MathDB

2

Part of 2002 Bosnia Herzegovina Team Selection Test

Problems(2)

Show that 2MN=BM+CN

Source: P2: Bosnia and Herzegovina TST 2002

9/30/2014
Triangle ABCABC is given in a plane. Draw the bisectors of all three of its angles. Then draw the line that connects the points where the bisectors of angles ABCABC and ACBACB meet the opposite sides of the triangle. Through the point of intersection of this line and the bisector of angle BACBAC, draw another line parallel to BCBC. Let this line intersect ABAB in MM and ACAC in NN. Prove that 2MN=BM+CN2MN = BM+CN.
geometrytrapezoidgeometry unsolved
The vertices of the convex quad ABCD are integer points

Source: P6: Bosnia and Herzegovina TST 2002

10/3/2014
The vertices of the convex quadrilateral ABCDABCD and the intersection point SS of its diagonals are integer points in the plane. Let PP be the area of ABCDABCD and P1P_1 the area of triangle ABSABS. Prove that \sqrt{P} \ge \sqrt{P_1}+\frac{\sqrt2}2
geometrygeometry unsolved