IT-CESENATICO 2023

Let $a, b$ be positive integers such that $54^a=a^b$ . Prove that $a$ is a power of $54$ .

Let $n$ be a positive integer. On a blackboard, Bobo writes a list of $n$ non-negative integers. He then performs a sequence of moves, each of which is as follows:

-for each $i = 1, . . . , n$ , he computes the number $a_i$ of integers currently on the board that are at most $i$ ,

-he erases all integers on the board,

-he writes on the board the numbers $a_1, a_2,\ldots , a_n$ .

For instance, if $n = 5$ and the numbers initially on the board are $0, 7, 2, 6, 2$ , after the first move the numbers on the board will be $1, 3, 3, 3, 3$ , after the second they will be $1, 1, 5, 5, 5$ , and so on.

(a) Show that, whatever $n$ and whatever the initial configuration, the numbers on the board will eventually not change any more.

(b) As a function of $n$ , determine the minimum integer $k$ such that, whatever the initial configuration, moves from the $k$ -th onwards will not change the numbers written on the board.

Let $s(n)$ denote the sum of the digits of $n$ . a) Do there exist distinct positive integers $a, b$ , such that $2023a+s(a)=2023b+s(b)$ ? b) Do there exist distinct positive integers $a, b$ , such that $a+2023s(a)=b+2023s(b)$ ?

Fix circle with center $O$ , diameter $AB$ and a point $C$ on it, different from $A, B$ . Let a point $D$ , different from $A, B$ , vary on the arc $AB$ not containing $C$ . Let $E$ lie on $CD$ such that $BE \perp CD$ . Prove that $CE \cdot ED$ is maximal exactly when $BOED$ is cyclic.

Let $a, b, c$ be reals satisfying $a^2+b^2+c^2=6$ . Find the maximal values of the expressions a) $(a-b)^2+(b-c)^2+(c-a)^2$ ; b) $(a-b)^2 \cdot (b-c)^2 \cdot (c-a)^2$ . In both cases, describe all triples for which equality holds.

Dedalo buys a finite number of binary strings, each of finite length and made up of the binary digits 0 and 1. For each string, he pays $(\frac{1}{2})^L$ drachmas, where $L$ is the length of the string. The Minotaur is able to escape the labyrinth if he can find an infinite sequence of binary digits that does not contain any of the strings Dedalo bought. Dedalo's aim is to trap the Minotaur. For instance, if Dedalo buys the strings $00$ and $11$ for a total of half a drachma, the Minotaur is able to escape using the infinite string $01010101 \ldots$ . On the other hand, Dedalo can trap the Minotaur by spending $75$ cents of a drachma: he could for example buy the strings $0$ and $11$ , or the strings $00, 11, 01$ . Determine all positive integers $c$ such that Dedalo can trap the Minotaur with an expense of at most $c$ cents of a drachma.