MathDB

1

Part of 2008 China National Olympiad

Problems(2)

China Mathematics Olympiads (National Round) 2008 Problem 1

Source:

11/28/2010
Suppose △ABC\triangle ABC is scalene. OO is the circumcenter and A′A' is a point on the extension of segment AOAO such that ∠BA′A=∠CA′A\angle BA'A = \angle CA'A. Let point A1A_1 and A2A_2 be foot of perpendicular from A′A' onto ABAB and ACAC. HAH_{A} is the foot of perpendicular from AA onto BCBC. Denote RAR_{A} to be the radius of circumcircle of △HAA1A2\triangle H_{A}A_1A_2. Similiarly we can define RBR_{B} and RCR_{C}. Show that: 1RA+1RB+1RC=2R\frac{1}{R_{A}} + \frac{1}{R_{B}} + \frac{1}{R_{C}} = \frac{2}{R} where R is the radius of circumcircle of △ABC\triangle ABC.
geometrycircumcircleincentergeometric transformationreflectionperpendicular bisectorgeometry unsolved
China Mathematics Olympiads (National Round) 2008 Problem 4

Source:

11/28/2010
Let AA be an infinite subset of N\mathbb{N}, and nn a fixed integer. For any prime pp not dividing nn, There are infinitely many elements of AA not divisible by pp. Show that for any integer m>1,(m,n)=1m >1, (m,n) =1, There exist finitely many elements of AA, such that their sum is congruent to 1 modulo mm and congruent to 0 modulo nn.
number theory unsolvednumber theory