6

Part of 2000 AIME Problems

Problems(2)

AM is two more than GM

Source:

For how many ordered pairs

(x,y)

of integers is it true that

0<x<y<10^{6}

and that the arithmetic mean of

x

y

2

more than the geometric mean of

x

y?

quadraticsAMCAIMEalgebraquadratic formula

Source:

One base of a trapezoid is 100 units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio

2: 3.

x

be the length of the segment joining the legs of the trapezoid that is parallel to the bases and that divides the trapezoid into two regions of equal area. Find the greatest integer that does not exceed

x^2/100.

geometrytrapezoidratioquadraticsfloor functionAMCAIME