MathDB

4

Part of 2005 Croatia National Olympiad

Problems(4)

let circumradius, find angles

Source: Croatian NMC 2005, 1st Grade

5/8/2007
The circumradius RR of a triangle with side lengths a,b,ca, b, c satisfies R=abcb+cR =\frac{a\sqrt{bc}}{b+c}. Find the angles of the triangle.
geometrycircumcircletrigonometry
eleven numbers choose six numbers

Source: Croatian NMC 2005, 2nd Grade

5/8/2007
Show that in any set of eleven integers there are six whose sum is divisible by 66.
color the vertices of a regular $2005-$ gon

Source: Croatian NMC 2005, 3rd Grade

5/8/2007
The vertices of a regular 20052005-gon are colored red, white and blue. Whenever two vertices of different colors stand next to each other, we are allowed to recolor them into the third color. (a) Prove that there exists a finite sequence of allowed recolorings after which all the vertices are of the same color. (b) Is that color uniquely determined by the initial coloring?
combinatorics proposedcombinatorics
area of two triangles and line joining their orthocenters

Source: Croatian NMC 2005, 4 th Grade

5/9/2007
Let PP and QQ be points on the sides BCBC and CDCD of a convex quadrilateral ABCDABCD, respectively, such that ∠BAP=∠DAQ\angle{BAP}=\angle{ DAQ}. Prove that the triangles ABPABP and ADQADQ have equal area if and only if the line joining their orthocenters is perpendicular to AC.AC.
geometryvectorgeometry proposed