There is circle $\omega$ and $A, B$ on it. Circle $\gamma_1$ tangent to $\omega$ on $T$ and $AB$ on $D$. Circle $\gamma_2$ tangent to $\omega$ on $S$ and $AB$ on $E$. (like the figure below) Let $AB\cap TS=C$. Prove that $CA=CB$ iff $CD=CE$

Given an integer $n\geq 3$. For each $3\times3$ squares on the grid, call this $3\times3$ square isolated if the center unit square is white and other 8 squares are black, or the center unit square is black and other 8 squares are white. Now suppose one can paint an infinite grid by white or black, so that one can select an $a\times b$ rectangle which contains at least $n^2-n$ isolated $3\times 3$ square. Find the minimum of $a+b$ that such thing can happen. (Note that $a,b$ are positive reals, and selected $a\times b$ rectangle may have sides not parallel to grid line of the infinite grid.)

Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following. $f(xf(y)+yf(z)+zf(x))=yf(x)+zf(y)+xf(z)$

$p(x)$ is an irreducible polynomial with integer coefficients, and $q$ is a fixed prime number. Let $a_n$ be a number of solutions of the equation $p(x)\equiv 0\mod q^n$. Prove that we can find $M$ such that $\{a_n\}_{n\ge M}$ is constant.

Find all $\{a_n\}_{n\ge 0}$ that satisfies the following conditions. (1) $a_n\in \mathbb{Z}$ (2) $a_0=0, a_1=1$ (3) For infinitly many $m$, $a_m=m$ (4) For every $n\ge2$, $\{2a_i-a_{i-1} | i=1, 2, 3, \cdots , n\}\equiv \{0, 1, 2, \cdots , n-1\}$ $\mod n$

Let $a_i, b_i$ ($1 \le i \le n$, $n \ge 2$) be positive real numbers such that $\sum_{i=1}^n a_i = \sum_{i=1}^n b_i$. Prove that $\sum_{i=1}^n \frac{(a_{i+1}+b_{i+1})^2}{n(a_i-b_i)^2+4(n-1)\sum_{j=1}^n a_jb_j} \ge \frac{1}{n-1}$

There are three circles $w_1, w_2, w_3$. Let $w_{i+1} \cap w_{i+2} = A_i, B_i$, where $A_i$ lies insides of $w_i$. Let $\gamma$ be the circle that is inside $w_1,w_2,w_3$ and tangent to the three said circles at $T_1, T_2, T_3$. Let $T_iA_{i+1}T_{i+2}$'s circumcircle and $T_iA_{i+2}T_{i+1}$'s circumcircle meet at $S_i$. Prove that the circumcircles of $A_iB_iS_i$ meet at two points. ($1 \le i \le 3$, indices taken modulo $3$) If one of $A_i,B_i,S_i$ are collinear - the intersections of the other two circles lie on this line. Prove this as well.

There are $n$ lattice points in a general position. (no three points are collinear) A convex polygon $P$ covers the said $n$ points. (the borders are included) Prove that, for large enough $n$ and a positive real $\epsilon$, the perimeter of $P$ is no less than $(\sqrt{2}+\epsilon)n$.

Solve: $a, b, m, n\in \mathbb{N}$ $a^2+b^2=m^2-n^2, ab=2mn$

Let there be an acute triangle $ABC$, such that $\angle ABC < \angle ACB$. Let the perpendicular from $A$ to $BC$ hit the circumcircle of $ABC$ at $D$, and let $M$ be the midpoint of $AD$. The tangent to the circumcircle of $ABC$ at $A$ hits the perpendicular bisector of $AD$ at $E$, and the circumcircle of $MDE$ hits the circumcircle of $ABC$ at $F$. Let $G$ be the foot of the perpendicular from $A$ to $BD$, and $N$ be the midpoint of $AG$. Prove that $B, N, F$ are collinear.

$p, q, r$ are natural numbers greater than 1. There are $pq$ balls placed on a circle, and one number among $0, 1, 2, \cdots , pr-1$ is written on each ball, satisfying following conditions. (1) If $i$ and $j$ is written on two adjacent balls, $|i-j|=1$ or $|i-j|=pr-1$. (2) $i$ is written on a ball $A$. If we skip $q-1$ balls clockwise from $A$ and see $q^{th}$ ball, $i+r$ or $i-(p-1)r$ is written on it. (This condition is satisfied for every ball.) If $p$ is even, prove that the number of pairs of two adjacent balls with $1$ and $2$ written on it is odd.

Let $x,y,z \ge 0$ be real numbers such that $(x+y-1)^2+(y+z-1)^2+(z+x-1)^2=27$. Find the maximum and minimum of $x^4+y^4+z^4$

Let there be an acute triangle $ABC$ with orthocenter $H$. Let $BH, CH$ hit the circumcircle of $\triangle ABC$ at $D, E$. Let $P$ be a point on $AB$, between $B$ and the foot of the perpendicular from $C$ to $AB$. Let $PH \cap AC = Q$. Now $\triangle AEP$'s circumcircle hits $CH$ at $S$, $\triangle ADQ$'s circumcircle hits $BH$ at $R$, and $\triangle AEP$'s circumcircle hits $\triangle ADQ$'s circumcircle at $J (\not=A)$. Prove that $RS$ is the perpendicular bisector of $HJ$.

Find all pairs of positive integers $(n,t)$ such that $6^n+1=n^2t$, and $(n,29 \times 197)=1$

Let there be a triangle $\triangle ABC$ with $BC=a$, $CA=b$, $AB=c$. Let $T$ be a point not inside $\triangle ABC$ and on the same side of $A$ with respect to $BC$, such that $BT-CT=c-b$. Let $n=BT$ and $m=CT$. Find the point $P$ that minimizes $f(P)=-a \cdot AP + m \cdot BP + n \cdot CP$.

Let $a_1, a_2, \cdots a_{100}$ be a permutation of $1,2,\cdots 100$. Define $l(k)$ as the maximum $m$ such that there exists $i_1, i_2 \cdots i_m$ such that $a_{i_1} > a_{i_2} > \cdots > a_{i_m}$ or $a_{i_1} < a_{i_2} < \cdots < a_{i_m}$, where $i_1=k$ and $i_1<i_2< \cdots <i_m$ Find the minimum possible value for $\sum_{i=1}^{100} l(i)$.