Maths Olympiad Prep

Library / /57 of 520

Algebra Difficulty 4.7 AIME Find the answer

3. If [a][a] denotes the integer part of the real number aa, then [116−67]\left[\frac{1}{\sqrt{16-6 \sqrt{7}}}\right] equals:

Pick one

Solution

3. B.
∵16−67=4+7−263=(9−7)2,∴116−67=19−7=3+72. \begin{array}{l} \because 16-6 \sqrt{7}=4+7-2 \sqrt{63} \\ \quad=(\sqrt{9}-\sqrt{7})^{2}, \\ \therefore \frac{1}{\sqrt{16-6 \sqrt{7}}}=\frac{1}{\sqrt{9}-\sqrt{7}}=\frac{3+\sqrt{7}}{2} . \end{array}

And 2<7<32<\sqrt{7}<3,
∴52<3+72<3.  \therefore \frac{5}{2}<\frac{3+\sqrt{7}}{2}<3 \text {. }

Thus [116−67]=2\left[\frac{1}{\sqrt{16-6 \sqrt{7}}}\right]=2.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

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