MathDB

9

Part of 2005 AIME Problems

Problems(2)

Aime Problem for Rubix Cubers

Source:

11/11/2005
Twenty seven unit cubes are painted orange on a set of four faces so that two non-painted faces share an edge. The 2727 cubes are randomly arranged to form a 3×3×33\times 3 \times 3 cube. Given the probability of the entire surface area of the larger cube is orange is paqbrc,\frac{p^a}{q^br^c}, where pp,qq, and rr are distinct primes and aa,bb, and cc are positive integers, find a+b+c+p+q+ra+b+c+p+q+r.
AMCAIMEgeometry3D geometryprobabilityrotation
Working with i

Source:

11/13/2005
For how many positive integers nn less than or equal to 10001000 is (sin⁡t+icos⁡t)n=sin⁡nt+icos⁡nt(\sin t + i \cos t)^n=\sin nt + i \cos nt true for all real tt?
trigonometryfloor functionmodular arithmeticcomplex numbers