MathDB

14

Part of 2003 AIME Problems

Problems(2)

Digits 2, 5, and 1

Source:

5/12/2006
The decimal representation of m/nm/n, where mm and nn are relatively prime positive integers and m<nm < n, contains the digits 2, 5, and 1 consecutively, and in that order. Find the smallest value of nn for which this is possible.
AMCAIMEinequalitiesnumber theoryrelatively prime
Area of Hexagon

Source:

12/26/2006
Let A=(0,0)A=(0,0) and B=(b,2)B=(b,2) be points on the coordinate plane. Let ABCDEFABCDEF be a convex equilateral hexagon such that ∠FAB=120∘,\angle FAB=120^\circ, AB‾∥DE‾,\overline{AB}\parallel \overline{DE}, BC‾∥EF,‾\overline{BC}\parallel \overline{EF,} CD‾∥FA‾,\overline{CD}\parallel \overline{FA}, and the y-coordinates of its vertices are distinct elements of the set {0,2,4,6,8,10}.\{0,2,4,6,8,10\}. The area of the hexagon can be written in the form mn,m\sqrt{n}, where mm and nn are positive integers and n is not divisible by the square of any prime. Find m+n.m+n.
geometryanalytic geometrytrigonometrysymmetry