MathDB

4

Part of 2015 AIME Problems

Problems(2)

Lots of Equilateral Triangles

Source: 2015 AIME I Problem 4

3/20/2015
Point BB lies on line segment AC‾\overline{AC} with AB=16AB=16 and BC=4BC=4. Points DD and EE lie on the same side of line ACAC forming equilateral triangles △ABD\triangle ABD and △BCE\triangle BCE. Let MM be the midpoint of AE‾\overline{AE}, and NN be the midpoint of CD‾\overline{CD}. The area of △BMN\triangle BMN is xx. Find x2x^2.
AMCAIMEAIME ISpiral Similaritygeometrygeometry solved2015 AIME I
Trapezoid Perimeter Given Lengths in Logs

Source: 2015 AIME 2 Problem 4

3/26/2015
In an isosceles trapezoid, the parallel bases have lengths log⁡3\log3 and log⁡192\log192, and the altitude to these bases has length log⁡16\log16. The perimeter of the trapezoid can be written in the form log⁡2p3q\log2^p3^q, where pp and qq are positive integers. Find p+qp+q.
geometrytrapezoidperimeter