OIM 2021

Let $P = \{p_1,p_2,\ldots, p_{10}\}$ be a set of $10$ different prime numbers and let $A$ be the set of all the integers greater than $1$ so that their prime decomposition only contains primes of $P$ . The elements of $A$ are colored in such a way that:

• each element of $P$ has a different color,
• if $m,n \in A$ , then $mn$ is the same color of $m$ or $n$ ,
• for any pair of different colors $\mathcal{R}$ and $\mathcal{S}$ , there are no $j,k,m,n\in A$ (not necessarily distinct from one another), with $j,k$ colored $\mathcal{R}$ and $m,n$ colored $\mathcal{S}$ , so that $j$ is a divisor of $m$ and $n$ is a divisor of $k$ , simultaneously.

Prove that there exists a prime of $P$ so that all its multiples in $A$ are the same color.

Consider an acute-angled triangle $ABC$ , with $AC>AB$ , and let $\Gamma$ be its circumcircle. Let $E$ and $F$ be the midpoints of the sides $AC$ and $AB$ , respectively. The circumcircle of the triangle $CEF$ and $\Gamma$ meet at $X$ and $C$ , with $X\neq C$ . The line $BX$ and the tangent to $\Gamma$ through $A$ meet at $Y$ . Let $P$ be the point on segment $AB$ so that $YP = YA$ , with $P\neq A$ , and let $Q$ be the point where $AB$ and the parallel to $BC$ through $Y$ meet each other. Show that $F$ is the midpoint of $PQ$ .

Let $a_1,a_2,a_3, \ldots$ be a sequence of positive integers and let $b_1,b_2,b_3,\ldots$ be the sequence of real numbers given by $$b_n = \dfrac{a_1a_2\cdots a_n}{a_1+a_2+\cdots + a_n},\ \text{for}\ n\geq 1$$ Show that, if there exists at least one term among every million consecutive terms of the sequence $b_1,b_2,b_3,\ldots$ that is an integer, then there exists some $k$ such that $b_k > 2021^{2021}$ .

Let $a,b,c,x,y,z$ be real numbers such that

$$a^2+x^2=b^2+y^2=c^2+z^2=(a+b)^2+(x+y)^2=(b+c)^2+(y+z)^2=(c+a)^2+(z+x)^2$$

Show that $a^2+b^2+c^2=x^2+y^2+z^2$ .

For a finite set $C$ of integer numbers, we define $S(C)$ as the sum of the elements of $C$ . Find two non-empty sets $A$ and $B$ whose intersection is empty, whose union is the set $\{1,2,\ldots, 2021\}$ and such that the product $S(A)S(B)$ is a perfect square.

Consider a $n$ -sided regular polygon, $n \geq 4$ , and let $V$ be a subset of $r$ vertices of the polygon. Show that if $r(r-3) \geq n$ , then there exist at least two congruent triangles whose vertices belong to $V$ .