HMIC 2019

Let $A B C$ be an acute scalene triangle with incenter $I$. Show that the circumcircle of $B I C$ intersects the Euler line of $A B C$ in two distinct points. (Recall that the Euler line of a scalene triangle is the line that passes through its circumcenter, centroid, orthocenter, and the nine-point center.)

Annie has a permutation $\left(a_{1}, a_{2}, \ldots, a_{2019}\right)$ of $S=\{1,2, \ldots, 2019\}$, and Yannick wants to guess her permutation. With each guess Yannick gives Annie an $n$-tuple ( $y_{1}, y_{2}, \ldots, y_{2019}$ ) of integers in $S$, and then Annie gives the number of indices $i \in S$ such that $a_{i}=y_{i}$. (a) Show that Yannick can always guess Annie's permutation with at most 1200000 guesses. (b) Show that Yannick can always guess Annie's permutation with at most 24000 guesses.

Do there exist four points $P_{i}=\left(x_{i}, y_{i}\right) \in \mathbb{R}^{2}(1 \leq i \leq 4)$ on the plane such that:

• for all $i=1,2,3,4$, the inequality $x_{i}^{4}+y_{i}^{4} \leq x_{i}^{3}+y_{i}^{3}$ holds, and
• for all $i \neq j$, the distance between $P_{i}$ and $P_{j}$ is greater than 1?

A cactus is a finite simple connected graph where no two cycles share an edge. Show that in a nonempty cactus, there must exist a vertex which is part of at most one cycle.

Let $p=2017$ be a prime and $\mathbb{F}_{p}$ be the integers modulo $p$. A function $f: \mathbb{Z} \rightarrow \mathbb{F}_{p}$ is called good if there is $\alpha \in \mathbb{F}_{p}$ with $\alpha \not \equiv 0(\bmod p)$ such that

$$f(x) f(y)=f(x+y)+\alpha^{y} f(x-y) (\bmod p)$$

for all $x, y \in \mathbb{Z}$. How many good functions are there that are periodic with minimal period 2016 ?