Prove that there exists a positive integer N such that there are at least 2005 ordered pairs (x, y), of non-negative integers x and y, satisfying x^2 + y^2 = N.

Express the following as a function of a single angle, cos(60) cos(-20) - sin(60) sin(-20)

Afurniture manufacturer produces chairs and sofas. each chair requires 10 yards of fabric, and each sofa requires 20 yards of fabric. the manufacturer has 300 yards of fabric available. to fulfill orders, the number of sofas must be at least twice the number of chairs. let x be the number of chairs and y the number of sofas. which inequalities are described in the problem? check all of the boxes that apply

Complete the equation of the line through (−10,3) and (−8,−8). use exact numbers.