OMM 2021

The real positive numbers $a_1, a_2,a_3$ are three consecutive terms of an arithmetic progression, and similarly, $b_1, b_2, b_3$ are distinct real positive numbers and consecutive terms of an arithmetic progression. Is it possible to use three segments of lengths $a_1, a_2, a_3$ as bases, and other three segments of lengths $b_1, b_2, b_3$ as altitudes, to construct three rectangles of equal area ?

Let $ABC$ be a triangle with $\angle ACB > 90^{\circ}$ , and let $D$ be a point on $BC$ such that $AD$ is perpendicular to $BC$ . Consider the circumference $\Gamma$ with with diameter $BC$ . A line $\ell$ passes through $D$ and is tangent to $\Gamma$ at $P$ , cuts $AC$ at $M$ (such that $M$ is in between $A$ and $C$ ), and cuts the side $AB$ at $N$ . Prove that $M$ is the midpoint of $DP$ if and only if $N$ is the midpoint of $AB$ .

Let $ABC$ be an acutangle scalene triangle with $\angle BAC = 60^{\circ}$ and orthocenter $H$ . Let $\omega_b$ be the circumference passing through $H$ and tangent to $AB$ at $B$ , and $\omega_c$ the circumference passing through $H$ and tangent to $AC$ at $C$ .

• Prove that $\omega_b$ and $\omega_c$ only have $H$ as common point.
• Prove that the line passing through $H$ and the circumcenter $O$ of triangle $ABC$ is a common tangent to $\omega_b$ and $\omega_c$ .

Note: The orthocenter of a triangle is the intersection point of the three altitudes, whereas the circumcenter of a triangle is the center of the circumference passing through it's three vertices.

If $n=\overline{a_1a_2\cdots a_{k-1}a_k}$ , be $s(n)$ such that . If $k$ is even, $s(n)=\overline{a_1a_2}+\overline{a_3a_4}\cdots+\overline{a_{k-1}a_k}$ . If $k$ is odd, $s(n)=a_1+\overline{a_2a_3}\cdots+\overline{a_{k-1}a_k}$ For example $s(123)=1+23=24$ and $s(2021)=20+21=41$ Be $n$ is $digital$ if $s(n)$ is a divisor of $n$ . Prove that among any 198 consecutive positive integers, all of them less than 2000021 there is one of them that is $digital$ .

Determine all non empty sets $C_1, C_2, C_3, \cdots$ such that each one of them has a finite number of elements, all their elements are positive integers, and they satisfy the following property: For any positive integers $n$ and $m$ , the number of elements in the set $C_n$ plus the number of elements in the set $C_m$ equals the sum of the elements in the set $C_{m + n}$ .

Note: We denote $\lvert C_n \lvert$ the number of elements in the set $C_n$ , and $S_k$ as the sum of the elements in the set $C_n$ so the problem's condition is that for every $n$ and $m$ : $$\lvert C_n \lvert + \lvert C_m \lvert = S_{n + m}$$ is satisfied.