CN-SMO 2008

Given a set $S=\{1,2,3,\ldots,3n\},(n\in N^*)$ , let $T$ be a subset of $S$ , such that for any $x, y, z\in T$ (not necessarily distinct) we have $x+y+z\not \in T$ . Find the maximum number of elements $T$ can have.

Let $\{a_n\}$ be a sequence satisfying: $a_1=1$ and $a_{n+1}=2a_n+n\cdot (1+2^n),(n=1,2,3,\cdots)$ . Determine the general term formula of $\{a_n\}$ .

In $\triangle ABC$ , side $BC>AB$ . Point $D$ lies on side $AC$ such that $\angle ABD=\angle CBD$ . Points $Q,P$ lie on line $BD$ such that $AQ\bot BD$ and $CP\bot BD$ . $M,E$ are the midpoints of side $AC$ and $BC$ respectively. Circle $O$ is the circumcircle of $\triangle PQM$ intersecting side $AC$ at $H$ . Prove that $O,H,E,M$ lie on a circle.

Let $m, n$ be positive integers $(m, n>=2)$ . Given an $n$ -element set $A$ of integers $(A=\{a_1,a_2,\cdots ,a_n\})$ , for each pair of elements $a_i, a_j(j>i)$ , we make a difference by $a_j-a_i$ . All these $C^2_n$ differences form an ascending sequence called "derived sequence" of set $A$ . Let $\bar{A}$ denote the derived sequence of set $A$ . Let $\bar{A}(m)$ denote the number of terms divisible by $m$ in $\bar{A}$ . Prove that $\bar{A}(m)\ge \bar{B}(m)$ where $A=\{a_1,a_2,\cdots ,a_n\}$ and $B=\{1,2,\cdots ,n\}$ .

Let $\lambda$ be a positive real number. Inequality $|\lambda xy+yz|\le \dfrac{\sqrt5}{2}$ holds for arbitrary real numbers $x, y, z$ satisfying $x^2+y^2+z^2=1$ . Find the maximal value of $\lambda$ .

Circle $I$ is the incircle of $\triangle ABC$ . Circle $I$ is tangent to sides $BC$ and $AC$ at $M,N$ respectively. $E,F$ are midpoints of sides $AB$ and $AC$ respectively. Lines $EF, BI$ intersect at $D$ . Show that $M,N,D$ are collinear.

Captain Jack and his pirate men plundered six chests of treasure $(A_1,A_2,A_3,A_4,A_5,A_6)$ . Every chest $A_i$ contains $a_i$ coins of gold, and all $a_i$ s are pairwise different $(i=1,2,\cdots ,6)$ . They place all chests according to a layout (see the attachment) and start to alternately take out one chest a time between the captain and a pirate who serves as the delegate of the captain's men. A rule must be complied with during the game: only those chests that are not adjacent to other two or more chests are allowed to be taken out. The captain will win the game if the coins of gold he obtains are not less than those of his men in the end. Let the captain be granted to take chest firstly, is there a certain strategy for him to secure his victory?

Attachments:

Let $n$ be a positive integer. $f(n)$ denotes the number of $n$ -digit numbers $\overline{a_1a_2\cdots a_n}$ (wave numbers) satisfying the following conditions : (i) for each $a_i \in\{1,2,3,4\}$ , $a_i \not= a_{i+1}$ , $i=1,2,\cdots$ ; (ii) for $n\ge 3$ , $(a_i-a_{i+1})(a_{i+1}-a_{i+2})$ is negative, $i=1,2,\cdots$ . (1) Find the value of $f(10)$ ; (2) Determine the remainder of $f(2008)$ upon division by $13$ .