IMO 1983 SL 20

Origin: ROM

Problem

Solve the system of equations x1|x1| = x2|x2| + (x1 −a)|x1 −a|, x2|x2| = x3|x3| + (x2 −a)|x2 −a|, \cdot \cdot \cdot xn|xn| = x1|x1| + (xn −a)|xn −a|, in the set of real numbers, where a > 0.

Solution

If (x1, x2, . . . , xn) satisfies the system with parameter a, then (−x1, −x2, . . . , −xn) satisfies the system with parameter −a. Hence it is sufficient to consider only a \geq0. Let (x1, . . . , xn) be a solution. Suppose x1 \leqa, x2 \leqa, . . . , xn \leqa. Summing the equations we get (x1 −a)2 + \cdot \cdot \cdot + (xn −a)2 = 0 and see that (a, a, . . . , a) is the only such solution. Now suppose that xk \geqa for some k. According to the kth equation,

xk+1|xk+1| = x2 k −(xk −a)2 = a(2xk −a) \geqa2, which implies that xk+1 \geqa as well (here xn+1 = x1). Consequently, all x1, x2, . . . , xn are greater than or equal to a, and as above (a, a, . . . , a) is the only solution.