Find all functions $ f:\mathbb{R} \mapsto \mathbb{R} $ such that $f[x^2+f(y)] = y +[f(x)]^2$ I read this question in the book 'Problem Solving Strategies ' by Arthur Engel. This question was asked in IMO of 1992. Here's how it goes
Find all the functions  $ f:\mathbb{R} \mapsto \mathbb{R} $ that satisfy
$f[x^2+f(y)] = y +[f(x)]^2$ $ \ \ \ \ x,y \in \mathbb{R}$
The book simply states that it has no solution but it is not a sufficient answer. How can I prove that no such function exists ?


In the case that $ [ \cdot ] $ means the floor function (also denoted by $ \lfloor \cdot \rfloor $), you can show that there is no $ f $ satisfying $ f \big( \big\lfloor x ^ 2 + f ( y ) \big\rfloor \big) = y + \big\lfloor f \big( x ^ 2 \big) \big\rfloor $. Let $ x = 0 $ and you'll get $ f \big( \lfloor f ( y ) \rfloor \big) = y + \lfloor f ( 0 ) \rfloor $. This shows that $ f $ maps a subset of $ \mathbb Z $ onto $ \mathbb R $, which is imposible, since $ \mathbb Z $ is countable and $ \mathbb R $ is uncountable. The case is similar for the functional equation $ f \big( \big\lfloor x ^ 2 + f ( y ) \big\rfloor \big) = y + \lfloor f ( x ) \rfloor ^ 2 $, which corresponds with the IMO problem.
