AIME 2015 Problem 7

$PQ$ is parallel to $BC$ , the longest side. Whatever percentage of the triangle’s base (i.e. $x \%$ ) is the width of the rectangle gets subtracted from the triangle’s height ( $h(100-x)\%$ ). The base sacrifices for the height and vice versa. The base of the triangle has length $BC = 25$ . The fractional part lost is thus $\frac{w}{25}$ . The ratio of the rectangles height to the triangle’s altitude is then $1-\frac{w}{25}$ . To get the height of the triangle, we need to know the area: Heron’s Formula $A=\sqrt{s(s-a)(s-b)(s-c)}$ . The semiperimeter $s$ is $\frac{12+25+17}{2} = \frac{54}{2} = 27$ . The area of $\triangle ABC$ is $\sqrt{27(27-12)(27-25)(27-17)} = \sqrt{27 \cdot 15 \cdot 2 \cdot 10} = \sqrt{8100} = 90$ . Since the base is $25$ , the height is $2 \cdot \frac{90}{25} = \frac{36}{5}$ . So the rectangle’s height is $\frac{36}{5} \cdot \left( 1-\frac{w}{25} \right) = \frac{36}{5} - \frac{36w}{125}$ . The are of the rectangle is the height multiplied by the width, or $\frac{36w}{5} - \frac{36w^{2}}{125}$ . The coefficient of $w^2$ is $\beta = \frac{36}{125}$ , and $36 + 125 = \boxed{161}$

$CosABC=\dfrac {12^2+25^2-17^2}{2*12*25}=\dfrac 4 5. \ \ \therefore\ SinABC=\dfrac 3 5.\\ PQ || BC\ \therefore\ \dfrac {PB}{AB}=\dfrac {25-w} {25}.\ \ \implies PB=AB*\dfrac {25-w} {25}.\\ Area\ \ PQRS=w*PS=w*PB*SinABC=w*(AB*\dfrac {25-w} {25})*\dfrac 3 5.\\ \therefore\ b=\dfrac{12}{25}*\dfrac 3 5=\dfrac{36}{125}\\ m+n=36+125=\Large \ \ \color{#D61F06}{161}$

By Heron's Formula , we have that the area of the triangle is

$\sqrt{(27)(2)(10)(15)}=90$

This means that the area of the triangle, if we treat the side of length 25 as the base, is

$25h=180\\h=\frac{36}{5}$

Segment $PQ$ , because it is parallel to the base of length $25$ , makes a triangle $\triangle PAQ$ which is similar to $\triangle ABC$ . Let $PQ=w$ , and the height of $\triangle PAQ=v$ .

$\frac{25}{w}=\frac{36}{5v}\\v=\frac{36w}{125}$

Since the height of $\triangle PAQ+$ height of the rectangle $= \frac{36}{5}$ , we know that the area of the rectangle is

$\frac{36w}{5}-\frac{36w^{2}}{125}$

And so our answer is $36+125=\boxed{161}$