Determine all functions $f:\mathbb{R}^{+} \to \mathbb{R}^{+}$ such that for any positive reals $x,y$, $$f(xy+f(xy)) = xf(y) + yf(x)$$

Let $P$ and $Q$ be non-constant integer-coefficient monic polynomials, and let $a$ and $b$ be integers satisfying $| a | \geq 3$ and $ | b | \geq 3$. These satisfy the following conditions for all positive integers $n$: $$ P(n) \mid Q(n)^2 + aQ(n) + 1, \quad Q(n) \mid P(n)^2 + bP(n) + 1. $$Determine all possible ordered pairs $(a+b, \deg P)$. Original wording상수다항식이 아닌 최고차항의 계수가 1인 정수계수다항식 $P$, $Q$와 정수 $a$, $b$($| a |, | b | \geq 3$)가 모든 양의 정수 $n$에 대해 $$P(n) \mid Q(n)^2 +aQ(n)+1, \quad Q(n) \mid P(n)^2+bP(n)+1$$을 만족한다. 이때 가능한 모든 $(a+b, \deg P)$ 순서쌍을 구하여라.

n assistants start simultaneously from one vertex of a cube-shaped planet with edge length 1. Each assistant moves along the edges of the cube at a constant speed of $2, 4, 8, \cdots, 2^n$, and can only change their direction at the vertices of the cube. The assistants can pass through each other at the vertices, but if they collide at any point that is not a vertex, they will explode. Determine the maximum possible value of $n$ such that the assistants can move indefinitely without any collisions.

Let $\omega$ be the circumcircle of triangle $ABC$ with center $O$, and the $A$ inmixtilinear circle is tangent to $AB, AC, \omega$ at $D,E,T$ respectively. $P$ is the intersection of $TO$ and $DE$ and $X$ is the intersection of $AP$ and $\omega$. Prove that the isogonal conjugate of $P$ lies on the line passing through the midpoint of $BC$ and $X$.

In a convex quadrilateral $ABCD$, $\angle ABC = \angle CDA$. Let $X \neq C$ be the intersection of the circumcircle of $\triangle BCD$ and circle with diameter $AC$. Prove that the tangent to the circumcircle of $\triangle BCD$ at $X$, the tangent to the circumcircle of $\triangle ABD$ at $A$ concur on $BD$.

There are $n$ parallel lines on a plane, and there is a set $S$ of distinct points. Each point in $S$ lies on one of the $n$ lines and is colored either red or blue. Determine the minimum value of $n$ such that if $S$ satisfies the following condition, it is guaranteed that there are infinitely many red points and infinitely many blue points. Each line contains at least one red point and at least one blue point from $S$. Consider a triangle formed by three elements of $S$ located on three distinct lines. If two of the vertices of the triangle are red, there must exist a blue point, not one of the vertices, either inside or on the boundary of the triangle. Similarly, if two of the vertices are blue, there must exist a red point, not one of the vertices, either inside or on the boundary of the triangle.

There are $2025$ positive integers $a_1, a_2, \cdots, a_{2025}$ are placed around a circle. For any $k = 1, 2, \cdots, 2025$, $a_k \mid a_{k-1} + a_{k+1}$ where indices are considered modulo $n$. Prove that there exists a positive integer $N$ such that satisfies the following condition. (Condition) For any positive integer $n > N$, when $a_1 = n^n$, $a_1, a_2, \cdots, a_{2025}$ are all multiples of $n$.

Determine all triplets of positive integers $(p,m,n)$ such that $p$ is a prime, $m \neq n < 2p$ and $2 \nmid n$. Also, the following polynomial is reducible in $\mathbb{Z}[x]$ $$x^{2p} - 2px^m - p^2x^n - 1$$