Let $M, \alpha, \beta \in \mathbb{R} $ with $M > 0$ and $\alpha, \beta \in (0,1)$. If $R>1$ is a real number, we say that a sequence of positive real numbers $\{ C_n \}_{n\geq 0}$ is $R$-inoceronte if $ \sum_{i=1}^n R^{n-i}C_i \leq R^n \cdot M$ for all $n \geq 1$. Determine the smallest real $R>1$ for which exists a $R$-inoceronte sequence $ \{ C_n \}_{n\geq 0}$ such that $\sum_{n=1}^{\infty} \beta ^n C_n^{\alpha}$ diverges.

Davi and George are taking a city tour through Fortaleza, with Davi initially leading. Fortaleza is organized like an $n \times n$ grid. They start in one of the grid's squares and can move from one square to another adjacent square via a street (for each pair of neighboring squares on the grid, there is a street connecting them). Some streets are dangerous. If Davi or George pass through a dangerous street, they get scared and swap who is leading the city tour. Their goal is to pass through every block of Fortaleza exactly once. However, if the city tour ends with George in command, the entire world becomes unemployed and everyone starves to death. Given that there is at least one street that is not dangerous, prove that Davi and George can achieve their goal without everyone dying of hunger.

Let $A_1A_2 \dots A_n$ a cyclic $n$-agon with center $O$ and $P$, $Q$ being two isogonal conjugates of it (i.e, $\angle PA_{i+1}A_i = \angle QA_{i+1}A_{i+2}$ for all $i$). Let $P_i$ be the circumcenter of $\triangle PA_iA_{i+1}$ and $Q_i$ the circumcenter of $\triangle QA_iA_{i+1}$ for all $i$. Prove that: $a) ~P_1P_2 \dots P_n$ and $Q_1Q_2 \dots Q_n$ are cyclic, with centers $O_P$ and $O_Q$, respectively. $b)~O, O_P$ and $O_Q$ are collinears. $c)~O_PO_Q \mid \mid PQ.$ Remark: indices are taken modulo $n$.

Find all positive integers $n$ such that \[2n = \varphi(n)^{\frac{2}{3}}(\varphi(n)^{\frac{2}{3}}+1)\]

Régis, Ed and Rafael are at the IMO. They are going to play a game in Bath, and there are $2^n$ houses in the city. Régis and Ed will team up against Rafael. The game operates as follows: First, Régis and Ed think on a strategy and then let Rafael know it. After this, Régis and Ed no longer communicate, and the game begins. Rafael decides on an order to visit the houses and then starts taking Régis to them in that order. At each house, except for the last one, Régis choose a number between $1$ and $n$ and places it in the house. In the last house, Rafael chooses a number from $1$ to $n$ and places it there. Afterwards, Ed sees all the houses and the numbers in them, and he must guess in which house Rafael placed the number. Ed is allowed $k$ guesses. What is the smallest $k$ for which there exists a strategy for Ed and Régis to ensure that Ed correctly guess the house where Rafael placed the number?