MathDB

13

Part of 2012 AIME Problems

Problems(2)

Cocentric Circles of radii 3,4,5

Source: 2012 AIME I Problem 13

3/16/2012
Three concentric circles have radii 33, 44, and 55. An equilateral triangle with one vertex on each circle has side length ss. The largest possible area of the triangle can be written as a+bcda+\frac{b}{c}\sqrt{d}, where a,b,ca,b,c and dd are positive integers, bb and cc are relatively prime, and dd is not divisible by the square of any prime. Find a+b+c+da+b+c+d.
geometrygeometric transformationreflectiontrigonometryrotationanalytic geometry3D geometry
Five Congruent Equilateral Triangles

Source: 2012 AIME II Problem 13

3/29/2012
Equilateral △ABC\triangle ABC has side length 111\sqrt{111}. There are four distinct triangles AD1E1AD_1E_1, AD1E2AD_1E_2, AD2E3AD_2E_3, and AD2E4AD_2E_4, each congruent to △ABC\triangle ABC, with BD1=BD2=11BD_1 = BD_2=\sqrt{11}. Find ∑k=14(CEk)2\sum^4_{k=1}(CE_k)^2.
trigonometryvectorsymmetrytrig identitiesLaw of Cosinescomplex numbersAMC