KMO 2004

For arbitrary real number $x$ , the function $f : \mathbb R \to \mathbb R$ satisfies $f(f(x))-x^2+x+3=0$ . Show that the function $f$ does not exist.

$x$ and $y$ are positive and relatively prime and $z$ is an integer. They satisfy $(5z-4x)(5z-4y)=25xy$ . Show that at least one of $10z+x+y$ or quotient of this number divided by $3$ is a square number (i.e. prove that $10z+x+y$ or integer part of $\frac{10z+x+y}{3}$ is a square number).

Positive real numbers, $a_1, .. ,a_6$ satisfy $a_1^2+..+a_6^2=2$ . Think six squares that has side length of $a_i$ ( $i=1,2,\ldots,6$ ). Show that the squares can be packed inside a square of length $2$ , without overlapping.

Let $k$ and $N$ be positive real numbers which satisfy $k\leq N$ . For $1\leq i \leq k$ , there are subsets $A_i$ of $\{1,2,3,\ldots,N\}$ that satisfy the following property.

For arbitrary subset of $\{ i_1, i_2, \ldots , i_s \} \subset \{ 1, 2, 3, \ldots, k \}$ , $A_{i_1} \triangle A_{i_2} \triangle ... \triangle A_{i_s}$ is not an empty set.

Show that a subset $\{ j_1, j_2, .. ,j_t \} \subset \{ 1, 2, ... ,k \}$ exist that satisfies $n(A_{j_1} \triangle A_{j_2} \triangle \cdots \triangle A_{j_t}) \geq k$ . ( $A \triangle B=A \cup B-A \cap B$ )

$A, B, C$ , and $D$ are the four different points on the circle $O$ in the order. Let the centre of the scribed circle of triangle $ABC$ , which is tangent to $BC$ , be $O_1$ . Let the centre of the scribed circle of triangle $ACD$ , which is tangent to $CD$ , be $O_2$ .

(1) Show that the circumcentre of triangle $ABO_1$ is on the circle $O$ .

(2) Show that the circumcircle of triangle $CO_1O_2$ always pass through a fixed point on the circle $O$ , when $C$ is moving along arc $BD$ .