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max [\sqrt{[\sqrt{x+y}]}] if [\sqrt{x}] = 10 and [\sqrt{x}] =14

Source: 2003 Austria Beginners' Competition p1

October 4, 2022

algebrafloor function

Problem Statement

For the real numbers

x

and

y

[\sqrt{x}] = 10

and

[\sqrt{y}] =14

. How large is

\left[\sqrt{[ \sqrt{x+y} ]}\right]

?
(Note: the square roots are the positive values and

[x]

is the largest integer less than or equal to x.)