Decide whether there exists a positive real number \( a < 1 \) such that, for any positive real numbers \( x \) and \( y \), the inequality \[ \frac{2xy^2}{x^2 + y^2} \leq (1 - a)x + ay \]holds true.

Let \( m \) and \( n \) be positive integers. Kellem and Carmen play the following game: initially, the number $0$ is on the board. Starting with Kellem and alternating turns, they add powers of \( m \) to the previous number on the board, such that the new value on the board does not exceed \( n \). The player who writes \( n \) wins. Determine, for each pair \( (m, n) \), who has the winning strategy. Note: A power of \( m \) is a number of the form \( m^k \), where \( k \) is a non-negative integer.

Let \( ABC \) be an acute scalene triangle with orthocenter \( H \), and consider \( M \) to be the midpoint of side \( BC \). Define \( P \neq A \) as the intersection point of the circle with diameter \( AH \) and the circumcircle of triangle \( ABC \), and let \( Q \) be the intersection of \( AP \) with \( BC \). Let \( G \neq M \) be the intersection of the circumcircle of triangle \( MPQ \) with the circumcircle of triangle \( AHM \). Show that \( G \) lies on the circle that passes through the feet of the altitudes of triangle \( ABC \).

The infinite sequence \( a_1, a_2, \ldots \) is defined by \( a_1 = 1 \) and, for each \( n \geq 1 \), the number \( a_{n+1} \) is the smallest positive integer greater than \( a_n \) that has the following property: for each \( k \in \{1, 2, \ldots, n\} \), the number \( a_{n+1} + a_k \) is not a perfect square. Prove that, for all \( n \), it holds that \( a_n \leq (n - 1)^2 + 1 \).

Let \( \mathbb{N} \) be the set of all positive integers. We say that a function \( f: \mathbb{N} \to \mathbb{N} \) is Georgian if \( f(1) = 1 \) and, for every positive integer \( n \), there exists a positive integer \( k \) such that \[ f^{(k)}(n) = 1, \quad \text{where } f^{(k)} = f \circ f \cdots \circ f \quad \text{(applied } k \text{ times)}. \]If \( f \) is a Georgian function, we define, for each positive integer \( n \), \( \text{ord}(n) \) as the smallest positive integer \( m \) such that \( f^{(m)}(n) = 1 \). Determine all positive real numbers \( c \) for which there exists a Georgian function such that, for every positive integer \( n \geq 2024 \), it holds that \( \text{ord}(n) \geq cn - 1 \).

Let \( n, k \geq 1 \). In a school, there are \( n \) students and \( k \) clubs. Each student participates in at least one of the clubs. One day, a school uniform was established, which could be either blue or red. Each student chose only one of these colors. Every day, the principal visited one of the clubs, forcing all the students in it to switch the colors of the uniforms they wore. Assuming that the students are distributed in clubs in such a way that any initial choice of uniforms they make, after a certain number of days, it is possible to have at most one student with one of the colors. Show that \[ n \geq 2^{n-k-1} - 1. \]

Consider 90 distinct positive integers. Show that there exist two of them whose least common multiple is greater than 2024.

Let $ABCD$ be a cyclic quadrilateral with all distinct sides that has an inscribed circle. The incircle of $ABCD$ has center $I$ and is tangent to $AB$, $BC$, $CD$, and $DA$ at points $W$, $X$, $Y$, and $Z$, respectively. Let $K$ be the intersection of the lines $WX$ and $YZ$. Prove that $KI$ is tangent to the circumcircle of triangle $AIC$.