MathDB

13

Part of 2011 AIME Problems

Problems(2)

Suspended cube

Source:

3/18/2011
A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labelled AA. The three vertices adjacent to vertex AA are at heights 10, 11, and 12 above the plane. The distance from vertex AA to the plane can be expressed as r−st\tfrac{r-\sqrt{s}}{t}, where rr, ss, and tt are positive integers, and r+s+t<1000r+s+t<1000. Find r+s+tr+s+t.
geometry3D geometryvectoranalytic geometrygraphing linessloperatio
Points in a square [2011.II.13]

Source:

3/31/2011
Point PP lies on the diagonal ACAC of square ABCDABCD with AP>CPAP>CP. Let O1O_1 and O2O_2 be the circumcenters of triangles ABPABP and CDPCDP respectively. Given that AB=12AB=12 and ∠O1PO2=120∘\angle O_1 P O_2 = 120^\circ, then AP=a+bAP=\sqrt{a}+\sqrt{b} where aa and bb are positive integers. Find a+ba+b.
geometrycircumcircletrigonometrysymmetryinequalitiestrig identitiesLaw of Sines