Find all triplets of positive integers $(a,b,c)$ for which the number $3^a+3^b+3^c$ is a perfect square.

In a chess festival that is held in a school with $2017$ students, each pair of students played at most one match versus each other. In the end, it is seen that for any pair of students which have played a match versus each other, at least one of them has played at most $22$ matches. What is the maximum possible number of matches in this event?

In a convex quadrilateral $ABCD$ whose diagonals intersect at point $E$, the equalities$$\dfrac{|AB|}{|CD|}=\dfrac{|BC|}{|AD|}=\sqrt{\dfrac{|BE|}{|ED|}}$$hold. Prove that $ABCD$ is either a paralellogram or a cyclic quadrilateral

If real numbers $a>b>1$ satisfy the inequality$$(ab+1)^2+(a+b)^2\leq 2(a+b)(a^2-ab+b^2+1)$$what is the minimum possible value of $\dfrac{\sqrt{a-b}}{b-1}$