Found problems: 916
For a positive integer $n$, denote $p(n)$ to be the number of nonnegative integer tuples $(x,y,z,w)$ such that $x+2y+2z+3w=n$. Also, denote $q(n)$ to be the number of nonnegative integer tuples $(a,b,c,d)$ such that
(i) $a+b+c+d=n$
(ii) $a \ge b \ge d$
(iii) $a \ge c \ge d$
Prove that for all $n$, $p(n) = q(n)$.
For each natural number $n$, determine the number $A(n)$ of all integer nonnegative solutions the equation $$5x + 2y + z = 10n.$$
Find all integer solutions to the equation $y^2=2x^4+17$.
Find all pairs of positive integers $(m, n)$ satisfying the equation $$m!+n!=m^n+1.$$
Find all pairs $(x,y)$ of positive integers that satisfy the equation \[y^{2}=x^{3}+16.\]
Find all integers \(a,b,c\) such that \(a^{2}=bc+4\) and \(b^{2}=ca+4\).
For all positive integers $n$, prove that there are integers $x, y$ relatively prime to $5$ such that $x^2 + y^2 = 5^n$.
Find all positive integer solutions of the equation $10(m+n)=mn$.
Solve in positive integers
$x^yy^x=(x+y)^z$
Find all integers $x,y,z$ such that $x^2(x^2+y)=y^{z+1}$.
Find all triples of integers $(x, y, z)$ such that $x^4 + 5y^4 = z^4$.
Solve in positive integers $x,y$ the equation $2^x - 3^y = 7$.
Can you find three natural numbers $a, b, c$ whose greatest common divisor is $1$ and which satisfy the equality
$$ab + bc + ac = (a + b -c)(b + c - a)(c + a - b) ?$$
Find all positive integers $a$ for which the equation $7an -3n! = 2020$ has a positive integer solution $n$.
(Richard Henner)
Find all pairs of positive integers $x,y$ such that $$x^3+y^3=4(x^2y+xy^2-5).$$
Find all the pairs of integers $(x,y)$ for which $(x^2+y)(y^2+x)=(x+1)(y+1).$
Find all non-negative integers $a, b, c, d$ such that $7^a = 4^b + 5^c + 6^d$
Find all prime numbers $p$ for which one can find a positive integer $m$ and nonnegative integers $a_0,a_1,...,a_m$ less than $p$ such that $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\
a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$
Let $k$ be a positive integer. Prove that there exist integers $x$ and $y$, neither of which is divisible by $3$, such that
$x^2+2y^2 = 3^k$.
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.
Let n is a natural number,for which $\sqrt{1+12n^2}$ is a whole number.Prove that $2+2\sqrt{1+12n^2}$ is perfect square.
Prove that for equation $$x^{2015} + y^{2015} = z^{2016}$$ there are infinitely many solutions where $x,y$ and $z$ are different natural numbers.
Find all positive integers $a, b, c, d,$ and $n$ satisfying $n^a + n^b + n^c = n^d$ and prove that these are the only such solutions.
Does there exist a solution to the equation \[x^{2}+y^{2}+z^{2}+u^{2}+v^{2}=xyzuv-65\] in integers with $x, y, z, u, v$ greater than $1998$?
Determine all pairs $(x, y)$ of positive integers satisfying
$x + y + 1 | 2xy$ and $ x + y - 1 | x^2 + y^2 - 1$.