CWMO 2005

It is known that $a^{2005} + b^{2005}$ can be expressed as the polynomial of $a + b$ and $ab$ . Find the coefficients' sum of this polynomial.

Given three points $P$ , $A$ , $B$ and a circle such that the lines $PA$ and $PB$ are tangent to the circle at the points $A$ and $B$ , respectively. A line through the point $P$ intersects that circle at two points $C$ and $D$ . Through the point $B$ , draw a line parallel to $PA$ ; let this line intersect the lines $AC$ and $AD$ at the points $E$ and $F$ , respectively. Prove that $BE = BF$ .

Given is the positive integer $n > 2$ . Real numbers $\mid x_i \mid \leq 1$ ( $i = 1, 2, ..., n$ ) satisfying $\mid \sum_{i=1}^{n}x_i \mid > 1$ . Prove that there exists positive integer $k$ such that $\mid \sum_{i=1}^{k}x_i - \sum_{i=k+1}^{n}x_i \mid \leq 1$ .

Circles $C(O_1)$ and $C(O_2)$ intersect at points $A$ , $B$ . $CD$ passing through point $O_1$ intersects $C(O_1)$ at point $D$ and tangents $C(O_2)$ at point $C$ . $AC$ tangents $C(O_1)$ at $A$ . Draw $AE \bot CD$ , and $AE$ intersects $C(O_1)$ at $E$ . Draw $AF \bot DE$ , and $AF$ intersects $DE$ at $F$ . Prove that $BD$ bisects $AF$ .

In isosceles right-angled triangle $ABC$ , $CA = CB = 1$ . $P$ is an arbitrary point on the sides of $ABC$ . Find the maximum of $PA \cdot PB \cdot PC$ .

If $a,b,c$ are positive reals such that $a+b+c=1$ , prove that $$10(a^3+b^3+c^3)-9(a^5+b^5+c^5)\geq 1 .$$

For $n$ people, if it is known that

(a) there exist two people knowing each other among any three people, and

(b) there exist two people not knowing each other among any four people.

Find the maximum of $n$ .

Here, we assume that if $A$ knows $B$ , then $B$ knows $A$ .