# Day $$1$$

Tuesday, July 10, 2012    9:00 am-1:30 pm

Problem 1.  Given triangle $$ABC$$ the point $$J$$ is the centre of the excircle opposite the vertex $$A$$. This excircle is tangent to the side $$BC$$ at $$M$$, and to the lines $$AB$$ and $$AC$$ at $$K$$ and $$L$$, respectively. The lines $$LM$$  and $$BJ$$ meet at $$F$$, and the lines $$KM$$ and $$CJ$$ meet at $$G$$. Let $$S$$ be the point of intersection of the lines $$AF$$ and $$BC$$, and let $$T$$ be the point of intersection of the lines $$AG$$ and $$BC$$.
Prove that $$M$$ is the midpoint of $$ST$$.

(The excircle of $$ABC$$ opposite the vertex $$A$$ is the circle that is tangent to the line segment $$BC$$, to the ray $$AB$$ beyond $$B$$, and to the ray $$AC$$ beyond $$C$$.)

Problem 2.  Let  $$n\geqslant 3$$ be an integer, and let  $$a_2,a_2,\dotsc,a_n$$ be positive real numbers such that $$a_2a_3\dotsm a_n=1.$$ Prove that

$\left(1+a_2\right)^2\left(1+a_3\right)^3\dotsm\left(1+a_n\right)^n>n^n.$
Problem 3.   The liar’s guessing game is a game played between two players $$A$$ and $$B$$. The rules of the game depend on two positive integers $$k$$ and $$n$$ which are known to both players.
At the start of the game $$A$$ chooses integers $$x$$ and $$N$$ with $$1\leqslant x \leqslant N$$. Player $$A$$  keeps $$x$$ secret, and truthfully tells $$N$$ to player $$B$$. Player $$B$$ now tries to obtain information about $$x$$ by asking player $$A$$ questions as follows: each question consists of $$B$$ specifying an arbitrary set $$S$$ of positive integers (possibly one specified in some previous question), and asking $$A$$ whether $$x$$ belongs to $$S$$. Player $$B$$ may ask as many such questions as he wishes. After each question, player $$A$$ must immediately answer it with yes or no, but is allowed to lie as many times as she wants; the only restriction is that, among any $$k+1$$ consecutive answers, at least one answer must be truthful.
After $$B$$ has asked as many questions as he wants, he must specify a set $$X$$ of at most $$n$$ positive integers. If $$x$$ belongs to $$X$$, then $$B$$ wins; otherwise, he loses. Prove that:

1. If  $$n\geqslant 2^k$$, then $$B$$ can guarantee a win.
2. For all sufficiently large $$k$$, there exists an integer $$n\geqslant1.99^k$$ such that $$B$$ cannot guarantee a win.

# Day $$2$$

Wednesday, July 11, 2012     9:00 am-1:30 pm

Problem 4.  Find all functions $$f:\Bbb Z \rightarrow\Bbb Z$$ such that, for all integers $$a,b,c$$ that satisfy $$a+b+c=0$$, the following equality holds:

$f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).$

(Here $$\Bbb Z$$ denotes the set of integers.)

Problem 5.  Let $$ABC$$ be a triangle with $$\angle BCA=90^\circ$$, and let $$D$$ be the foot of the altitude from $$C$$. Let $$X$$ be a point in the interior of the segment $$CD$$. Let $$K$$ be the point on the segment $$AX$$ such that $$BK=BC$$. Similarly, let $$L$$ be the point on the segment $$BX$$ such that $$AL=AC$$. Let $$M$$ be the point of intersection of $$AL$$ and $$BK$$.
Show that $$MK=ML$$.

Problem 6.  Find all positive integers $$n$$ for which there exist non-negative integers $$a_1,a_2,\dotsc,a_n$$ such that

$\frac1{2^{a_1}}+\frac1{2^{a_2}}+\dotsb+\frac1{2^{a_n}}=\frac1{3^{a_1}}+\frac2{3^{a_2}}+\dotsb+\frac n{3^{a_n}}= 1.$

# 第一天

2012年7月10日, 星期二

1.  设 $$J$$ 为三角形 $$ABC$$ 顶点 $$A$$ 所对旁切圆的圆心. 该旁切圆与边 $$BC$$ 相切于点 $$M$$, 与直线 $$AB$$ 和 $$AC$$ 分别相切于点$$K$$ 和 $$L$$. 直线 $$LM$$ 和 $$BJ$$ 相交于点 $$F$$, 直线 $$KM$$ 与 $$CJ$$ 相交于点 $$G$$. 设 $$S$$ 是直线 $$AF$$ 和 $$BC$$ 的交点, $$T$$ 是直线 $$AG$$ 和 $$BC$$ 的交点.

(三角形 $$ABC$$ 的顶点 $$A$$ 所对的旁切圆是指与边 $$BC$$ 相切, 并且与边 $$AB,AC$$ 的延长线相切的圆.)

2. 设整数 $$n\geqslant 3$$, 正实数$$a_2,a_2,\dotsc,a_n$$ 满足 $$a_2a_3\dotsm a_n=1.$$ 证明:

$\left(1+a_2\right)^2\left(1+a_3\right)^3\dotsm\left(1+a_n\right)^n>n^n.$
3.  “欺诈猜数游戏”在两个玩家甲和乙之间进行, 游戏依赖于两个甲和乙都知道的正整数 $$k$$ 和 $$n$$.

1. 若 $$n\geqslant 2^k$$, 则乙可保证获胜;
2. 对所有充分大的整数 $$k$$, 存在整数 $$n\geqslant1.99^k$$, 使得乙无法保证获胜.

# 第二天

2012年7月11日, 星期三

4.  求所有的函数 $$f:\Bbb Z \rightarrow\Bbb Z$$, 使得对所有满足 $$a+b+c=0$$ 的整数 $$a,b,c$$, 都有

$f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).$

(这里 $$\Bbb Z$$ 表示整数集.)

5.  已知三角形 $$ABC$$ 中,  $$\angle BCA=90^\circ$$,  $$D$$ 是过顶点 $$C$$ 的高的垂足. 设 $$X$$ 是线段 $$CD$$ 内部的一点.  $$K$$ 是线段 $$AX$$ 上一点, 使得 $$BK=BC$$.  $$L$$ 是线段 $$BX$$ 上一点, 使得 $$AL=AC$$. 设 $$M$$ 是 $$AL$$ 与 $$BK$$ 的交点.

6.  求所有的正整数 $$n,$$ 使得存在非负整数  $$a_1,a_2,\dotsc,a_n,$$ 满足

$\frac1{2^{a_1}}+\frac1{2^{a_2}}+\dotsb+\frac1{2^{a_n}}=\frac1{3^{a_1}}+\frac2{3^{a_2}}+\dotsb+\frac n{3^{a_n}}= 1.$

Tagged with:

This site uses Akismet to reduce spam. Learn how your comment data is processed.