IMO 1988 SL 12

Origin: GRE

Problem

In a triangle ABC, choose any points K \inBC, L \inAC, M \inAB, N \inLM, R \inMK, and F \inKL. If E1, E2, E3, E4, E5, E6, and E denote the areas of the triangles AMR, CKR, BKF, ALF, BNM, CLN, and ABC respectively, show that E \geq8 E1E2E3E4E5E6. Remark. Points K, L, M, N, R, F lie on segments BC, AC, AB, LM, MK, KL respectively.

Solution

Let E(XY Z) stand for the area of a triangle XY Z. We have E1 E = E(AMR) E(AMK) \cdot E(AMK) E(ABK) \cdot E(ABK) E(ABC) = MR MK \cdot AM AB \cdot BK BC ⇒ E1 E 1/3 \leq1  MR MK + AM AB + BK BC  . We similarly obtain E2 E 1/3 \leq1  KR MK + BM AB + CK BC  .

Therefore (E1/E)1/3 + (E2/E)1/3 \leq1, i.e., 3\sqrtE1 + 3\sqrtE2 \leq 3\sqrt E. Analo- gously, 3\sqrtE3 + 3\sqrtE4 \leq 3\sqrt E and 3\sqrtE5 + 3\sqrtE6 \leq 3\sqrt E; hence E1E2E3E4E5E6 = 2( E1 E2)1/2 \cdot 2( E3 E4)1/2 \cdot 2( E5 E6)1/2 \leq( E1 + E2) \cdot ( E3 + E4) \cdot ( E5 + E6) \leqE.