MathDB

2

Part of 2004 Federal Math Competition of S&M

Problems(3)

Geometry problem

Source: Federal Mathematical Competition of Serbia and Montenegro 2004

5/14/2018
Let rr be the inradius of an acute triangle. Prove that the sum of the distances from the orthocenter to the sides of the triangle does not exceed 3r3r
geometry
Geometry

Source: Federal Mathematical Competition of Serbia and Montenegro 2004

5/14/2018
In a triangle ABCABC, points DD and EE are taken on rays CBCB and CACA respectively so that CD=CE=AC+BC2CD=CE = \frac{AC+BC}{2}. Let HH be the orthocenter of the triangle, and PP be the midpoint of the arc ABAB of the circumcircle of ABCABC not containing CC. Prove that the line DEDE bisects the segment HPHP.
geometry
Sequences and Number theory

Source: Federal Mathematical Competition of Serbia and Montenegro 2004

5/14/2018
The sequence (an)(a_n) is determined by a1=0a_1 = 0 and (n+1)3an+1=2n2(2n+1)an+2(3n+1)(n+1)^3a_{n+1} = 2n^2(2n+1)a_n+2(3n+1) for n≥1n \geq 1. Prove that infinitely many terms of the sequence are positive integers.
number theory