Day \(1\)

  Tuesday, July 23, 2013

Problem 1. Prove that for any pair of  positive integers \(k\) and \(n\), there exist \(k\) positive integers \(m_1,m_2,\dotsc, m_k\)(not necessarily different) such that

\[1+\frac{2^k-1}n=\left(1+\frac{1}{m_1}\right)\left(1+\frac{1}{m_2}\right)\dotsm\left(1+\frac{1}{m_k}\right).\]

Problem 2. A configuration of \(4027\) points in the plane is called Colombian if it consists of \(2013\) red points and \(2014\) blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is good for a Colombian conguration if the following two conditions are satisfied:

• no line passes through any point of the conguration;
  
• no region contains points of both colours.

Find the least value of \(k\) such that for any Colombian conguration of \(4027\) points, there is a good arrangement of \( k\) lines.

Problem 3.  Let the excircle of triangle \(ABC\) opposite the vertex \(A\) be tangent to the side \(BC\) at the point \(A_1\). Define the points \(B_1\) on \(CA\) and \(C_1\) on \(AB\) analogously, using the excircles opposite \(B\) and \(C\), respectively. Suppose that the circumcentre of triangle \(A_1B_1C_1\) lies on the circumcircle of triangle \(ABC\).Prove that triangle \(ABC\) is right-angled.

The excircle of triangle \( ABC\) 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 \(C\). The excircles opposite \(B\) and \(C\) are similarly defined.

                                      Day \(2\)

Wednesday, July 24, 2013

Problem 4. Let \(ABC\) be an acute-triangle with orthocenter \(H\), and let \(W\) be a point on the side \(BC\), lying strictlybetween \(B\) and \(C\). The points \(M\) and \(N\) are the feet of the altitudes from \(B\) and \(C\), respectively. Denote by \(\omega_1\) the circumcircle of \(BWN\), and let \(X\) be the point on \(\omega_1\) such that \(WX\) is a diameter of \(\omega_1\). Similarly, denote by \(\omega_2\) the circumcircle of triangle \(CWM\), and let \(Y\) be the point on \(\omega_2\) such that \(WY\) is a diameter of \(\omega_2\). Prove that the points \(X, Y\) and \(H\) are collinear.

Problem 5. Let \(\Bbb Q_{>0}\) be the set of positive rational numbers. Let \(f\colon\Bbb Q_{>0}\to\Bbb R\) be a function satisfying the following three conditions:

(i)   for all \(x,y\in\Bbb Q_{>0}\), we have \(f(x)f(y)\geqslant f(xy)\);
(ii)   for all \(x,y\in\Bbb Q_{>0}\), we have \( f(x+y)\geqslant f(x)+f(y)\);
(iii)  there exists a rational number \(a>1\) such that \(f(a)=a\).

Prove  that \(f(x)=x\) for all \(x\in\Bbb Q_{>0}\).

Problem 6. Let \(n\geqslant 3\) be an integer, and consider a circle with \(n+1\) equally spaced points marked on it. Consider all labellings of these points with the numbers \(0,1,\dotsc, n\) such that each label is used exactly once; two such labellings are considered to be the same if one can be obtained from the other by a rotation of the circle. A labelling is called beautiful if, for any four labels \(a<b<c<d\) with \(a+d=b+c\), the chord joining the points labelled \(a\) and \(d\) does not intersect the chord joining the points labelled \(b\) and \(c\).

