CL-MO 2021

Consider the sequence of numbers defined by $a_1 = 7$ , $a_2 = 7^7$ , $...$ , $a_n = 7^{a_{n-1}}$ for $n \ge 2$ . Determine the last digit of the decimal representation of $a_{2021}$ .

Find all polynomials $p(x)$ with real coefficients that satisfy $$4p(x^2) = 4(p(x))^2 + 4p(x)- 1$$

Consider quadrilateral $ABCD$ with $|DC| > |AD|$ . Let $P$ be a point on $DC$ such that $PC = AD$ and let $Q$ be the midpoint of $DP$ . Let $L_1$ be the line perpendicular on $DC$ passing through $Q$ and let $L_2$ be the bisector of the angle $\angle ABC$ . Let us call $X = L_1 \cap L_2$ . Show that if quadrilateral is cyclic then $X$ lies on the circumcircle of $ABCD.$