Jul 082014

                                      Day \(1\)

  2014 年 7 月 8 日, 星期二

第 1 题. 设  \( a_0\lt a_1\lt a_2\lt\dotsb \) 是一个无穷正整数列. 证明: 存在惟一的整数 \(n\geq1\) 使得

\[ a_n \lt\frac{a_0+a_1+a_2+\dotsb+a_n}n\leq a_{n+1}. \]

第 2 题. 设 \(n\geq2\) 是一个整数. 考虑由 \(n^2\) 个单位正方形组成的一个 \(n\times n\) 棋盘. 一种放置 \(n\) 个棋子”车” 的方案被称为是和平的, 如果每一行和每一列上都恰好有一个”车”. 求最大的正整数 \(k\), 使得对于任何一种和平放置 \(n\) 个 “车” 的方案, 都存在一个 \(k\times k\) 的正方形, 它的 \(k^2\) 个单位正方形里都没有”车”.

第 3 题. 在凸四边形 \(ABCD\) 中 \(\angle ABC=\angle CDA=90^\circ\). 点 \(H\) 是 \(A\) 向 \(BD\) 引的垂线的垂足. 点 \(S\) 和点 \(T\) 分别在边 \(AB\) 和边 \(AD\) 上, 使得 \(H\) 在三角形 \(SCT\) 内部, 且

\[ \angle CHS-\angle CSB = 90^\circ,\quad\angle THC-\angle DTC = 90^\circ. \]

证明: 直线 \(BD\) 和三角形 \(TSH\) 的外接圆相切.

                                      Day \(2\)

  2014 年 7 月 9 日, 星期三

第 4 题. 点 \(P\) 和 \(Q\) 在锐角三角形 \(ABC \) 的边 \(BC \) 上, 满足 \(\angle PAB =\angle BCA\) 且 \(\angle CAQ = \angle ABC\). 点 \(M\) 和 \(N\) 分别在直线 \(AP\) 和 \(AQ\) 上, 使得 \(P\) 是 \(AM\) 的中点, 且 \(Q\) 是 \(AN\) 的中点. 证明: 直线 \(BM\) 和 \(CN\) 的交点在三角形 \(ABC\) 的外接圆上.

第 5 题.  对每一个正整数 \(n\), 开普敦银行都发行面值为 \(\dfrac1n\) 的硬币. 给定总额不超过 \(99+\dfrac12\) 的有限多个这样的硬币(面值不必两两不同) , 证明可以把它们分为至多 \(100\) 组, 使得每一组中硬币的面值之和最多是 \(1\).

第 6 题.  平面上的一族直线被称为是处于一般位置的, 如果其中没有两条直线平行, 没有三条直线共点. 一族处于一般位置的直线把平面分割成若干区域, 我们把其中面积有限的区域称为这族直线的有限区域. 证明: 对于充分大的 \(n\) 和任意处于一般位置的 \(n\) 条直线, 我们都可以把其中至少\(\sqrt n\) 条直线染成蓝色, 使得每一个有限区域的边界都不全是蓝色的.

: 如果你的答卷上证明的是 \(c\sqrt n\)(而不是 \(\sqrt n\)) 的情形, 那么将会根据常数 \(c\) 的值给分.

                                    Day \(1\)

 Tuesday, July 8, 2014

Problem 1.  Let \( a_0\lt a_1\lt a_2\lt\dotsb \)  be an infinite sequence of positive integers. Prove that there exists a unique integer \(n\geq1\) such that

\[ a_n \lt\frac{a_0+a_1+a_2+\dotsb+a_n}n\leq a_{n+1}. \]

Problem 2. Let \(n\geq2\) be an integer. Consider an \(n\times n\) chessboard consisting of  \(n^2\) unit squares. A configuration of \(n\) rooks on this board is  peaceful  if every row and every column contains exactly one rook. Find the greatest positive integer \(k\) such that, for each peaceful configuration of \(n\) rooks, there is a \(k\times k\) square which does not contain a rook on any of its  \(k^2\) unit squares.

Problem 3.  Convex quadrilateral \(ABCD\) has \(\angle ABC=\angle CDA=90^\circ\). Point \(H\) is the foot of the perpendicular from \(A\)  to \(BD\). Points \(S\) and \(T\)  lie on sides \(AB\) and \(AD\) respectively, such that \(H\) lies inside triangle \(SCT\) and

\[ \angle CHS-\angle CSB = 90^\circ,\quad\angle THC-\angle DTC = 90^\circ. \]

Prove that line \(BD\) is tangent to the circumcircle of triangle \(TSH\).

                                    Day \(2\)

Wednesday, July 9, 2014

Problem 4. Points \(P\) and \(Q\) lie on side \(BC\) of acute-angled triangle \(ABC\) so that \(\angle PAB=\angle BCA\) and \(\angle CAQ=\angle ABC\). Points \(M\) and \(N\) lie on lines \(AP\) and \(AQ\), respectively, such that \(P\) is the midpoint of \(AM\) and \(Q\) is the midpoint of \(AN\). Prove that lines \(BM\) and \(CN\) intersect on the circumcircle of triangle \(ABC\).

Problem 5. For each positive integer \(n\), the Bank of Cape Town issues coins of denomination \(\frac1n\). Given a finite collection of such coins (of not neccesarily different denominations) with total value at most \(99+\frac12\). Prove that it is possible to split this collection into \(100\) or fewer groups, such that each group has total value at most \(1\).

Problem 6.A set of lines in the plane is in general position if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite area; we call these its finite regions. Prove that for all sufficiently large \(n\), in any set of \(n\) lines in general position it is possible to colour at least \(\sqrt n\) of the lines blue in such a way that none of its finite regions has a completely blue boundary.

Note: Results with \(\sqrt n\) replaced by \(c\sqrt n\) will be awarded points depending on the value of the constant \(c\).

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