Problem
ALG-B2-M10-P020 Separated numbers
Five real numbers \(u_1,\ldots,u_5\) are such that any two of them differ by at least \(2\). For some real \(m\), \[\sum_{i=1}^5u_i=3m,\qquad \sum_{i=1}^5u_i^2=3m^2.\] Prove that \(m^2\ge\frac{100}{3}\).
Hint 1. Order the numbers: \(v_1<\cdots Hint 2. Bound \(\sum_{i
Order the numbers as \(v_1<\cdots
A. Source analysis. Main objects: real numbers with distance constraints, a sum, and a sum of squares.
B. Insufficient first move. Cauchy using only the sum and square sum is too weak and ignores all pairwise distances.
C. Hidden observation. One must sum the squares of all pairwise differences.
D. Required move. After ordering, the distances are bounded below by an arithmetic progression.
E. Number of ideas. Three ideas: ordering, lower bound on pairwise distances, and the identity for squared pairwise differences.
F. Difficulty justification. Final level 8: the method is hidden and combines discrete spacing with a quadratic identity.
G. Why it is not one-step. It is not one-step: neither the sum conditions nor the distance condition alone gives the required bound.