MathDB

2

Part of 2008 Iran MO (2nd Round)

Problems(2)

Phone Numbers - Iran NMO 2008 - Problem5

Source:

9/22/2010
We want to choose telephone numbers for a city. The numbers have 1010 digits and 00 isn’t used in the numbers. Our aim is: We don’t choose some numbers such that every 22 telephone numbers are different in more than one digit OR every 22 telephone numbers are different in a digit which is more than 11. What is the maximum number of telephone numbers which can be chosen? In how many ways, can we choose the numbers in this maximum situation?
combinatorics unsolvedcombinatorics
Iran NMO 2008 (Second Round) - Problem2

Source:

9/22/2010
Let IaI_a be the AA-excenter of ΔABC\Delta ABC and the AA-excircle of ΔABC\Delta ABC be tangent to the lines AB,ACAB,AC at B′,C′B',C', respectively. IaB,IaC I_aB,I_aC meet B′C′B'C' at P,QP,Q, respectively. MM is the meet point of BQ,CPBQ,CP. Prove that the length of the perpendicular from MM to BCBC is equal to rr where rr is the radius of incircle of ΔABC\Delta ABC.
geometryincentertrigonometrypower of a pointradical axisgeometry proposed