MathDB

5

Part of 2013 Kosovo National Mathematical Olympiad

Problems(4)

Kosovo National Mathematical 2013 grade 10 #5

Source:

3/17/2013
Let PP be a point inside or outside (but not on) of a triangle ABCABC. Prove that PA+PB+PCPA +PB +PC is greater than half of the perimeter of the triangle
geometryperimeterinequalitiestriangle inequality
Kosovo National Mathematical 2013 grade 9 #5

Source:

3/17/2013
Let ABCABC be an equilateral triangle, with sidelength equal to aa. Let PP be a point in the interior of triangle ABCABC, and let D,ED,E and FF be the feet of the altitudes from PP on AB,BCAB, BC and CACA, respectively. Prove that ∣PD∣+∣PE∣+∣PF∣3a=36\frac{|PD|+|PE|+|PF|}{3a}=\frac{\sqrt{3}}{6}
geometry
Kosovo National Mathematical 2013 grade 11 #5

Source:

3/17/2013
Let ABCDABCD be a convex quadrilateral with perpendicular diagonals. . Assume that ABCDABCD has been inscribed in the circle with center OO. Prove that AOCAOC separates ABCDABCD into two quadrilaterals of equal area
geometry
Kosovo National Mathematical 2013 grade 12 #5

Source:

3/17/2013
A trapezium has parallel sides of length equal to aa and bb (a<ba <b), and the distance between the parallel sides is the altitude hh. The extensions of the non-parallel lines intersect at a point that is a vertex of two triangles that have as sides the parallel sides of the trapezium. Express the areas of the triangles as functions of a,ba,b and hh.
geometrytrapezoidfunctionsimilar triangles