INMO 1998

In a circle $C_1$ with centre $O$ , let $AB$ be a chord that is not a diameter. Let $M$ be the midpoint of this chord $AB$ . Take a point $T$ on the circle $C_2$ with $OM$ as diameter. Let the tangent to $C_2$ at $T$ meet $C_1$ at $P$ . Show that $PA^2 + PB^2 = 4 \cdot PT^2$ .

Let $a$ and $b$ be two positive rational numbers such that $\sqrt[3] {a} + \sqrt[3]{b}$ is also a rational number. Prove that $\sqrt[3]{a}$ and $\sqrt[3] {b}$ themselves are rational numbers.

Let $p , q, r , s$ be four integers such that $s$ is not divisible by $5$ . If there is an integer $a$ such that $pa^3 + qa^2+ ra +s$ is divisible be 5, prove that there is an integer $b$ such that $sb^3 + rb^2 + qb + p$ is also divisible by 5.

Suppose $ABCD$ is a cyclic quadrilateral inscribed in a circle of radius one unit. If $AB \cdot BC \cdot CD \cdot DA \geq 4$ , prove that $ABCD$ is a square.

Suppose $a,b,c$ are three rela numbers such that the quadratic equation $$x^2 - (a +b +c )x + (ab +bc +ca) = 0$$ has roots of the form $\alpha + i \beta$ where $\alpha > 0$ and $\beta \not= 0$ are real numbers. Show that (i) The numbers $a,b,c$ are all positive. (ii) The numbers $\sqrt{a}, \sqrt{b} , \sqrt{c}$ form the sides of a triangle.

It is desired to choose $n$ integers from the collection of $2n$ integers, namely, $0,0,1,1,2,2,\ldots,n-1,n-1$ such that the average of these $n$ chosen integers is itself an integer and as minimum as possible. Show that this can be done for each positive integer $n$ and find this minimum value for each $n$ .