Found problems: 988
Prove that the equation \[6(6a^{2}+3b^{2}+c^{2}) = 5n^{2}\] has no solutions in integers except $a=b=c=n=0$.
Find all pairs of positive integers $(a,b)$ such that $5a^b - b = 2004$.
Find all integers $m$ for which the equation $$x^3 - mx^2 + mx - (m^2 + 1) = 0$$ has an integer solution.
Find all odd integers $k$ for which there exists a positive integer $m$ satisfying the equation
$k + (k + 5) + (k + 10) + ... + (k + 5(m - 1)) = 1372$.
Determine all pairs of integer numbers $(a, b)$ that satisfy the relation:
$$(a + b)(a + b + 6) = 34 \cdot 3^{|2a-b|}- 7$$
$14.$ If $3^x+2^y=985.$ and $3^x-2^y=473.$ What is the value of $xy ?$
Find all pairs $(x, y)$ of positive integers satisfying the equation $(x + y)^x = x^y$.
Show that $\vert 12^m -5^n\vert \ge 7$ for all $m, n \in \mathbb{N}$.
Let $a, b, c$ be positive integers such that $a$ and $b$ are relatively prime and $c$ is relatively prime either to $a$ or $b$. Prove that there exist infinitely many triples $(x, y, z)$ of distinct positive integers such that \[x^{a}+y^{b}= z^{c}.\]
Show that the following equation has finitely many solutions $(t,A,x,y,z)$ in positive integers
$$\sqrt{t(1-A^{-2})(1-x^{-2})(1-y^{-2})(1-z^{-2})}=(1+x^{-1})(1+y^{-1})(1+z^{-1})$$
Find all positive integers $a, b, c, d$ with $a \le b$ and $c \le d$ such that $\begin{cases} a + b = cd \\
c + d = ab \end{cases}$ .
$11.$ Let $a,$ $b,$ and $c$ be real numbers such that $a-7b+8c=4.$ and $8a+4b-c=7.$ What is the value of $a^2-b^2+c^2 ?$
Determine all pairs $(x,y)$ of positive integers satisfying the equation \[(x+y)^{2}-2(xy)^{2}=1.\]
Determine all integers $x,y,z$ such that \[x^2+y^2=3z^2.\]
Let a, b, c, d, and e be positive integers. Are there any solutions to $a^2+b^3+c^5+d^7=e^{11}$?
Find all solutions of the equation $$p ^ 2 + q ^ 2 + 49r ^ 2 = 9k ^ 2-101$$ with $ p$, $q$ and $r$ positive prime numbers and $k$ a positive integer.
Prove that the system \begin{align*}
x^6+x^3+x^3y+y & = 147^{157} \\
x^3+x^3y+y^2+y+z^9 & = 157^{147}
\end{align*} has no solutions in integers $x$, $y$, and $z$.
The systems of equations
\[\left\{ \begin{array}{l}
2x_1 - x_2 = 1 \\
-x_1 + 2x_2 - x_3 = 1 \\
-x_2 + 2x_3 - x_4 = 1 \\
-x_3 + 3x_4 - x_5 =1 \\
\cdots\cdots\cdots\cdots\\
-x_{n-2} + 2x_{n-1} - x_n = 1 \\
-x_{n-1} + 2x_n = 1 \\
\end{array} \right.
\]
has a solution in positive integers $x_i$. Show that $n$ must be even.
Find all positive integers $a,b,c$ such that $$\begin{cases} a^2 = 4(b+c) \\ a^3 -2b^3 -4c^3 =\frac12 abc \end {cases}$$
There is a unique triple $(a,b,c)$ of two-digit positive integers $a,\,b,$ and $c$ that satisfy the equation $$a^3+3b^3+9c^3=9abc+1.$$ Compute $a+b+c$.
Let $a$ and $b$ be positive integers such that $2a^2 + a = 3b^2 + b$. Prove that $a-b$ is a perfect square.
How many different integer solutions to the inequality $|x| + |y| < 100$ are there?
Given $a$ and $b$ distinct positive integers, show that the system of equations
$x y +zw = a$
$xz + yw = b$
has only finitely many solutions in integers $x, y, z,w$.
Natural numbers $ x, y, z $ whose greatest common divisor is equal to 1 satisfy the equation
$$\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$$
Prove that $ x + y $ is the square of a natural number.
Let $a$, $b$, $c$, $d$, $e$ be distinct positive integers such that $a^4+b^4=c^4+d^4=e^5$. Show that $ac+bd$ is a composite number.