If $a_0, a_1,\cdots, a_n \in \{0, 1, 2,\cdots, 9\}, n\ge 1, a_0\ge 1$, then the zeros of $f(x)=a_0 x^n + a_1x^{n-1} +\cdots +a_n$ have real parts less then 4.
A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe.
Let $a$, $b$, and $c$ be three odd integers. Prove the equation $ax^2 + bx + c=0$ does not have rational roots.
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 $x^2+y^2+z^2-2xyz=0$
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.
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 $x^2 + y^2 +z^2 - 2xyz=0$.
For every point on the plane, one of $ n$ colors are colored to it such that:
$ (1)$ Every color is used infinitely many times.
$ (2)$ There exists one line such that all points on this lines are colored exactly by one of two colors.
Find the least value of $ n$ such that there exist four concyclic points with pairwise distinct colors.
Show that the sum and difference of two squares cannot be both squares themselves.
If for a given positive integer $n$, the equation $x^n + y^n = z^n$ is not solvable in positive integer. Show that the equation $$x^{2n} + y^{2n} = z^{2n}$$
is not solvable in positive integers either.
Show that the equation $$x^2 + y^2 -19xy - 19 =0$$ is not solvable in integers.
Solve in positive integers $$x^3 + y^3 + z^3 = 3xyz$$
What is the smallest integer n for which any subset of {1, 2, 3, . . . , 20} of size $n$ must contain two numbers that differ by 8?
Let $a$ and $b$ be non-negative real numbers such that $a + b = 2$. Show that: $$\frac{1}{a^2+1}+\frac{1}{b^2 +1} \le \frac{2}{ab+1}$$
Show that if $a$, $b$ and $c$ are odd integers, then the equation $ax^2 + bx + c=0$ has no integer solution.