IMO Shortlist 1968 problem 9


Kvaliteta:
  Avg: 0,0
Težina:
  Avg: 0,0
Dodao/la: arhiva
2. travnja 2012.
LaTeX PDF
Let ABC be an arbitrary triangle and M a point inside it. Let d_a, d_b, d_c be the distances from M to sides BC,CA,AB; a, b, c the lengths of the sides respectively, and S the area of the triangle ABC. Prove the inequality
abd_ad_b + bcd_bd_c + cad_cd_a \leq \frac{4S^2}{3}.
Prove that the left-hand side attains its maximum when M is the centroid of the triangle.
Izvor: Međunarodna matematička olimpijada, shortlist 1968