AIME II 2009 #3

[stupidityismygam]

In rectangle [unparseable or potentially dangerous latex formula] Let [unparseable or potentially dangerous latex formula] be the midpoint of [unparseable or potentially dangerous latex formula] Given that line [unparseable or potentially dangerous latex formula] and line [unparseable or potentially dangerous latex formula] are perpendicular, find the greatest integer less than [unparseable or potentially dangerous latex formula]

[jahu]

umm well, i dont know how to draw it on here =( so cant really explain. is the answer 141?

well i drew the rectangle the way the problem stated (i labeled AE and ED x), then i drew an extra line connecting point E to point C. it makes 5 small right triangles within the rectangle and i just labeled the sides of them a, b, c, d, and e. with no order. set up a lot of pythagorean formulas and substituted, getting the final equation x^2 = 5000. then x = sqrt(5000). AD is 2x so AD is 2sqrt(5000) which is sqrt(20000) and the closest square under it is 141^2 (19881).

thus the answer is 141?

[stupidityismygam]

That is right. Here's my solution.

Let [unparseable or potentially dangerous latex formula] Let [unparseable or potentially dangerous latex formula] be the intersection of [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] Notice that [unparseable or potentially dangerous latex formula] So, [unparseable or potentially dangerous latex formula]

From Pythagorean Theorem we have that [unparseable or potentially dangerous latex formula] Also, since [unparseable or potentially dangerous latex formula] is a right triangle, we have that [unparseable or potentially dangerous latex formula] Substituting in values, [unparseable or potentially dangerous latex formula] So, [unparseable or potentially dangerous latex formula]