2002$.
Find all natural numbers $n$, and all integers $x,y$ ($x\ne y$) for which the following equation is satisfied:
$$x + x^2 + x^4 + ...+ x^{2^n} = y + y^2 + y^4 + ... + y^{2^n} .$$
Find all square-free integers $d$ for which there exist positive integers $x, y$ and $n$ satisfying
$x^2+dy^2=2^n$
Submitted by Kada Williams, Cambridge
Find all triples of primes $(p,q,r)$ satisfying $3p^{4}-5q^{4}-4r^{2}=26$.
Source: Lusophon MO 2016
Prove that any positive power of $2$ can be written as:
$$5xy-x^2-2y^2$$
where $x$ and $y$ are odd numbers.
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}.$
Given a prime $p>3$ and an odd integer $n>0$, show that the equation $$xyz=p^n(x+y+z)$$ has at least $3(n+1)$ different solutions up to symmetry. (That is, if $(x',y',z')$ is a solution and $(x'',y'',z'')$ is a permutation of the previous, they are considered to be the same solution.)
Show that there are only finitely many integral solutions to
\[
3^m - 1 = 2^n
\]
and find them.
Find all the natural numbers $x, y, z$ that satisfy simultaneously
$$\begin{cases} x y z=4104 \\ x+y+z=77 \end{cases}$$
Find all triples of positive integers $(a,b,c)$ such that:
$a^2bc-2ab^2c-2abc^2+b^3c+bc^3+2b^2c^2=11$
Solve the equation $7^x -3^y =4$ in positive integers.
Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^3$
Find all quadruples $(x,y,z,w)$ of integers satisfying the system of equations
$$x + y + z + w = xy + yz + zx + w^2 - w = xyz - w^3 = - 1$$
Show that there are no 2-tuples $ (x,y)$ of positive integers satisfying the equation $ (x+1) (x+2)\cdots (x+2014)= (y+1) (y+2)\cdots (y+4028).$
Prove that if positive integers $x,y,z$ satisfy the equation $$x^n + y^n = z^n,$$
then $\min\, (x,y) \ge n$.
Find all natural numbers $a$, $b$, $c$ and prime numbers $p$ and $q$, such that:
$\blacksquare$ $4\nmid c$
$\blacksquare$ $p\not\equiv 11\pmod{16}$
$\blacksquare$ $p^aq^b-1=(p+4)^c$
Find all triples of integers $(x, y, z)$ such that
$$x^2 + y^2 + z^2 = 16(x + y + z).$$
Find all positive integers $l$ for which the equation
\[
a^3+b^3+ab=(lab+1)(a+b)
\]
has a solution over positive integers $a,b$.
Determine all pairs of integers $(x, y)$ satisfying the equation
\[y(x + y) = x^3- 7x^2 + 11x - 3.\]
Show that there exist infinitely many positive integers $n$ such that $n^{2}+1$ divides $n!$.
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?
Prove that there are no positive integers $x, y$ such that: $(x + 1)^2 + (x + 2)^2 +...+ (x + 9)^2 = y^2$