CMO 2003

Let $I$ and $H$ be the incentre and orthocentre of triangle $ABC$ respectively. Let $P,Q$ be the midpoints of $AB,AC$ . The rays $PI,QI$ intersect $AC,AB$ at $R,S$ respectively. Suppose that $T$ is the circumcentre of triangle $BHC$ . Let $RS$ intersect $BC$ at $K$ . Prove that $A,I$ and $T$ are collinear if and only if $[BKS]=[CKR]$ .

Shen Wunxuan

Determine the maximal size of the set $S$ such that: i) all elements of $S$ are natural numbers not exceeding $100$ ; ii) for any two elements $a,b$ in $S$ , there exists $c$ in $S$ such that $(a,c)=(b,c)=1$ ; iii) for any two elements $a,b$ in $S$ , there exists $d$ in $S$ such that $(a,d)>1,(b,d)>1$ .

Yao Jiangang

Given a positive integer $n$ , find the least $\lambda>0$ such that for any $x_1,\ldots x_n\in \left(0,\frac{\pi}{2}\right)$ , the condition $\prod_{i=1}^{n}\tan x_i=2^{\frac{n}{2}}$ implies $\sum_{i=1}^{n}\cos x_i\le\lambda$ .

Huang Yumin

Find all integer triples $(a,m,n)$ such that $a^m+1|a^n+203$ where $a,m>1$ .

Chen Yonggao

Ten people apply for a job. The manager decides to interview the candidates one by one according to the following conditions: i) the first three candidates will not be employed; ii) from the fourth candidates onwards, if a candidate's comptence surpasses the competence of all those who preceded him, then that candidate is employed; iii) if the first nine candidates are not employed, then the tenth candidate will be employed. We assume that none of the $10$ applicants have the same competence, and these competences can be ranked from the first to tenth. Let $P_k$ represent the probability that the $k$ th-ranked applicant in competence is employed. Prove that: i) $P_1>P_2>\ldots>P_8=P_9=P_{10}$ ; ii) $P_1+P_2+P_3>0.7$ iii) $P_8+P_9+P_{10}\le 0.1$ .

Su Chun

Suppose $a,b,c,d$ are positive reals such that $ab+cd=1$ and $x_i,y_i$ are real numbers such that $x_i^2+y_i^2=1$ for $i=1,2,3,4$ . Prove that $$(ax_1+bx_2+cx_3+dx_4)^2+(ay_4+by_3+cy_2+dy_1)^2\le 2\left(\frac{a^2+b^2}{ab}+\frac{c^2+d^2}{cd}\right).$$

Li Shenghong