MathDB

5

Part of 2000 Canada National Olympiad

Problems(1)

Max of the sum of the square.

Source: Canada 2000, Problem 5

3/4/2006
Suppose that the real numbers a1,a2,…,a100a_1, a_2, \ldots, a_{100} satisfy \begin{eqnarray*} 0 \leq a_{100} \leq a_{99} \leq \cdots \leq a_2 &\leq& a_1 , \\ a_1+a_2 & \leq & 100 \\ a_3+a_4+\cdots+a_{100} &\leq & 100. \end{eqnarray*} Determine the maximum possible value of a12+a22+⋯+a1002a_1^2 + a_2^2 + \cdots + a_{100}^2, and find all possible sequences a1,a2,…,a100a_1, a_2, \ldots , a_{100} which achieve this maximum.
inequalities unsolvedinequalities