SORRY, I made a typo. it should be $D \not= 0$,not $D>0$.
It is a bit like Vandermonde determinant $D=$ $\begin{vmatrix} 1 & 2 & 3&\cdots &2008&2009 & 2010 & 2011\\ 2^2 & 3^2 &4^2& \cdots&2009^2&2010^2 & 2011^2 &2012^2 \\ 3^3 & 4^3 &5^3&\cdots &2010^3&2011^3&2012^3 &2012^3 \\ \cdots &\cdots &\cdots &\cdots&\cdots&\cdots &\cdots &\cdots\\ k^k&(k+1)^k&\cdots&2011^k&2012^k&\cdots&2012^k&2012^k\\ \cdots &\cdots &\cdots &\cdots &\cdots &\cdots\\ 2010^{2010}&2011^{2010}&2012^{2010}&\cdots&2012^{2010}&2012^{2010}&2012^{2010}&2012^{2010}\\ 2011^{2011} &2012^{2011} &2012^{2011}&\cdots &2012^{2011} &2012^{2011} &2012^{2011}&2012^{2011} \end{vmatrix}$
Is the above determinant $D\not= 0$?
the exam is only need to show $D \not= 0$,maybe figure out $D$ is impossible. and I edit it, maybe more clearly.
thanks for comments and answer.
FIRST PAGE OF EXAM: