INMO 1992

In a triangle $ABC$ , $\angle A = 2 \cdot \angle B$ . Prove that $a^2 = b (b+c)$ .

If $x , y, z \in \mathbb{R}$ such that $x+y +z =4$ and $x^2 + y^2 +z^2 = 6$ , then show that each of $x, y, z$ lies in the closed interval $\left[ \dfrac{2}{3} , 2 \right]$ . Can $x$ attain the extreme value $\dfrac{2}{3}$ or $2$ ?

Find the remainder when $19^{92}$ is divided by 92.

Find the number of permutations $( p_1, p_2, p_3 , p_4 , p_5 , p_6)$ of $1, 2 ,3,4,5,6$ such that for any $k, 1 \leq k \leq 5$ , $(p_1, \ldots, p_k)$ does not form a permutation of $1 , 2, \ldots, k$ .

Two circles $C_1$ and $C_2$ intersect at two distinct points $P, Q$ in a plane. Let a line passing through $P$ meet the circles $C_1$ and $C_2$ in $A$ and $B$ respectively. Let $Y$ be the midpoint of $AB$ and let $QY$ meet the cirlces $C_1$ and $C_2$ in $X$ and $Z$ respectively. Show that $Y$ is also the midpoint of $XZ$ .

Let $f(x)$ be a polynomial in $x$ with integer coefficients and suppose that for five distinct integers $a_1, \ldots, a_5$ one has $f(a_1) = f(a_2) = \ldots = f(a_5) = 2$ . Show that there does not exist an integer $b$ such that $f(b) = 9$ .