Problem
ALG-B2-M11-P020 Five separated numbers
#20
★★★★★ Level 5 of 5
Let \(x_1,\ldots,x_5\) be real numbers such that any two of them differ by at least \(d>0\). If \(\sum_{i=1}^5x_i=0\), prove \[\sum_{i=1}^5x_i^2\ge10d^2.\]
Hint. Order the numbers and estimate the sum of squares of all pairwise differences.
Order the numbers as \(y_1
Teaching goal: the student should first recognise the type of estimate, then choose the tool.