▪ Hong Kong Mathematical Olympiad 2006 Test 1

 1. Find the integral solutions of the equation

$7(x+y) = 3(x^2-xy+y^2)$.

2. The function $f(x,y)$, defined on the set of all non-negative integers, satisfies 

(i) $f(0,y)=y+1$
(ii) $f(x+1,0)$=$f(x,1)$
(iii) $f(x+1, y+1)= f(x, f(x+1, y))$

Find $f(3,2005)$, $f(4,2005)$. 

3. In triangle $ABC$, the altitude, angle bisector and median from $C$ divide the angle $C$ into four equal angles. Find angle $B$. 

4. Let $x,y,z$ be positive real numbers such that

$x+y+z=1$. 

For positive integer $n$, define $S_n =x^{n}+y^{n}+z^{n}$.

Furthermore, let $P = S_{2}S_{2}005$ and $Q = S_{3}S_{2}004$. 

(a) Find the smallest possible value of $Q$. 

(b) If $x,y,z$ are pairwise distinct, determine whether $P$ or $Q$ is larger.

 5. Given finitely many points in a plane, it is known that the area of the triangle formed by any three points of the set is less than $1$. Show that all points of the set lie inside or on boundary of a triangle with area less than $4$. 

6. Find $2^{2006}$ positive integers satisfying the following conditions. 

(i) Each positive integer has $2^{2005}$digits. 

(ii) Each positive integer only has $7$ or $8$ in its digits. 

(iii) Among any two chosen integers, at most half of their corresponding digits are the same. 

▪ Hong Kong Mathematical Olympiad  2006 Test 2