Problem
ALG-B2-M01-P023 Sum of three fractions
Do there exist distinct real numbers \(x,y,z\) such that \[\frac1{x^2+x+1}+\frac1{y^2+y+1}+\frac1{z^2+z+1}=4?\]
Hint 1. Find the maximum of one fraction.
Hint 2. Equality of the sum requires equality in all three terms.
For every \(t\), \(t^2+t+1=\left(t+\frac12\right)^2+\frac34\ge\frac34\). Hence \(\frac1{t^2+t+1}\le\frac43\), with equality only at \(t=-\frac12\). The sum of three such fractions is at most \(4\). If it equals \(4\), each fraction must equal \(\frac43\), so \(x=y=z=-\frac12\), impossible for distinct numbers. Answer: no.
A. Source analysis. Main objects: inequalities, order, an extremal element, or an invariant. The obvious first move usually gives only a local estimate. The hidden observation is to choose the right nondecreasing quantity, or to add/multiply inequalities only after signs are controlled. The needed step is an ordering, an invariant, a product transformation, or a boundary case.
F. Difficulty justification. Regional level 7: besides the estimate, a strict equality-case check is needed.
G. Why this is not a one-step exercise. The termwise bound alone does not exclude sum \(4\) until the equality case is analyzed.