Algebra · Floor function · Inequalities · Quadratic inequality · Integer solutions
For a real number \(a\), let \([a]\) denote the largest integer that is not greater than \(a\). Find all integers \(y\) for which there exists a real number \(x\) satisfying
\[ \left[\frac{x+23}{8}\right] = \left[\sqrt{x}\,\right] = y . \]
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Slovenian High School Mathematics Competition for Vega Awards (MaSSA), drzavno (national) round 2013, 1. letnik, category A, problem 2. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source