# 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

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

• 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.

# Day $$2$$

2013 年 7 月 24 日, 星期三

(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 $$f(a) = a$$.

$M=N+1.$

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