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Problem 637

AIME late

Algebra Difficulty 5.1 Find the answer

G2.2 Let $[x]$ be the largest integer not greater than $x$ .
If $B=[10+\sqrt{10+\sqrt{10+\sqrt{10+\cdots}}}]$ , find the value of $B$ .

Your answer

Official solution

Reference: $\mathbf{2 0 0 7}$ FG2.2 $\ldots x \geq 3+\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3}}}}} \ldots$
Let $y=\sqrt{10+\sqrt{10+\sqrt{10+\cdots}}}$
$\begin{array}{l} y^{2}=10+\sqrt{10+\sqrt{10+\sqrt{10+\cdots}}}=10+y \\ y^{2}-y-10=0 \\ y=\frac{1+\sqrt{41}}{2} \text { or } \frac{1-\sqrt{41}}{2} \quad \text { (rejected) } \\ 6<\sqrt{41}<7 \Rightarrow \frac{7}{2}<\frac{1+\sqrt{41}}{2}<4 \\ 13.5<10+\sqrt{10+\sqrt{10+\sqrt{10+\cdots}}}<14 ; B=13 \end{array}$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.