Let \(M\) be the number of beautiful labellings, and let \(N\) be the number of ordered pairs \((x,y)\) of positive integers such that \(x+y\leqslant n\) and \(\gcd(x,y)=1\). Prove that

\[M=N+1.\]

                                       Day \(1\)

  2013 年 7 月 23 日, 星期二

第 1 题.  证明对于任意一对正整数 \(k\) 和 \(n\), 都存在 \(k\) 个(不必不相同的)正整数 \(m_1,m_2,\dotsc, m_k\) 使得

\[1+\frac{2^k-1}n=\left(1+\frac{1}{m_1}\right)\left(1+\frac{1}{m_2}\right)\dotsm\left(1+\frac{1}{m_k}\right).\]

第 2 题.  平面上的 \(4027\) 个点称为是一个哥伦比亚式点集, 如果其中任意三点不共线, 且有 \(2013\) 个点是红色的, \(2014\) 个点是蓝色的. 在平面上画出一组直线, 可以将平面分成若干区域. 如果一组直线对于一个哥伦比亚式点集满足下述两个条件, 我们就称这是一个好直线组:

• 这些直线不经过该哥伦比亚式点集中的任何一个点;
  
• 每个区域中都不会同时出现两种颜色的点.

求 \(k\) 的最小值, 使得对于任意的哥伦比亚式点集, 都存在由 \(k\) 条直线构成的好直线组.

第 3 题. 设三角形\(ABC\) 的顶点 \(A\) 所对的旁切圆与边 \(BC\) 相切于点 \(A_1\). 类似地, 分别用顶点 \(B\) 和顶点 \(C\) 所对的旁切圆定义 \(CA \) 边上的点 \(B_1\) 和 \(AB\) 边上的点 \(C_1\). 假设三角形 \(A_1B_1C_1\) 的外接圆圆心在三角形 \(ABC\)  的外接圆上. 证明: 三角形\(ABC\) 是直角三角形.

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. 顶点 \(B, C\)所对的旁切圆可类似定义．

 

                                       Day \(2\)

  2013 年 7 月 24 日, 星期三

第 4 题. 设三角形 \(ABC\) 是一个锐角三角形, 其垂心为 \(H\), 设 \(W\) 是边 \(BC\) 上一点, 与顶点 \(B,C\) 均不重合. \(M\) 和 \(N\) 分别是过顶点 \(B\) 和 \(C\) 的高的垂足. 记三角形 \(BWN\) 的外接圆为 \(\omega_1\), 设 \(X\) 是 \(\omega_1\) 上一点, 且 \(WX\) 是  \(\omega_1\) 的直径. 类似地, 记三角形 \(CWM\)  的外接圆为  \(\omega_2\), 设 \(Y\) 是  \(\omega_2\) 上一点, 且 \(WY\) 是  \(\omega_2\) 的直径. 证明: \(点 X, Y\) 和 \(H\) 共线.

第 5 题.  记 \(\Bbb Q_{>0}\)是所有正有理数组成的集合. 设函数 \(f\colon\Bbb Q_{>0}\to \Bbb R\) 满足如下三个条件:

(i)  对所有的 \(x,y\in\Bbb Q_{>0}\), 都有 \(f(x)f(y)\geqslant f(xy)\);
(ii)  对所有的 \(x,y\in\Bbb Q_{>0}\), 都有 \(f(x + y) \geqslant f(x) + f(y)\);
(iii) 存在有理数 \(a > 1\), 使得 \(f(a) = a\).

证明: 对所有的 \(\Bbb Q_{>0}\), 都有 \(f(x) = x\).

第 6 题.  设整数 \(n \geqslant 3\) , 在圆周上有 \(n + 1\) 个等分点. 用数 \(0, 1,\dotsc, n\) 标记这些点, 每个数字恰好用一次. 考虑所有可能的标记方式; 如果一种标记方式可以由另一种标记方式通过圆的旋转得到, 那么认为这两种标记方式是同一个. 一种标记方式称为是漂亮的, 如果对于任意满足 \(a + d = b + c\) 的四个标记数 \(a < b < c < d\), 连接标 \(a\) 和 \(d\) 点的弦与连接标 \(b\) 和 \(c\) 的点的弦都不相交.

设 \(M\) 是漂亮的标记方式的总数, 又设 \(N\) 是满足 \(x + y \leqslant n\), 且 \(\gcd(x,y)=1\) 的有序正整数对 \((x,y)\)的个数. 证明:

\[M=N+1.\]