# Day $$1$$

2014 年 7 月 8 日, 星期二

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

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

# Day $$2$$

2014 年 7 月 9 日, 星期三

: 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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