India National Olympiad 2002 Problems
Problem 1
For a convex hexagon $ ABCDEF$ in which each pair of opposite sides is unequal, consider the following statements.
($ a_1$) $ AB$ is parallel to $ DE$. ($ a_2$)$ AE = BD$.
($ b_1$) $ BC$ is parallel to $ EF$. ($ b_2$)$ BF = CE$.
($ c_1$) $ CD$ is parallel to $ FA$. ($ c_2$) $ CA = DF$.
$ (a)$ Show that if all six of these statements are true then the hexagon is cyclic.
$ (b)$ Prove that, in fact, five of the six statements suffice.
Problem 2
Find the smallest positive value taken by $a^3 + b^3 + c^3 – 3abc$ for positive integers $a$, $b$, $c$ .
Find all $a$, $b$, $c$ which give the smallest value
Problem 3
If $x$, $y$ are positive reals such that $x + y = 2$ show that $x^3y^3(x^3+ y^3) \leq 2$.If $a,b,c$ are positive real numbers such that $abc= 1$, Prove that[ a^{b+c} b^{c+a} c^{a+b} leq 1 . ]
Problem 4
Is it true that there exist 100 lines in the plane, no three concurrent, such that they intersect in exactly 2002 points?
Problem 5
Do there exist distinct positive integers $a$, $b$, $c$ such that $a$, $b$, $c$, $-a+b+c$, $a-b+c$, $a+b-c$, $a+b+c$ form an arithmetic progression (in some order).
Problem 6
The numbers $1, 2, 3$, $\ldots$, $n^2$ are arranged in an $n\times n$ array, so that the numbers in each row increase from left to right, and the numbers in each column increase from top to bottom. Let $a_{ij}$ be the number in position $i, j$. Let $b_j$ be the number of possible values for $a_{jj}$. Show that\[ b_1 + b_2 + \cdots + b_n = \frac{ n(n^2-3n+5) }{3} . \]
