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Πέμπτη 30 Ιανουαρίου 2025

Junior Balkan Mathematical Olympiad 2003 [Shortlists & Solutions]

  1. Is there is a convex quadrilateral which the diagonals divide into four triangles with areas of distinct primes?
  2. Is there a triangle with 12cm2 area and 12 cm perimeter?
  3. Let G be the centroid of triangle ABC, and A the symmetric of A wrt C. Show that G, B, C, A are concyclic if and only if GAGC.
  4. Three equal circles have a common point M and intersect in pairs at points A, B, C. Prove that that M is the orthocenter of triangle ABC.
  5. Let ABC be an isosceles triangle with AB=AC. A semi-circle of diameter [EF] with E,F[BC], is tangent to the sides AB, AC in M, N respectively and AE intersects the semicircle at P. Prove that PF passes through the midpoint of [MN].
  6. Parallels to the sides of a triangle passing through an interior point divide the inside of a triangle into 6 parts with the marked areas as in the figure. Show that aα+bβ+cγ32
  7. Let D, E, F be the midpoints of the arcs BC, CA, AB on the circumcircle of a triangle ABC not containing the points A, B, C, respectively. Let the line DE meets BC and CA at G and H, and let M be the midpoint of the segment GH. Let the line FD meet BC and AB at K and J, and let N be the midpoint of the segment KJ.
    a) Find the angles of triangle DMN;
    b) Prove that if P is the point of intersection of the lines AD and EF, then the circumcenter of triangle DMN lies on the circumcircle of triangle PMN.