JMO 1997

Take 10 points inside the circle with diameter 5. Prove that for any these 10 points there exist two points whose distance is less than 2.

Prove that

$\frac{\left(b+c-a\right)^{2}}{\left(b+c\right)^{2}+a^{2}}+\frac{\left(c+a-b\right)^{2}}{\left(c+a\right)^{2}+b^{2}}+\frac{\left(a+b-c\right)^{2}}{\left(a+b\right)^{2}+c^{2}}\geq\frac35$

for any positive real numbers $a$, $b$, $c$.

Call the Graph the set which composed of several vertices $P_1,\ \cdots P_2$ and several edges $($ s egment s $)$ connecting two points among these vertices. Now let $G$ be a graph with 9 vertices and satisfies the following condition.

Condition: Even if we select any five points from the vertices in $G,$ there exist at least two edges whose endpoints are included in the set of 5 points.

What is the minimum possible numbers of edges satisfying the condition?

Let $A,B,C,D$ be points in space which are not on a same plane and no any 3 points are not colinear. Suppose that the sum of the segments $AX+BX+CX+DX$ is minimized at $X=X_0$ which is different from $A,B,C,D.$ Prove that $\angle{AX_0B}=\angle{CX_0D}.$

The letters $A$ or $B$ are assigned on the points divided equally into $2^{n}\ (n=1,\ 2,\cdots)$ parts of a circumference.If you choose $n$ letters from any succesively arranging points directed clockwise, prove that there exists the way of assignning for which the line of letters are mutually distinct.