Jul 102012

                                       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

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





                                                                                   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.
证明: \(M\) 是线段 \(ST\) 的中点.

(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 \(n\).
游戏开始时甲先选定两个整数 \(x\) 和 \(N,1\leqslant x \leqslant N.\) 甲如实告诉乙 \(N\) 的值, 但对 \(x\) 守口如瓶. 乙现在试图通过如下方式的提问来获得关于 \(x\) 的信息: 每次提问, 乙任选一个由若干正整数组成的集合 \(S\) (可以重复使用之前提问中使用过的集合), 问甲”\(x\) 是否属于 \(S\)?”. 乙可以提任意数量的问题. 在乙每次提问之后, 甲必须对乙的提问立刻回答 “是” 或 “否”, 甲可以说谎话, 并且说谎的次数没有限制, 唯一的限制是甲在任意连续 \(k+1\) 次回答中至少有一次回答是真话.
在乙问完所有想问的问题之后, 乙必须指出一个至多包含 \(n\) 个正整数的集合 \(X\), 若 \(x\) 属于 \(X\), 则乙获胜; 否则甲获胜. 证明:

  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.



                                                                                     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.
证明 : \(MK=ML\).

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

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

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