VMO 1981

Show that the triangle ABC is right-angled iff sin A + sin B + sin C = cos A + cos B + cos C + 1.

Let $p, q$ be real numbers with $0 < p < q$ and let $t_1, t_2, \cdots, t_n$ be real numbers in the interval $[p, q]$ . Denote by $A$ and $B$ the arithmetic means of $t_1, t_2, \cdots, t_n$ and of $t_1^2, t_2^2,\cdots , t_n^2$ , respectively. Prove that $$\frac{A^2}{B}\ge\frac{4pq}{(p + q)^2}.$$

A plane $\rho$ and two points $M, N$ outside it are given. Determine the point $A$ on $\rho$ for which $\frac{AM}{AN}$ is minimal.

Find all real solutions to: $w^2$ + $x^2$ + $y^2$ + $z^2$ = 50 $w^2$ - $x^2$ + $y^2$ - $z^2$ = -24 wx = yz w - x + y - z = 0.

$x_1$, $x_2$, $x_3$, ... , $x_n$ are reals in the interval [a, b]. M = ($x_1$ + $x_2$ + ... + $x_n$)/n, V = ($x_1^2$ + $x_2^2$ + ... + $x_n^2$)/n. Show that $M^2$ ≥ 4Vab/(a + b)$^2$.

Two circles touch externally at A. P is a point inside one of the circles, not on the line of centers. A variable line L through P meets one circle at B (and possibly another point) and the other circle at C (and possibly another point). Find L such that the circumcircle of ABC touches the line of centers at A.