CONO-SUR 2021

We say that a positive integer is guarani if the sum of the number with its reverse is a number that only has odd digits. For example, $249$ and $30$ are guarani, since $249 + 942 = 1191$ and $30 + 03 = 33$ . a) How many $2021$ -digit numbers are guarani? b) How many $2023$ -digit numbers are guarani?

Let $ABC$ be a triangle and $I$ its incenter. The lines $BI$ and $CI$ intersect the circumcircle of $ABC$ again at $M$ and $N$ , respectively. Let $C_1$ and $C_2$ be the circumferences of diameters $NI$ and $MI$ , respectively. The circle $C_1$ intersects $AB$ at $P$ and $Q$ , and the circle $C_2$ intersects $AC$ at $R$ and $S$ . Show that $P$ , $Q$ , $R$ and $S$ are concyclic.

In a tennis club, each member has exactly $k > 0$ friends, and a tournament is organized in rounds such that each pair of friends faces each other in matches exactly once. Rounds are played in simultaneous matches, choosing pairs until they cannot choose any more (that is, among the unchosen people, there is not a pair of friends which has its match pending). Determine the maximum number of rounds the tournament can have, depending on $k$ .

In a heap there are $2021$ stones. Two players $A$ and $B$ play removing stones of the pile, alternately starting with $A$ . A valid move for $A$ consists of remove $1, 2$ or $7$ stones. A valid move for B is to remove $1, 3, 4$ or $6$ stones. The player who leaves the pile empty after making a valid move wins. Determine if some of the players have a winning strategy. If such a strategy exists, explain it.

Given an integer $n \geq 3$ , determine if there are $n$ integers $b_1, b_2, \dots , b_n$ , distinct two-by-two (that is, $b_i \neq b_j$ for all $i \neq j$ ) and a polynomial $P(x)$ with coefficients integers, such that $P(b_1) = b_2, P(b_2) = b_3, \dots , P(b_{n-1}) = b_n$ and $P(b_n) = b_1$ .

Let $ABC$ be a scalene triangle with circle $\Gamma$ . Let $P,Q,R,S$ distinct points on the $BC$ side, in that order, such that $\angle BAP = \angle CAS$ and $\angle BAQ = \angle CAR$ . Let $U, V, W, Z$ be the intersections, distinct from $A$ , of the $AP, AQ, AR$ and $AS$ with $\Gamma$ , respectively. Let $X = UQ \cap SW$ , $Y = PV \cap ZR$ , $T = UR \cap VS$ and $K = PW \cap ZQ$ . Suppose that the points $M$ and $N$ are well determined, such that $M = KX \cap TY$ and $N = TX \cap KY$ . Show that $M, N, A$ are collinear.