CMO 1986

We are given $n$ reals $a_1,a_2,\cdots , a_n$ such that the sum of any two of them is non-negative. Prove that the following statement and its converse are both true: if $n$ non-negative reals $x_1,x_2,\cdots ,x_n$ satisfy $x_1+x_2+\cdots +x_n=1$ , then the inequality $a_1x_1+a_2x_2+\cdots +a_nx_n\ge a_1x^2_1+ a_2x^2_2+\cdots + a_nx^2_n$ holds.

In $\triangle ABC$ , the length of altitude $AD$ is $12$ , and the bisector $AE$ of $\angle A$ is $13$ . Denote by $m$ the length of median $AF$ . Find the range of $m$ when $\angle A$ is acute, orthogonal and obtuse respectively.

Let $Z_1,Z_2,\cdots ,Z_n$ be complex numbers satisfying $|Z_1|+|Z_2|+\cdots +|Z_n|=1$ . Show that there exist some among the $n$ complex numbers such that the modulus of the sum of these complex numbers is not less than $1/6$ .

Given a $\triangle ABC$ with its area equal to $1$ , suppose that the vertices of quadrilateral $P_1P_2P_3P_4$ all lie on the sides of $\triangle ABC$ . Show that among the four triangles $\triangle P_1P_2P_3, \triangle P_1P_2P_4, \triangle P_1P_3P_4, \triangle P_2P_3P_4$ there is at least one whose area is not larger than $1/4$ .

Given a sequence $1,1,2,2,3,3,\ldots,1986,1986$ , determine, with proof, if we can rearrange the sequence so that for any integer $1\le k \le 1986$ there are exactly $k$ numbers between the two " $k$ "s.

Suppose that each point on the plane is colored either white or black. Show that there exists an equilateral triangle with the side length equal to $1$ or $\sqrt{3}$ whose three vertices are in the same color.