1986 AIME Problem 2

Attempt Problem 2 of the 1986 AIME below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1986 AIME solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

2.

Evaluate the product

\begin{gathered} (\sqrt5+\sqrt6+\sqrt7)\\ {}\cdot(-\sqrt5+\sqrt6+\sqrt7)\\ {}\cdot(\sqrt5-\sqrt6+\sqrt7)\\ {}\cdot(\sqrt5+\sqrt6-\sqrt7). \end{gathered}

Answer

Answer: 104

Concepts:algebraic manipulation difference of squares radical

Difficulty rating: 1890

Small hint:

Pair factors so that each pair is a difference of squares

Big hint:

After the first pairing, the remaining radicals occur only through

\sqrt{42}

Solution:

Pair the first two factors, then the last two:

\begin{gathered} (\sqrt6+\sqrt7)^2-(\sqrt5)^2\\ {}=8+2\sqrt{42},\\ (\sqrt5)^2-(\sqrt7-\sqrt6)^2\\ {}=-8+2\sqrt{42}. \end{gathered}

Therefore the requested product is

\begin{aligned} &(8+2\sqrt{42})(-8+2\sqrt{42})\\ &\qquad=168-64=104. \end{aligned}