Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ such that if $a,b,c$ are the length sides of a triangle, and $r$ is the radius of its incircle, then $f(a),f(b),f(c)$ also form a triangle where its radius of the incircle is $f(r)$.

$\omega_1,\omega_2$ are orthogonal circles, and their intersections are $P,P'$. Another circle $\omega_3$ meets $\omega_1$ at $Q,Q'$, and $\omega_2$ at $R,R'$. (The points $Q,R,Q',R'$ are in clockwise order.) Suppose $P'R$ and $PR'$ meet at $S$, and let $T$ be the circumcenter of $\triangle PQR$. Prove that $T,Q,S$ are collinear if and only if $O_1,S,O_3$ are collinear. ($O_i$ is the center of $\omega_i$ for $i=1,2,3$.)

Find all polynomials $P(x)$ with integer coefficients such that for all positive number $n$ and prime $p$ satisfying $p\nmid nP(n)$, we have $ord_p(n)\ge ord_p(P(n))$.

A rabbit is placed on a $2n\times 2n$ chessboard. Every time the rabbit moves to one of the adjacent squares. (Adjacent means sharing an edge). It is known that the rabbit went through every square and came back to the place where the rabbit started, and the path of the rabbit form a polygon $\mathcal{P}$. Find the maximum possible number of the vertices of $\mathcal{P}$. For example the answer for the case $n=2$ would be $12$. [asy][asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(2cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -11.3, xmax = 27.16, ymin = -11.99, ymax = 10.79; /* image dimensions */ /* draw figures */ draw((5.14,3.19)--(8.43,3.22), linewidth(1)); draw((8.43,3.22)--(11.72,3.25), linewidth(1)); draw((11.72,3.25)--(11.75,-0.04), linewidth(1)); draw((11.75,-0.04)--(11.78,-3.33), linewidth(1)); draw((11.78,-3.33)--(8.49,-3.36), linewidth(1)); draw((8.49,-3.36)--(5.2,-3.39), linewidth(1)); draw((5.2,-3.39)--(5.17,-0.1), linewidth(1)); draw((5.17,-0.1)--(5.14,3.19), linewidth(1)); draw((6.785,3.205)--(6.845,-3.375), linewidth(1)); draw((8.43,3.22)--(8.49,-3.36), linewidth(1)); draw((10.075,3.235)--(10.135,-3.345), linewidth(1)); draw((5.155,1.545)--(11.735,1.605), linewidth(1)); draw((5.17,-0.1)--(11.75,-0.04), linewidth(1)); draw((11.765,-1.685)--(5.185,-1.745), linewidth(1)); draw((5.97,2.375)--(10.905,2.42), linewidth(1)); draw((10.905,2.42)--(10.92,0.775), linewidth(1)); draw((10.92,0.775)--(9.275,0.76), linewidth(1)); draw((9.275,0.76)--(9.29,-0.885), linewidth(1)); draw((9.29,-0.885)--(10.935,-0.87), linewidth(1)); draw((10.935,-0.87)--(10.95,-2.515), linewidth(1)); draw((10.95,-2.515)--(6.015,-2.56), linewidth(1)); draw((6.015,-2.56)--(6,-0.915), linewidth(1)); draw((6,-0.915)--(7.645,-0.9), linewidth(1)); draw((7.645,-0.9)--(7.63,0.745), linewidth(1)); draw((7.63,0.745)--(5.985,0.73), linewidth(1)); draw((5.985,0.73)--(5.97,2.375), linewidth(1)); /* dots and labels */ dot((5.97,2.375),linewidth(4pt) + dotstyle); dot((5.985,0.73),linewidth(4pt) + dotstyle); dot((6,-0.915),linewidth(4pt) + dotstyle); dot((6.015,-2.56),linewidth(4pt) + dotstyle); dot((7.645,-0.9),linewidth(4pt) + dotstyle); dot((7.63,0.745),linewidth(4pt) + dotstyle); dot((9.275,0.76),linewidth(4pt) + dotstyle); dot((9.29,-0.885),linewidth(4pt) + dotstyle); dot((10.95,-2.515),linewidth(4pt) + dotstyle); dot((10.935,-0.87),linewidth(4pt) + dotstyle); dot((10.92,0.775),linewidth(4pt) + dotstyle); dot((10.905,2.42),linewidth(4pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy][/asy]