14

Part of 2016 AIME Problems

Problems(2)

Tiny Circles and Squares in R^2

Source: 2016 AIME I #14

Centered at each lattice point in the coordinate plane are a circle of radius

\tfrac{1}{10}

and a square with sides of length

\tfrac{1}{5}

whose sides are parallel to the coordinate axes. The line segment from

(0, 0)

(1001, 429)

m

of the squares and

n

of the circles. Find

m + n

2016 AIME IAIME

All men are created equilateral

Source: 2016 AIME II #14

\triangle ABC

has side length

600

P

Q

lie outside of the plane of

\triangle ABC

and are on the opposite sides of the plane. Furthermore,

PA=PB=PC

QA=QB=QC

, and the planes of

\triangle PAB

\triangle QAB

120^{\circ}

dihedral angle (The angle between the two planes). There is a point

O

whose distance from each of

A,B,C,P

Q

d

d