12

Part of 2003 AIME Problems

Problems(2)

A convex quadrilateral

Source:

In convex quadrilateral

ABCD

\angle A \cong \angle C

AB = CD = 180

AD \neq BC

. The perimeter of

ABCD

\lfloor 1000 \cos A \rfloor

. (The notation

\lfloor x \rfloor

means the greatest integer that is less than or equal to

x

geometryperimeterfloor functiontrigonometrycircumcircletrig identitiesLaw of Cosines

Members of a Committee

Source:

The members of a distinguished committee were choosing a president, and each member gave one vote to one of the

27

candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least

1

than the number of votes for that candidate. What is the smallest possible number of members of the committee?

quadraticsceiling functionpigeonhole principleAMCAIME