Prove for any positives $a,b,c$ the inequality $$ \sqrt[3]{\dfrac{a}{b}}+\sqrt[5]{\dfrac{b}{c}}+\sqrt[7]{\dfrac{c}{a}}>\dfrac{5}{2}$$

The set Φ consists of a finite number of points on the plane. The distance between any two points from Φ is at least $\sqrt{2}$. It is known that a regular triangle with side lenght $3$ cut out of paper can cover all points of Φ. What is the greatest number of points that Φ can consist of?

Let $p$ be a prime number of the form $4k+1$ and $\frac{m}{n}$ is an irreducible fraction such that $$\sum_{a=2}^{p-2} \frac{1}{a^{(p-1)/2}+a^{(p+1)/2}}=\frac{m}{n}.$$Prove that $p|m+n$. (Fixed, thanks Pavel)

Find all positive integers $n,k,a_1,a_2,...,a_k$ so that $n^{k+1}+1$ is divisible by $(na_1+1)(na_2+1)...(na_k+1)$

Given a checkered rectangle of size n × m. Is it always possible to mark $3$ or $4$ nodes of a rectangle so that at least one of the marked nodes lay on each straight line containing the side of the rectangle, and the non-self-intersecting polygon with vertices at these nodes has an area equal to $$\dfrac{1}{2}\min \left ( \text{gcd}(n, m), \dfrac{n+m}{\text{gcd}(n, m)} \right)$$?

The tangent line $l$ to the circumcircle of an acute triangle $ABC$ intersects the lines $AB, BC$, and $CA$ at points $C', A'$ and $B'$, respectively. Let $H$ be the orthocenter of a triangle $ABC$. On the straight lines A'H, B′H and C'H, respectively, points $A_1, B_1$ and $C_1$ (other than $H$) are marked such that $AH = AA_1, BH = BB_1$ and $CH = CC_1$. Prove that the circumcircles of triangles $ABC$ and $A_1B_1C_1$ are tangent.