CONO-SUR 2018

Let $ABCD$ be a convex quadrilateral, where $R$ and $S$ are points in $DC$ and $AB$ , respectively, such that $AD=RC$ and $BC=SA$ . Let $P$ , $Q$ and $M$ be the midpoints of $RD$ , $BS$ and $CA$ , respectively. If $\angle MPC + \angle MQA = 90$ , prove that $ABCD$ is cyclic.

Prove that every positive integer can be formed by the sums of powers of 3, 4 and 7, where do not appear two powers of the same number and with the same exponent. Example: $2= 7^0 + 7^0$ and $22=3^2 + 3^2+4^1$ are not valid representations, but $2=3^0+7^0$ and $22=3^2+3^0+4^1+4^0+7^1$ are valid representations.

Define the product $P_n=1! \cdot 2!\cdot 3!\cdots (n-1)!\cdot n!$ a) Find all positive integers $m$ , such that $\frac {P_{2020}}{m!}$ is a perfect square. b) Prove that there are infinite many value(s) of $n$ , such that $\frac {P_{n}}{m!}$ is a perfect square, for at least two positive integers $m$ .

For each interger $n\geq 4$ , we consider the $m$ subsets $A_1, A_2,\dots, A_m$ of $\{1, 2, 3,\dots, n\}$ , such that $A_1$ has exactly one element, $A_2$ has exactly two elements,...., $A_m$ has exactly $m$ elements and none of these subsets is contained in any other set. Find the maximum value of $m$ .

Let $ABC$ be an acute-angled triangle with $\angle BAC = 60^{\circ}$ and with incenter $I$ and circumcenter $O$ . Let $H$ be the point diametrically opposite(antipode) to $O$ in the circumcircle of $\triangle BOC$ . Prove that $IH=BI+IC$ .

A sequence $a_1, a_2,\dots, a_n$ of positive integers is alagoana, if for every $n$ positive integer, one have these two conditions I- $a_{n!} = a_1\cdot a_2\cdot a_3\cdots a_n$ II- The number $a_n$ is the $n$ -power of a positive integer. Find all the sequence(s) alagoana.