If $x_1, x_2, ..., x_{100}$ and $y_1, y_2, ..., y_{100}$ are permutations of the set of numbers $1,2,3,...100$, what is the minimum value of $x_1y_1+x_2y_2+...x_{100}y_{100}$?
I'm pretty sure the minimum is when $x_1, x_2, ..., x_{100} = 1,2,...,100$ and $y_1, y_2, ..., y_{100}= 100,99,...,1$. However, I can't figure out the sum of the product $1 \cdot 100 + 2 \cdot 99 + ... + 100 \cdot 1$. Could someone tell me if my work is correct and how to calculate such sums?
The Chebyshev rearrangement inequality will confirm your answer as the right one.
If you want to calculate the required sum, use known formulas: $$\displaystyle\sum_{i=1}^{100} i(101-i)= 101\displaystyle\sum_{i=1}^{100} i - \displaystyle\sum_{i=1}^{100} i^2$$
This is then equal to: $$ \frac{101 \times 100 \times 101}{2} - \frac{100 \times 101 \times 201}{6} = 166650 $$
Using the fact that $\displaystyle\sum_{i=1}^{n} i = \frac {n(n+1)}2$ and $\displaystyle\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$