INMO 2002

For a convex hexagon $A B C D E F$ in which each pair of opposite sides is unequal, consider the following six statements:

$$\begin{array}{ll} \text { (a } \left.\mathrm{a}_{1}\right) A B \text { is parallel to } D E ; & \left(\mathrm{a}_{2}\right) A E=B D \\ \left(\mathrm{~b}_{1}\right) B C \text { is parallel to } E F ; & \left(\mathrm{b}_{2}\right) B F=C E \\ \text { (c } \left.\mathrm{c}_{1}\right) C D \text { is parallel to } F A ; & \left(\mathrm{c}_{2}\right) C A=D F \end{array}$$

(a) Show that if all the six statements are true, then the hexagon is cyclic(i.e., it can be inscribed in a circle).

(b) Prove that, in fact, any five of these six statements also imply that the hexagon is cyclic.

Determine the least positive value taken by the expression $a^{3}+b^{3}+c^{3}-3 a b c$ as $a, b, c$ vary over all positive integers. Find also all triples $(a, b, c)$ for which this least value is attained.

Let $x, y$ be positive reals such that $x+y=2$. Prove that

$$x^{3} y^{3}\left(x^{3}+y^{3}\right) \leq 2$$

Do there exist 100 lines in the plane, no three of them concurrent, such that they intersect exactly in 2002 points?

Do there exist three distinct positive real numbers $a, b, c$ such that the numbers $a, b, c, b+c-a, c+a-b, a+b-c$ and $a+b+c$ form a 7-term arithmetic progression in some order?

Suppose the $n^{2}$ numbers $1,2,3, \ldots, n^{2}$ are arranged to form an $n$ by $n$ array consisting of $n$ rows and $n$ columns such that the numbers in each row(from left to right) and each column(from top to bottom) are in increasing order. Denote by $a_{j k}$ the number in $j$-th row and $k$-th column. Suppose $b_{j}$ is the maximum possible number of entries that can occur as $a_{j j}, 1 \leq j \leq n$. Prove that

$$b_{1}+b_{2}+b_{3}+\cdots b_{n} \leq \frac{n}{3}\left(n^{2}-3 n+5\right)$$

(Example: In the case $n=3$, the only numbers which can occur as $a_{22}$ are 4,5 or 6 so that $b_{2}=3$.)