[obm-l] Geometria dos balcãs

[Alexandre Tessarollo <tessa@xxxxxxxxxxxx>]

    Da outra lista, um pouco de diversão...

[]'s

Alexandre Tessarollo

PS: Ainda não so li com a devida calma, mas acho que falta uma parte do
enunciado do primeiro prob retirado da 3a olimpíada...


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   Date: Tue, 23 Apr 2002 02:41:41 -0700
   From: "Antreas P. Hatzipolakis" <xpolakis@mac.com>
Subject: BMO

The Balkan Mathematical Olympiads:

     1st - 16th: http://bmo.ournet.md/previous.html

           17th: http://bmo.ournet.md/index.html

Geometry Problems (selection):

Let O be the circumcenter of a triangle ABC, D be the midpoint of
the side AB and E be the centroid of the triangle ACD. Prove that
the lines CD and OE are orthogonal if and only if AB=AC.

           The 2nd Balkan Mathematical Olympiad
           1985, Sofia, Bulgaria

-------

A line that passes through the incenter I of the triangle ABC meets
the incircle in D and E and the circumcircle of the triangle ABC in
F and G (D is between I and F). Prove that , where r is the radius
of the incircle. When does the equality hold?

Let ABCD be a tetrahedron and points E, F, G, H, K, L be situated
on the edges AB, BC, CA, DA, DB, DC respectively. Prove that if

           AE*BE = BF*CF = CG*AG = DH*AH = DK*BK = DL*CL

then the points E, F, G, H, K, L are placed on a sphere.

           The 3rd Balkan Mathematical Olympiad
           1986, Bucharest, Romania

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Let A1B1C1 be the orthic triangle of an acute-angled nonequilateral
triangle ABC and A2, B2, C2 be the contacts of the incircle of the
triangle A1B1C1 with its sides. Prove that the triangles A2B2C2 and
ABC have the same Euler line.

          The 7th Balkan Mathematical Olympiad
          1990, Sofia, Bulgaria

-------

Three circles Gamma, C1 and C2 are given in the plane. C1 and
C2 tangent Gamma internally at points B and C, respectively.
Moreover C1 and C2 tangent each other externally at a point D.
Let A be one point in which the common tangent of C1 and C2
intersects Gamma. Denote by M the second point of intersection
of the line AB and the circle C1 and by N the second point of
intersection of the line AC and the circle C2. Further denote by
K and L second points of intersections of the line BC with C1
and C2, respectively. Show that lines AD, MK and NL are
concurrent.

           The 14th Balkan Mathematical Olympiad
           Kalampaka, Greece, April 29, 1997

-------

Given an acute triangle ABC, let D be the midpoint of the arc BC of
the circumcircle around the triangle ABC, not containing the point A.
The points which are symmetric to D with respect to the line BC and
the circumcentre O are denoted by E and F, respectively. Finally,
let K be the midpoint of the segment EA. Prove that:
  a) The circle, passing through the midpoints of the sides of
the triangle ABC, also passes through K;
  b) The line, passing through K and the midpoint of the segment
BC is perpendicular to the line AF.

          The 16th Balkan Mathematical Olympiad
          Ohrid, FYR Macedonia, May 7th, 1999

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APH

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