Found problems: 916
Let $a,b$ be nonnegative integers. Prove that $5a>7b$ if and only if there exist nonnegative integers $x,y,z,t$ such that
\begin{align*}
x+2y+3z+7t&=a,\\
y+2z+5t&=b.
\end{align*}
Find:
a) Any quadruple of positive integers $(a, k, l, m)$ such that $a^k = a^l + a^m,$
b) Any quintuple of positive integers $(a, k, l, m, n)$ for which $a^k = a^l + a^m+a^n$
Show that there are only finitely many triples $(a, b, c)$ of positive integers such that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{1000}$.
Prove that for any integer $n>1$ there exist infinitely many pairs $(x,y)$ of integers $1<x<y$, such that $x^n+y \mid x+y^n$.
([i]Dan Schwarz[/i])
Solve in positive integers:
\[x^{y^2+1}+y^{x^2+1}=2^z\]
Let $m,n > 1$ be integer numbers. Solve in positive integers $x^n+y^n = 2^m$.
Determine all positive integers $a,b,c,d$ such that$$\begin{cases} a<b \\ a^2c =b^2d \\ ab+cd =2^{99}+2^{101} \end{cases}$$
Find the number of positive integer solutions to the equation $(x_1+x_2+x_3)^2(y_1+y_2) = 2548$.
Find all solutions of the equation $(a^a)^5 = b^b$ in positive integers.
Solve the equation $2^x -5 =11^{y}$ in positive integers.
Prove that there exist infinitely many positive integers $m$ such that there exist $m$ consecutive perfect squares with sum $m^3$. Specify one solution with $m>1$.
Product of two integers is $1$ less than three times of their sum. Find those integers.
Prove that the only integer solution of the following system of equations is $u=v=x=y=z=0$: $$uv=x^2-5y^2, (u+v)(u+2v)=x^2-5z^2$$
In $\triangle PQR$, $PQ=8$, $QR=13$, and $RP=15$. Prove that there is a point $S$ on line segment $\overline{PR}$, but not at its endpoints, such that $PS$ and $QS$ are also integers.
[asy]
size(200);
defaultpen(linewidth(0.8));
pair P=origin,Q=(8,0),R=(7,10),S=(3/2,15/7);
draw(P--Q--R--cycle);
label("$P$",P,W);
label("$Q$",Q,E);
label("$R$",R,NE);
draw(Q--S,linetype("4 4"));
label("$S$",S,NW);
[/asy]
Find all pairs $(x,y)$ of positive integers such that $x^{x+y} =y^{y-x}$.
Find all nonnegative integers $x, y, z$ satisfying the equation $$2^x+31^y=z^2.$$
Find all triples of integers $(a, b, c)$ satisfying $a^2 + b^2 + c^2 =3(ab + bc + ca).$
There exists a unique pair of positive integers $k,n$ such that $k$ is divisible by $6$, and $\sum_{i=1}^ki^2=n^2$. Find $(k,n)$.
Let $ a$ be a fix natural number . Prove that the set of prime divisors of $ 2^{2^{n}} \plus{} a$ for $ n \equal{} 1,2,\cdots$ is infinite
Find all pairs of prime numbers $(p,q)$ such that for each pair $(p,q)$, there is a positive integer m satisfying
$\frac{pq}{p + q}=\frac{m^2 + 6}{m + 1}$.
Let $x, y$, and $z$ be integers with $z>1$. Show that \[(x+1)^{2}+(x+2)^{2}+\cdots+(x+99)^{2}\neq y^{z}.\]
Find all pairs of integers $(a, b)$ such that $$a^2 + ab - b = 2018.$$
Let $k$ be a positive integer. Prove that there exist integers $x$ and $y$, neither of which divisible by $7$, such that
\begin{align*}
x^2 + 6y^2 = 7^k.
\end{align*}
The number $21982145917308330487013369$ is the thirteenth power of a positive integer. Which positive integer?
Find all pairs of integers $(x,y)$ that satisfy the equation $3^4 2^3(x^2+y^2)=x^3y^3$