Abdul
Number Theory
Level
4
Given that a,b,d,u,l are real numbers such that
a+b+d+u+l=8 a^{2}+b^{2}+d^{2}+u^{2}+l^{2}=16.
Let M be the maximum value of l. Find the number of factors of M×30.
The answer is 12.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
(8−l)^2 = (a+ b + d + u)^2 = a^2 + b^2 + d^2 + u^2 +2(ab + ad+ au + bd + bu + du) ≤ 4(a^2 + b^2 + d^2 + u^2) (16−l^2) = a^2 + b^2 + d^2 + u^2 4(a^2 + b^2 + d^2 + u^2)≥(a+ b + d+ u)^2 4(16−l^2)≥(8−l)^2 64−4l^2 ≥64 − 16l + l^2 5l^2 −16l ≤0 l(5l−16)≤0, l ≤ 16/5