CONO-SUR 2022

A positive integer is happy if:

1. All its digits are different and not $0$ ,
2. One of its digits is equal to the sum of the other digits.

For example, 253 is a happy number. How many happy numbers are there?

Given is a triangle $ABC$ with incircle $\omega$ , tangent to $BC, CA, AB$ at $D, E, F$ . The perpendicular from $B$ to $BC$ meets $EF$ at $M$ , and the perpendicular from $C$ to $BC$ meets $EF$ at $N$ . Let $DM$ and $DN$ meet $\omega$ at $P$ and $Q$ . Prove that $DP=DQ$ .

Prove that for every positive integer $n$ there exists a positive integer $k$ , such that each of the numbers $k, k^2, \dots, k^n$ have at least one block of $2022$ in their decimal representation.

For example, the numbers 4202213 and 544202212022 have at least one block of $2022$ in their decimal representation.

Ana and Beto play on a grid of $2022 \times 2022$ . Ana colors the sides of some squares on the board red, so that no square has two red sides that share a vertex. Next, Bob must color a blue path that connects two of the four corners of the board, following the sides of the squares and not using any red segments. If Beto succeeds, he is the winner, otherwise Ana wins. Who has a winning strategy?

An integer $n>1$ , whose positive divisors are $1=d_1<d_2< \cdots <d_k=n$ , is called $\textit{southern}$ if all the numbers $d_2-d_1, d_3- d_2 , \cdots, d_k-d_{k-1}$ are divisors of $n$ .

a) Find a positive integer that is $\textit{not southern}$ and has exactly $2022$ positive divisors that are $\textit{southern}$ .

b) Show that there are infinitely many positive integers that are $\textit{not southern}$ and have exactly $2022$ positive divisors that are $\textit{southern}$ .

On a blackboard the numbers $1,2,3,\dots,170$ are written. You want to color each of these numbers with $k$ colors $C_1,C_2, \dots, C_k$ , such that the following condition is satisfied: for each $i$ with $1 \leq i < k$ , the sum of all numbers with color $C_i$ divide the sum of all numbers with color $C_{i+1}$ . Determine the largest possible value of $k$ for which it is possible to do that coloring.