MathDB

P1

Part of 2014 BMT Spring

Problems(3)

BMT 2014 Spring - Geometry P1

Source:

12/29/2021
Let ABCABC be a triangle. Let r r denote the inradius of △ABC\vartriangle ABC. Let rar_a denote the AA-exradius of △ABC\vartriangle ABC. Note that the AA-excircle of △ABC\vartriangle ABC is the circle that is tangent to segment BCBC, the extension of ray ABAB beyond B B and the extension of ACAC beyond CC. The AA-exradius is the radius of the AA-excircle. Define rb r_b and rc r_c analogously. Prove that 1r=1ra+1rb+1rc\frac{1}{r}=\frac{1}{r_a}+\frac{1}{r_b}+\frac{1}{r_c}
geometry
BMT 2014 Spring - Analysis P1

Source:

1/6/2022
Suppose that a,b,c,da,b,c,d are non-negative real numbers such that a2+b2+c2+d2=2a^2+b^2+c^2+d^2=2 and ab+bc+cd+da=1ab+bc+cd+da=1. Find the maximum value of a+b+c+da+b+c+d and determine all equality cases.
inequalities
BMT 2014 Spring - Discrete P1

Source:

1/6/2022
Let a simple polygon be defined as a polygon in which no consecutive sides are parallel and no two non-consecutive sides share a common point. Given that all vertices of a simple polygon PP are lattice points (in a Cartesian coordinate system, each vertex has integer coordinates), and each side of PP has integer length, prove that the perimeter must be even.
geometry