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These shapes have four sides and 360° of interior angle goodness.

Suppose \(ABCD\) is an isosceles trapezoid with bases \(AB\) and \(CD\) and sides \(AD\) and \(BC\) such that \(|CD| \gt |AB|.\) Also suppose that \(|CD| = |AC|\) and that the altitude of the trapezoid is equal to \(|AB|.\)

If \(\dfrac{|AB|}{|CD|} = \dfrac{a}{b},\) where \(a\) and \(b\) are positive coprime integers, then find \(\large a^{b}.\)

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What is the area of the yellow region?

**Details and Assumptions**

The numbers \(3,4,\) and \(6\) denote the area enclosed by their respective triangles.

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On square \(ABCD\), points \(E,F,G\), and \(H\) lie on sides \(\overline{AB},\overline{BC},\overline{CD},\) and \(\overline{DA},\) respectively, so that \(\overline{EG} \perp \overline{FH}\) and \(EG=FH = 34\). Segments \(\overline{EG}\) and \(\overline{FH}\) intersect at a point \(P\), and the areas of the quadrilaterals \(AEPH, BFPE, CGPF,\) and \(DHPG\) are in the ratio \(269:275:405:411.\) Find the area of square \(ABCD\).

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I want to make a litter box for my newly adopted pet kitten Admiral. The box is a cuboid with the top removed. I want the volume of the box to be 32 but the surface area to be minimized.

What is the minimum internal surface area I can achieve?

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\( ABCD \) is a parallelogram. Point \(P\) on \(AB\) divides it in the ratio \( AP : PB = 3 : 2 \), and point \(Q\) on \(CD\) divides it in the ratio \( CQ : QD = 7 : 3 \). Let \(R\) be the intersection of \( PQ \) and \(AC\). Then, \( AR : AC = a : b \), where \(a\) and \(b\) are positive coprime integers. What is \(a + b \)?

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