Determine for what $n\ge 3$ integer numbers, it is possible to find positive integer numbers $a_1 < a_2 < ...< a_n$ such $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}=1$ and $a_1 a_2\cdot\cdot\cdot a_n$ is a perfect square.

Find all the real numbers $k$ that have the following property: For any non-zero real numbers $a$ and $b$, it is true that at least one of the following numbers: $$a, b,\frac{5}{a^2}+\frac{6}{b^3}$$is less than or equal to $k$.

In the trapezoid $ABCD$ , the base $AB$ is smaller than the $CD$ base. The point $K$ is chosen such that $AK$ is parallel to BC and $BK$ is parallel to $AD$. The points $P$ and $Q$ are chosen on the $AK$ and $BK$ rays respectively, such that $\angle ADP = \angle BCK$ and $\angle BCQ = \angle ADK$. (a) Show that the lines $AD, BC$ and $PQ$ go through the same point. (b) Assuming that the circumscribed circumferences of the $APD$ and $BCQ$ triangles intersect at two points, show that one of those points belongs to the line $PQ$.

A board that has some of its squares painted black is called acceptable if there are no four black squares that form a $2 \times 2$ subboard. Find the largest real number $\lambda$ such that for every positive integer $n$ the following proposition holds: mercy: if an $n \times n$ board is acceptable and has fewer than $\lambda n^2$ dark squares, then an additional square black can be painted so that the board is still acceptable.