Find the smallest positive integer $n$ which can not be expressed as $n =\frac{2^a - 2^b}{2^c - 2^d}$ for some positive integers $a, b, c, d$

Find all functions $f : R \to R$ such that $f( 2x^3 + f (y)) = y + 2x^2 f (x)$ for all real numbers $x, y$.

The partition of $2n$ positive integers into $n$ pairs is called square-free if the product of numbers in each pair is not a perfect square.Prove that if for $2n$ distinct positive integers, there exists one square-free partition, then there exists at least $n!$ square-free partitions.

Let $ABC$ be an acute, non isosceles with $I$ is its incenter. Denote $D, E$ as tangent points of $(I)$ on $AB,AC$, respectively. The median segments respect to vertex $A$ of triangles $ABE$ and $ACD$ meet$ (I)$ at$ P,Q,$ respectively. Take points $M, N$ on the line $DE$ such that $AM \perp BE$ and $AN \perp C D$ respectively. a) Prove that $A$ lies on the radical axis of $(MIP)$ and $(NIQ)$. b) Suppose that the orthocenter $H$ of triangle $ABC$ lies on $(I)$. Prove that there exists a line which is tangent to three circles of center $A, B, C$ and all pass through $H$.

Let $ABC$ be a triangle with $M, N, P$ as midpoints of the segments $BC, CA,AB$ respectively. Suppose that $I$ is the intersection of angle bisectors of $\angle BPM, \angle MNP$ and $J$ is the intersection of angle bisectors of $\angle CN M, \angle MPN$. Denote $(\omega_1)$ as the circle of center $I$ and tangent to $MP$ at $D$, $(\omega_2)$ as the circle of center $J$ and tangent to $MN$ at $E$. a) Prove that $DE$ is parallel to $BC$. b) Prove that the radical axis of two circles $(\omega_1), (\omega_2)$ bisects the segment $DE$.

Suppose that $2018$ numbers $1$ and $-1$ are written around a circle. For every two adjacent numbers, their product is taken. Suppose that the sum of all $2018$ products is negative. Find all possible values of sum of $2018$ given numbers.

Find all positive integers $n$ such that $\phi (n)$ is a divisor of $n^2+3$.

Find all functions $f : Z \to Z$ such that $x f (2f (y) - x) + y^2 f (2x - f (y)) = \frac{(f (x))^2}{x} + f (y f (y))$ , for all $x, y \in Z$, $x \ne 0$.