12

Part of 2014 AIME Problems

Problems(2)

Disjoint ranges of two functions

Source: 2014 AIME I #12

A=\{1,2,3,4\}

f

g

be randomly chosen (not necessarily distinct) functions from

A

A

. The probability that the range of

f

and the range of

g

are disjoint is

\tfrac{m}{n}

m

n

are relatively prime positive integers. Find

m

functionprobabilityAMCAIMEalgebradomainnumber theory

Cosine Equations

Source: 2014 AIME II Problem #12

Suppose that the angles of

\triangle ABC

\cos(3A) + \cos(3B) + \cos(3C) = 1

. Two sides of the triangle have lengths

10

13

. There is a positive integer

m

so that the maximum possible length for the remaining side of

\triangle ABC

\sqrt{m}

m

trigonometryAMCAIMEtrig identitiestangent half angleAIME II