BRATISLAVA MATHEMATICS CORRESPONDENCE SEMINAR
Faculty of Mathematics and Physics of Comenius University
Union of Slovak Mathematicians and Physicists
Centre of Spare Time - IUVENTA

BMCS - Problem set of the 2nd fall series 1997/98

Triangles

  1. Construct a triangle ABC if you know the sizes of angles alpha and beta (near vertices A and B, respectively) and the size of va + tb + R, where va is the height of the triangle from the vertex A, tb is the median from the vertex B and R is the radius of the circumscribed circle.

  2. Let D is the intersection of the side BC and the bisector of the angle alpha, which is near the vertex A of an acute-angle triangle ABC. The side AC has the same length as the line segment AD. In addition, AD is perpendicular to SO, where S is the center of the circumscribed circle of ABC and O is the intersection of heights of triangle ABC. Find the sizes of angles of the trianlge ABC.

  3. Find such point M inside the triangle ABC, that the value of expression |AM|2 + |BM|2 + |CM|2 will be minimal.

  4. Nine triangles met on the international conference on right parallelepipeds. They found out that in any group of three of them at least two had seen the same part of the TV-serial Mary. Furthermore, none of all nine triangles had seen more than three parts of the serial. Prove that there existed a group of three triangles in which all had seen the same part of the serial.

  5. The following holds for the inside point O of the triangle ABC. Vector sum of the vectors OK, OM, and ON is the zero vector, where K, M, and N are feet of the heights from the vertex O on sides AB, BC, and AC, respectively. Prove that:
    
        1       |OK| + |OM| + |ON|
      ----- >= -------------------
       23     |AB| + |BC| + |CA|