Solve in non-negative integers the equation $$x^3 + 2y^3 = 4z^3$$
Solve in nonnegative integers the equation $$2^x -1 = xy$$
Solve in integers the equation $x^2+y^2+z^2-2xyz=0$
For any given positive integer $n > 2$, show that there exists a right triangle with all sides' lengths are integers and one side's length equals $n$.
Show that the equation $x^4 + y^4 = z^2$ is not solvable in integers if $xyz\ne 0$.
Show that $\sqrt{2}$ is an irrational number.
Show that $x^4 + y^4 = z^2$ is not solvable in positive integers.
Find all primes $p$ for which there exist positive integers $x$, $y$, and $n$ such that $$p^n = x^3+y^3$$
Let $a_1, a_2, \cdots, a_{2n+1}$ be a set of integers such that, if any one of them is removed, the remaining ones can be divided into two sets of $n$ integers with equal sums. Prove $a_1 = a_2 = \cdots = a_{2n+1}$.
Prove that if positive integer $a$ and $b$ are such that $ab+1$ divides $a^2 + b^2$. then $$\frac{a^2+b^2}{ab+1}$$ is a square number.
Find the maximal value of $m^2+n^2$ if $m$ and $n$ are integers between $1$ and $1981$ satisfying $(n^2-mn-m^2)^2=1$.
Solve in positive integers $x^2 + y^2 + x+y+1 = xyz$
Solve in positive integers $x$, $y$, $u$, $v$ the system of equations
$$
\left\{
\begin{array}{ll}
x^2 +1 &= uy\\
y^2 + 1&= vx
\end{array}
\right.
$$
Show that if there is a triple $(x, y, z)$ of positive integers such that $$x^2 +y^2 +1 = xyz$$
then $z=3$, and find all such triples.
Find all solutions of $a^3 + b^3 = 2(s^2+t^2)$
Solve in integers $x^2 + y^2 +z^2 - 2xyz=0$.
Show that there exists an infinite sequence of positive integers $a_1, a_2, \cdots$ such that $$S_n=a_1^2 + a_2^2 + \cdots + a_n^2$$ is square for any positive integer $n$.
Show that the sides of a Pythagorean triangle in which the hypotenuse exceeds the larger leg by 1 are given by $\frac{n^2-1}{2}$, $n$ and $\frac{n^2+1}{2}$
Show that if the lengths of all the three sides in a right triangle are whole numbers, then radius of its incircle is always a whole number too.
Let $a$, $b$, $c$, $d$, and $e$ be five positive integers. If $ab+c=3115$, $c^2+d^2=e^2$, both $a$ and $c$ are prime numbers, $b$ is even and has $11$ divisors. Find these five numbers
25 boys and 8 girls sit in a circle. If there are at least two boys between any two girls, how many different sitting plans are there? (Two sitting plans will be considered as the same if they differ just by rotating.)
Solve in positive integers the equation $$m^2 - n^2 - 3n = 5$$
Let $ ABC$ be acute triangle. The circle with diameter $ AB$ intersects $ CA,\, CB$ at $ M,\, N,$ respectively. Draw $ CT\perp AB$ and intersects above circle at $ T$, where $ C$ and $ T$ lie on the same side of $ AB$. $ S$ is a point on $ AN$ such that $ BT = BS$. Prove that $ BS\perp SC$.
Let $ a,\, b,\, c$ be side lengths of a triangle and $ a+b+c = 3$. Find the minimum of \[ a^{2}+b^{2}+c^{2}+\frac{4abc}{3}\]
Sequence $ \{a_{n}\}$ is defined by $ a_{1}= 2007,\, a_{n+1}=\frac{a_{n}^{2}}{a_{n}+1}$ for $ n \ge 1.$ Prove that $ [a_{n}] =2007-n$ for $ 0 \le n \le 1004,$ where $ [x]$ denotes the largest integer no larger than $ x.$