1960 AMC 12 Problem 9

Attempt Problem 9 of the 1960 AMC 12 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 1960 AMC 12 solutions, or check the answer key.

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The fraction

\frac{a^2+b^2-c^2+2ab}{a^2+c^2-b^2+2ac}

is (with suitable restrictions on the values of

a,

b,

and

c

irreducible

reducible to

-1

reducible to a polynomial of three terms

reducible to

\dfrac{a-b+c}{a+b-c}

reducible to

\dfrac{a+b-c}{a-b+c}

Rewrite each quadratic expression as a difference of squares

The numerator and denominator share a factor

a+b+c

Factoring gives

\begin{gathered} a^2+b^2-c^2+2ab\\ {}=(a+b-c)(a+b+c),\\ a^2+c^2-b^2+2ac\\ {}=(a-b+c)(a+b+c). \end{gathered}

Cancelling the common factor, under the stated suitable restrictions, leaves

\frac{a+b-c}{a-b+c}.

Thus, the correct answer is E.

Problem 9 in other years