$\begin{aligned} & [\text { Algebraic equations and systems of equations (miscellaneous) }] \\ & {[\quad \underline{\text { Completing the square. Sums of squares }}]}\end{aligned}$

Solve the equation $a^{2}+b^{2}+c^{2}+d^{2}-a b-b c-c d-d+2 / 5=0$.
$0=a^{2}+b^{2}+c^{2}+d^{2}-a b-b c-c d-d+2 / 5=(a-b / 2)^{2}+3 / 4(b-2 c / 3)^{2}+2 / 3(c-3 d / 4)^{2}+5 / 8(d-4 / 5)^{2}$.

Therefore, all the terms are equal to 0.

## Answer

$a=1 / 5, b=2 / 5, c=3 / 5, d=4 / 5$.