IMO 1972 LL ROM35

(a) Prove that for a, b, c, d \inR, m \in[1, +\infty) with am + b =

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IMO 1972 LL ROM35

Origin: ROM

Problem

(a) Prove that for a, b, c, d \inR, m \in[1, +\infty) with am + b = −cm + d = m, (i) \sqrt a2 + b2 + \sqrt c2 + d2 + (a −c)2 + (b −d)2 \geq 4m2 1+m2 , and (ii) 2 \leq 4m2 1+m2 < 4. (b) Express a, b, c, d as functions of m so that there is equality in (1).