Given that $n> 2$ is a positive integer and a sequence of positive integers $a_1 <a_2 <...<a_n$. In the subsets of the set $\{1,2,..., n\} $, there a subset $X$ such that $| \sum_{i \notin X} a_i -\sum_{i \in X} a_i |$ is the smallest . Prove that there exists a sequence of positive integers $0<b_1 <b_2 <...<b_n$ such that $\sum_{i \notin X} b_i= \sum_{i \in X} b_i$. In case this doesn't make sense, have a look at original wording in Vietnamese.

In acute $\triangle ABC$, $O$ is the circumcenter, $I$ is the incenter. The incircle touches $BC,CA,AB$ at $D,E,F$. And the points $K,M,N$ are the midpoints of $BC,CA,AB$ respectively. a) Prove that the lines passing through $D,E,F$ in parallel with $IK,IM,IN$ respectively are concurrent. b) Points $T,P,Q$ are the middle points of the major arc $BC,CA,AB$ on $\odot ABC$. Prove that the lines passing through $D,E,F$ in parallel with $IT,IP,IQ$ respectively are concurrent.

Suppose $n$ is a positive integer, $4n$ teams participate in a football tournament. In each round of the game, we will divide the $4n$ teams into $2n$ pairs, and each pairs play the game at the same time. After the tournament, it is known that every two teams have played at most one game. Find the smallest positive integer $a$, so that we can arrange a schedule satisfying the above conditions, and if we take one more round, there is always a pair of teams who have played in the game.

Let $n$ be a positive integer. In a $(2n+1)\times (2n+1)$ board, each grid is dyed white or black. In each row and each column, if the number of white grids is smaller than the number of black grids, then we mark all white grids. If the number of white grids is bigger than the number of black grids, then we mark all black grids. Let $a$ be the number of black grids, and $b$ be the number of white grids, $c$ is the number of marked grids. In this example of $3\times 3$ table, $a=3$, $b=6$, $c=4$. (forget about my watermark) Proof that no matter how is the dyeing situation in the beginning, there is always $c\geq\frac{1}{2}\min\{a,b\}$.

Find all positive integers $k$, so that there are only finitely many positive odd numbers $n$ satisfying $n~|~k^n+1$.

In the scalene acute triangle $ABC$, $O$ is the circumcenter. $AD, BE, CF$ are three altitudes. And $H$ is the orthocenter. Let $G$ be the reflection point of $O$ through $BC$. Draw the diameter $EK$ in $\odot (GHE)$, and the diameter $FL$ in $\odot (GHF)$. a) If $AK, AL$ and $DE, DF$ intersect at $U, V$ respectively, prove that $UV\parallel EF$. b) Suppose $S$ is the intersection of the two tangents of the circumscribed circle of $\triangle ABC$ at $B$ and $C$. $T$ is the intersection of $DS$ and $HG$. And $M,N$ are the projection of $H$ on $TE,TF$ respectively. Prove that $M,N,E,F$ are concyclic.