MathDB

3

Part of 2011 Pre - Vietnam Mathematical Olympiad

Problems(2)

Two intersecting circles

Source: Pre-VMO 2012 - Problem 3

11/27/2011
Two circles (O)(O) and (O′)(O') intersect at AA and BB. Take two points P,QP,Q on (O)(O) and (O′)(O'), respectively, such that AP=AQAP=AQ. The line PQPQ intersects (O)(O) and (O′)(O') respectively at M,NM,N. Let E,FE,F respectively be the centers of the two arcs BPBP and BQBQ (which don't contains AA). Prove that MNEFMNEF is a cyclic quadrilateral.
geometryincenterangle bisectorgeometry proposed
Ineq of the selection in sets

Source: Pre-VMO 2012 - Problem 7

11/27/2011
There are nn students. Denoted the number of the selections to select two students (with their weights are aa and bb) such that ∣a−b∣≤1\left| {a - b} \right| \le 1 (kg) and ∣a−b∣≤2\left| {a - b} \right| \le 2 (kg) by A1A_1 and A2A_2, respectively. Prove that A2<3A1+nA_2<3A_1+n.
inductioncombinatorics proposedcombinatorics