The 24th Chinese Mathematical Olympiad (CMO) was held in Hainan province in January 2009. There are total 184 high school students from all over China participated in this contest. There are 11 students got all points (126 points), 46 students received gold medal (for score higher than or equal to 93 points, 75 students received silver medal (for score higher than or equal to 51 points, 77 students received bronze medal (for score higher than or equal to 6 points).

Here are the questions during the first day on January 9, 2009 from 8:00 a.m. to 12:30 p.m.

**1.** Given an acute triangle *PBC*, *PB* ≠ *PC*. Assume *A*, *D* are points on the sides *PB*, *PC* respectively. Connect *AC*, *BD* which cross each other at point *O*. Passing point *O*, draw *OE* ⊥ *AB*, *OF* ⊥ *CD* such that *OE* intercepts *AB* at point *E* and *OF* intercepts *CD* at point *F*. Denote the midpoint of line segments *BC*, *AD* as *M*, *N* respectively.

(1) If the four points *A*, *B*, *C*, *D* lie on a circle, show that *EM*·*FN* = *EN*·*FM*;

(2) If *EM*·*FN* = *EN*·*FM*, is it always true that the four points *A*, *B*, *C*, *D* lies on a circle? Prove your conclusion.

**2.** Find all pairs of prime numbers such that *pq* | 5^{p} + 5^{q}.

**3.** Assume m, n are given integers, 4 < *m* < *n*, *A*_{1}*A*_{2}…*A*_{2n+1} is a regular (2n+1)-polygon. Let *P* = {*A*_{1}, *A*_{2}, …, *A*_{2n+1}}. Find the number of convex m-polygons with all corners in *P* that have and only have two acute interior angles.

**Note:** This is my personal translation and I do not guarantee they are 100% correct. I do not have solutions to these problems.

The next set of questions follows here.

Link from Math Competitions National and International

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June 19, 2009 at 4:44 am |

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