CAPS 2021

Find all quadruples $(a, b, c, d)$ of positive integers satisfying $\gcd(a, b, c, d) = 1$ and $$a | b + c, ~ b | c + d, ~ c | d + a, ~ d | a + b.$$

Vítězslav Kala (Czech Republic)

In an acute triangle $ABC$ , the incircle $\omega$ touches $BC$ at $D$ . Let $I_a$ be the excenter of $ABC$ opposite to $A$ , and let $M$ be the midpoint of $DI_a$ . Prove that the circumcircle of triangle $BMC$ is tangent to $\omega$ .

Patrik Bak (Slovakia)

For any two convex polygons $P_1$ and $P_2$ with mutually distinct vertices, denote by $f(P_1, P_2)$ the total number of their vertices that lie on a side of the other polygon. For each positive integer $n \ge 4$ , determine $$\max \{ f(P_1, P_2) ~ | ~ P_1 ~ \text{and} ~ P_2 ~ \text{are convex} ~ n \text{-gons} \}.$$ (We say that a polygon is convex if all its internal angles are strictly less than $180^\circ$ .)

Josef Tkadlec (Czech Republic)

Determine the number of $2021$ -tuples of positive integers such that the number $3$ is an element of the tuple and consecutive elements of the tuple differ by at most $1$ .

Walther Janous (Austria)

The sequence $a_1, a_2, a_3, \ldots$ satisfies $a_1=1$ , and for all $n \ge 2$ , it holds that $$a_n= \begin{cases} a_{n-1}+3 ~~ \text{if} ~ n-1 \in \{ a_1,a_2,\ldots,,a_{n-1} \} ; a_{n-1}+2 ~~ \text{otherwise}. \end{cases}$$ Prove that for all positive integers n, we have $$a_n < n \cdot (1 + \sqrt{2}).$$

Dominik Burek (Poland) (also known as Burii)

Let $ABC$ be an acute triangle and suppose points $A, A_b, B_a, B, B_c, C_b, C, C_a,$ and $A_c$ lie on its perimeter in this order. Let $A_1 \neq A$ be the second intersection point of the circumcircles of triangles $AA_bC_a$ and $AA_cB_a$ . Analogously, $B_1 \neq B$ is the second intersection point of the circumcircles of triangles $BB_cA_b$ and $BB_aC_b$ , and $C_1 \neq C$ is the second intersection point of the circumcircles of triangles $CC_aB_c$ and $CC_bA_c$ . Suppose that the points $A_1, B_1,$ and $C_1$ are all distinct, lie inside the triangle $ABC$ , and do not lie on a single line. Prove that lines $AA_1, BB_1, CC_1,$ and the circumcircle of triangle $A_1B_1C_1$ all pass through a common point.

Josef Tkadlec (Czech Republic), Patrik Bak (Slovakia)