5

Part of 2013 AIME Problems

Problems(2)

Real Root of Cubic Equation

Source: 2013 AIME I Problem 5

The real root of the equation

8x^3 - 3x^2 - 3x - 1 = 0

can be written in the form

\frac{\sqrt[3]a + \sqrt[3]b + 1}{c}

a

b

c

are positive integers. Find

a+b+c

algebrapolynomialgeometry3D geometrydifference of squaresspecial factorizationsfactorization

Find sin DAE in equilateral ABC, where D and E trisect BC

Source: AIME II 2013, Problem 5

\triangle ABC

D

E

\overline{BC}

\sin \left( \angle DAE \right)

can be expressed in the form

\tfrac{a\sqrt{b}}{c}

a

c

are relatively prime positive integers, and

b

is an integer that is not divisible by the square of any prime. Find

a+b+c

trigonometrygeometrycircumcirclerationumber theoryrelatively primetrig identities