##
96 CHINESE MATH OLYMPIAD

**First Day 8:00--12:30 A.M 18 January 1996**
**Problem 1.**

H is the orthocenter of acute triangle ABC, from A, draw the two tangent
lines AP and AQ of the circle whose diameter is BC, the points of tangency
are P, Q respectively. Prove: P, H, Q are collinear.

**Problem 2.**

S={1,2,...,50}. Find the minimum natural number k, that of any k-element
subset of S, there are two different elements a and b, a+b|ab.

**Problem 3.**

Function F: R to R satisfies F(x^{3}+y^{3})=(x+y)((F(x))^{2}-F(x)F(y)+(F(y))^{2}),
where x, y are arbitrary natural numbers. Prove: For any real number x,
F(1996x)=1996F(x).

**Second Day 8:00--12:30 A.M 19 January 1996**
**Problem 4.**

8 singers take part in an art festivel. The organizer wants to plan
m concerts. Every concert there are 4 singers go on stage. Restrict that
the times of which every two singers go on stage in a concert are all same.
Please make a design that m is minimal.
**Problem 5.**

For natural number n, X_{0}=0, X_{i}>0, i=1,2,..,n,
and sum(i=1 to n, X_{i})=1. Prove: 1 £
sum(i=1 to n, X_{i}/(sqr(1+X_{0}+X_{1}+...+X_{(i-1)})sqr(Xi+...+Xn))

**Problem 6.**

In triangle ABC, angle C=90 degrees, angle A=30 degrees, BC=1. Find
the minimum value of all the longest sides of inscribed triangles of triangle
ABC. Where inscribed triangle means the three vertices of the triangle
belong to the three sides of triangle ABC respectively.

© 1997 *Enrique Valeriano
Cuba*