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Problem 676

AMC 10/12, early questions

Algebra Difficulty 3.9 Multiple choice GARA di FEBBRAIO · Italy

Given two positive real numbers $a, b$ we define
$a \star b=\frac{a b+1}{a+b}$
What is the value of $1 \star(2 \star(3 \star(\cdots(2017 \star 2018))))$ ?

Pick one

Official solution

Solution:

The answer is $\mathbf{( B )}$ . One can observe that for any positive real number $x$ we have $1 \star x= \frac{1 \cdot x+1}{1+x}=1$ ; taking as $x$ the value $2 \star(3 \star(\cdots(2017 \star 2018)))$ one obtains that the answer to the problem is $1 \star x=1$ .

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