MathDB

11

Part of 2014 AIME Problems

Problems(2)

Wandering Token

Source: 2014 AIME I Problem 11

3/14/2014
A token starts at the point (0,0)(0,0) of an xyxy-coordinate grid and them makes a sequence of six moves. Each move is 11 unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other moves. The probability the token ends at a point on the graph of ∣y∣=∣x∣|y|=|x| is mn\tfrac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
analytic geometryprobabilitysymmetryAMCAIMErotationgeometry
Nice Angles

Source: 2014 AIME II Problem #11

3/27/2014
In △RED,RD=1,∠DRE=75∘\triangle RED, RD =1, \angle DRE = 75^\circ and ∠RED=45∘\angle RED = 45^\circ. Let MM be the midpoint of segment RD‾\overline{RD}. Point CC lies on side ED‾\overline{ED} such that RC‾⊥EM‾\overline{RC} \perp \overline{EM}. Extend segment DE‾\overline{DE} through EE to point AA such that CA=ARCA = AR. Then AE=a−bc,AE = \tfrac{a-\sqrt{b}}{c}, where aa and cc are relatively prime positive integers, and bb is a positive integer. Find a+b+ca+b+c.
ratioanalytic geometrytrigonometrynumber theoryrelatively primePythagorean Theoremgeometry