A school has two classes $A$ and $B$ which have $m$ and $n$ students each. The students of the two classes sit in a circle. Each student is then given a number of candies equal to the number of consecutive students sitting to the left of him that are from his same class. After distributing the candies, the teacher decides to group the students such that in each group, all the students receive the same amount of candies, and any two students from two different groups should receive a different amount of candies. a) What is the maximum number of students that a group can have? b) Excluding the group where every student receives no candies, what is the maximum number of students that a group can have?

Given three functions $$P(x) = (x^2-1)^{2023}, Q(x) = (2x+1)^{14}, R(x) = \left(2x+1+\frac 2x \right)^{34}.$$ Initially, we pick a set $S$ containing two of these functions, and we perform some operations on it. Allowed operations include: - We can take two functions $p,q \in S$ and add one of $p+q, p-q$, or $pq$ to $S$. - We can take a function $p \in S$ and add $p^k$ to $S$ for $k$ is an arbitrary positive integer of our choice. - We can take a function $p \in S$ and choose a real number $t$, and add to $S$ one of the function $p+t, p-t, pt$. Show that no matter how we pick $S$ in the beginning, there is no way we can perform finitely many operations on $S$ that would eventually yield the third function not in $S$.

Let $ABC$ be an acute, non-isosceles triangle with circumcircle $(O)$. $BE, CF$ are the heights of $\triangle ABC$, and $BE, CF$ intersect at $H$. Let $M$ be the midpoint of $AH$, and $K$ be the point on $EF$ such that $HK \perp EF$. A line not going through $A$ and parallel to $BC$ intersects the minor arc $AB$ and $AC$ of $(O)$ at $P$, $Q$, respectively. Show that the tangent line of $(CQE)$ at $E$, the tangent line of $(BPF)$ at $F$, and $MK$ concur.

Given are two coprime positive integers $a, b$ with $b$ odd and $a>2$. The sequence $(x_n)$ is defined by $x_0=2, x_1=a$ and $x_{n+2}=ax_{n+1}+bx_n$ for $n \geq 1$. Prove that: $a)$ If $a$ is even then there do not exist positive integers $m, n, p$ such that $\frac{x_m} {x_nx_p}$ is a positive integer. $b)$ If $a$ is odd then there do not exist positive integers $m, n, p$ such that $mnp$ is even and $\frac{x_m} {x_nx_p}$ is a perfect square.

Let $ABCD$ be a convex quadrilateral with $\angle B < \angle A < 90^{o}$. Let $I$ be the midpoint of $AB$ and $S$ the intersection of $AD$ and $BC$. Let $R$ be a variable point inside the triangle $SAB$ such that $\angle ASR = \angle BSR$. On the straight lines $AR, BR$ , take the points $E, F$, respectively so that $BE , AF$ are parallel to $RS$. Suppose that $EF$ intersects the circumcircle of triangle $SAB$ at points $H, K$. On the segment $AB$, take points $M , N$ such that $\angle AHM =\angle BHI$ , $\angle BKN = \angle AKI$. a) Prove that the center $J$ of the circumcircle of triangle $SMN$ lies on a fixed line. b) On $BE, AF$ , take the points $P, Q$ respectively so that $CP$ is parallel to $SE$ and $DQ$ is parallel to $SF$. The lines $SE, SF$ intersect the circle $(SAB)$, respectively, at $U, V$. Let $G$ be the intersection of $AU$ and $BV$. Prove that the median of vertex $G$ of the triangle $GPQ$ always passes through a fixed point .

Let $n \ge 3$ be an integer and $S$ be a set of $n$ elements. Determine the largest integer $k_n$ such that: for each selection of $k_n$ $3-$subsets of $S$, there exists a way to color elements of $S$ with two colors such that none of the chosen $3-$subset is monochromatic.