AoPS Wiki talk:Problem of the Day/June 21, 2011
. Hence or . If and , then attains this maximum value on the circle .
Let and be real numbers such that . Note that thus, we may assume that and are positive. Furthermore, by the Cauchy-Schwarz Inequality, we have but since , the inequality is equivalent with or so the maximum is and it is reached when .
Imagine the equations graphed in the coordinate plane. is a circle centered at the origin with
radius . is a line. We want to find the largest value of such that the line
intersects the circle, giving real number solutions for and . This occurs when is tangent
to the circle, and thus when the distance from the line to the origin is . The distance from a point to a line is
Plugging in , and and setting the expression equal to yields
, or . We want the largest value of , so
is the highest possible value.
By AM-QM, , so , equality when .