2013 AIME I Problems/Problem 3

Problem 3

Let $ABCD$ be a square, and let $E$ and $F$ be points on $\overline{AB}$ and $\overline{BC},$ respectively. The line through $E$ parallel to $\overline{BC}$ and the line through $F$ parallel to $\overline{AB}$ divide $ABCD$ into two squares and two nonsquare rectangles. The sum of the areas of the two squares is $\frac{9}{10}$ of the area of square $ABCD.$ Find $\frac{AE}{EB} + \frac{EB}{AE}.$

Solution

It's important to note that $\dfrac{AE}{EB} + \dfrac{EB}{AE}$ is equivalent to $\dfrac{AE^2 + EB^2}{(AE)(EB)}$

We define $a$ as the length of the side of larger inner square, which is also $EB$ , $b$ as the length of the side of the smaller inner square which is also $AE$ , and $s$ as the side length of $ABCD$ . Since we are given that the sum of the areas of the two squares is $\frac{9}{10}$ of the the area of ABCD, we can represent that as $a^2 + b^2 = \frac{9s^2}{10}$ . The sum of the two nonsquare rectangles can then be represented as $2ab = \frac{s^2}{10}$ .

Looking back at what we need to find, we can represent $\dfrac{AE^2 + EB^2}{(AE)(EB)}$ as $\dfrac{a^2 + b^2}{ab}$ . We have the numerator, and dividing $\frac{s^2}{10}$ by two gives us the denominator $\frac{s^2}{20}$ . Dividing $\dfrac{\frac{9s^2}{10}}{\frac{s^2}{20}}$ gives us an answer of $018$ .

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2013 AIME I Problems/Problem 3

Problem 3

Solution