, 2013

Solution:

Let $f(a, b, c, d) = (a^{2}+b^{2}+c^{2}+d^{2})-(a b+\lambda b c+c d)$. For fixed $(b, c, d)$, $f$ is minimized at $a=\frac{b}{2}$, and for fixed $(a, b, c)$, $f$ is minimized at $d=\frac{c}{2}$, so simply we want the largest $\lambda$ such that $f\left(\frac{b}{2}, b, c, \frac{c}{2}\right)=\frac{3}{4}\left(b^{2}+c^{2}\right)-\lambda b c$ is always nonnegative. By AM-GM, this holds if and only if $\lambda \leq 2 \cdot \frac{3}{4}=\frac{3}{2}$.