MathDB

6

Part of 2017 Saint Petersburg Mathematical Olympiad

Problems(3)

a^2+b^2+c^2=1

Source: St. Petersburg

4/3/2018
Given three real numbers a,b,c∈[0,1)a,b,c\in [0,1) such that a2+b2+c2=1a^2+b^2+c^2=1. Find the smallest possible value of a1−a2+b1−b2+c1−c2.\frac{a}{\sqrt{1-a^2}}+\frac{b}{\sqrt{1-b^2}}+\frac{c}{\sqrt{1-c^2}}.
inequalities
Altitude and Median from two vertices

Source: Saint Petersburg 2017

8/18/2017
In acute-angled triangle ABCABC, the height AHAH and median BMBM were drawn. Point DD lies on the circumcircle of triangle BHMBHM such that AD∥BMAD \parallel BM and B,DB, D are on opposite sides of line ACAC. Prove that BC=BDBC=BD.
geometrycircumcircle
Mathematicians dont have friend?

Source: St.Petersburg 2017

1/8/2018
In the country some mathematicians know each other and any division of them into two sets contain 2 friends from different sets.It is known that if you put any set of four or more mathematicians at a round table so that any two neighbours know each other , then at the table there are two friends not sitting next to each other.We denote by cic_i the number of sets of ii pairwise familiar mathematicians(by saying "familiar" it means know each other).Prove that c1−c2+c3−c4+...=1c_1-c_2+c_3-c_4+...=1
combinatoricsgraph theory