chof96i

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(x³+y³)=(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, X₀=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+X₀+X₁+...+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.