CGMO 2006

Let $a>0$ , the function $f: (0,+\infty) \to R$ satisfies $f(a)=1$ , if for any positive reals $x$ and $y$ , there is $$f(x)f(y)+f \left( \frac{a}{x}\right)f \left( \frac{a}{y}\right) =2f(xy)$$ then prove that $f(x)$ is a constant.

Let $O$ be the intersection of the diagonals of convex quadrilateral $ABCD$ . The circumcircles of $\triangle{OAD}$ and $\triangle{OBC}$ meet at $O$ and $M$ . Line $OM$ meets the circumcircles of $\triangle{OAB}$ and $\triangle{OCD}$ at $T$ and $S$ respectively.

Prove that $M$ is the midpoint of $ST$ .

Show that for any $i=1,2,3$ , there exist infinity many positive integer $n$ , such that among $n$ , $n+2$ and $n+28$ , there are exactly $i$ terms that can be expressed as the sum of the cubes of three positive integers.

$8$ people participate in a party.

(1) Among any $5$ people there are $3$ who pairwise know each other. Prove that there are $4$ people who paiwise know each other.

(2) If Among any $6$ people there are $3$ who pairwise know each other, then can we find $4$ people who pairwise know each other?

The set $S = \{ (a,b) \mid 1 \leq a, b \leq 5, a,b \in \mathbb{Z}\}$ be a set of points in the plane with integeral coordinates. $T$ is another set of points with integeral coordinates in the plane. If for any point $P \in S$ , there is always another point $Q \in T$ , $P \neq Q$ , such that there is no other integeral points on segment $PQ$ . Find the least value of the number of elements of $T$ .

Let $M= \{ 1, 2, \cdots, 19 \}$ and $A = \{ a_{1}, a_{2}, \cdots, a_{k}\}\subseteq M$ . Find the least $k$ so that for any $b \in M$ , there exist $a_{i}, a_{j}\in A$ , satisfying $b=a_{i}$ or $b=a_{i}\pm a_{i}$ ( $a_{i}$ and $a_{j}$ do not have to be different) .

Given that $x_{i}>0$ , $i = 1, 2, \cdots, n$ , $k \geq 1$ . Show that: $$\sum_{i=1}^{n}\frac{1}{1+x_{i}}\cdot \sum_{i=1}^{n}x_{i}\leq \sum_{i=1}^{n}\frac{x_{i}^{k+1}}{1+x_{i}}\cdot \sum_{i=1}^{n}\frac{1}{x_{i}^{k}}$$

Let $p$ be a prime number that is greater than $3$ . Show that there exist some integers $a_{1}, a_{2}, \cdots a_{k}$ that satisfy: $$-\frac{p}{2}< a_{1}< a_{2}< \cdots <a_{k}< \frac{p}{2}$$ making the product: $$\frac{p-a_{1}}{|a_{1}|}\cdot \frac{p-a_{2}}{|a_{2}|}\cdots \frac{p-a_{k}}{|a_{k}|}$$ equals to $3^{m}$ where $m$ is a positive integer.