CMO 1988

Let $r_1,r_2,\dots ,r_n$ be real numbers. Given $n$ reals $a_1,a_2,\dots ,a_n$ that are not all equal to $0$ , suppose that inequality $$r_1(x_1-a_1)+ r_2(x_2-a_2)+\dots + r_n(x_n-a_n)\leq\sqrt{x_1^2+ x_2^2+\dots + x_n^2}-\sqrt{a_1^2+a_2^2+\dots +a_n^2}$$ holds for arbitrary reals $x_1,x_2,\dots ,x_n$ . Find the values of $r_1,r_2,\dots ,r_n$ .

Given two circles $C_1,C_2$ with common center, the radius of $C_2$ is twice the radius of $C_1$ . Quadrilateral $A_1A_2A_3A_4$ is inscribed in $C_1$ . The extension of $A_4A_1$ meets $C_2$ at $B_1$ ; the extension of $A_1A_2$ meets $C_2$ at $B_2$ ; the extension of $A_2A_3$ meets $C_2$ at $B_3$ ; the extension of $A_3A_4$ meets $C_2$ at $B_4$ . Prove that $P(B_1B_2B_3B_4)\ge 2P(A_1A_2A_3A_4)$ , and in what case the equality holds? ( $P(X)$ denotes the perimeter of quadrilateral $X$ )

Given a finite sequence of real numbers $a_1,a_2,\dots ,a_n$ ( $\ast$ ), we call a segment $a_k,\dots ,a_{k+l-1}$ of the sequence ( $\ast$ ) a "long"(Chinese dragon) and $a_k$ "head" of the "long" if the arithmetic mean of $a_k,\dots ,a_{k+l-1}$ is greater than $1988$ . (especially if a single item $a_m>1988$ , we still regard $a_m$ as a "long"). Suppose that there is at least one "long" among the sequence ( $\ast$ ), show that the arithmetic mean of all those items of sequence ( $\ast$ ) that could be "head" of a certain "long" individually is greater than $1988$ .

(1) Let $a,b,c$ be positive real numbers satisfying $(a^2+b^2+c^2)^2>2(a^4+b^4+c^4)$ . Prove that $a,b,c$ can be the lengths of three sides of a triangle respectively. (2) Let $a_1,a_2,\dots ,a_n$ be $n$ ( $n>3$ ) positive real numbers satisfying $(a_1^2+a_2^2+\dots +a_n^2)^2>(n-1)(a_1^4+ a_2^4+\dots +a_n^4)$ . Prove that any three of $a_1,a_2,\dots ,a_n$ can be the lengths of three sides of a triangle respectively.

Given three tetrahedrons $A_iB_i C_i D_i$ ( $i=1,2,3$ ), planes $\alpha _i,\beta _i,\gamma _i$ ( $i=1,2,3$ ) are drawn through $B_i ,C_i ,D_i$ respectively, and they are perpendicular to edges $A_i B_i, A_i C_i, A_i D_i$ ( $i=1,2,3$ ) respectively. Suppose that all nine planes $\alpha _i,\beta _i,\gamma _i$ ( $i=1,2,3$ ) meet at a point $E$ , and points $A_1,A_2,A_3$ lie on line $l$ . Determine the intersection (shape and position) of the circumscribed spheres of the three tetrahedrons.

Let $n$ ( $n\ge 3$ ) be a natural number. Denote by $f(n)$ the least natural number by which $n$ is not divisible (e.g. $f(12)=5$ ). If $f(n)\ge 3$ , we may have $f(f(n))$ in the same way. Similarly, if $f(f(n))\ge 3$ , we may have $f(f(f(n)))$ , and so on. If $\underbrace{f(f(\dots f}_{k\text{ times}}(n)\dots ))=2$ , we call $k$ the "length" of $n$ (also we denote by $l_n$ the "length" of $n$ ). For arbitrary natural number $n$ ( $n\ge 3$ ), find $l_n$ with proof.