SMO 1995

Let $A_1A_2A_3$ be a triangle and $M$ an interior point. The straight lines $MA_1, MA_2, MA_3$ intersect the opposite sides at the points $B_1, B_2, B_3$ respectively (see Fig.). Show that if the areas of triangles $A_2B_1M, A_3B_2M$ and $A_1B_3M$ are equal, then $M$ coincides with the centroid of triangle $A_1A_2A_3$ .

Let $P$ be a point inside $\vartriangle ABC$ . Let $D, E, F$ be the feet of the perpendiculars from $P$ to the lines $BC, CA$ and $AB$ , respectively (see Fig. ). Show that (i) $EF = AP \sin A$ , (ii) $PA+ PB + PC \ge 2(PE+ PD+ PF)$

Let $a, b$ and $c$ be positive integers such that $1 < a < b < c$ . Suppose that $(ab-l)(bc-1 )(ca-1)$ is divisible by $abc$ . Find the values of $a, b$ and $c$ . Justify your answer.

Let $a, b, c, d$ be four positive real numbers. Prove that $$a^{10} + b^{10}+c^{10} + d^{10} \ge (0.1a + 0.2b + 0.3c + 0.4d)^{10} + (0.4a + 0.3b + 0.2c + 0.ld)^{10} + (0.2a + 0.4b + 0.1c + 0.3d)^{10} + (0.3a + 0.1b + 0.4c + 0.2d)^{10}$$