Friday, January 12, 2007

2007, 12 and 30

Day Twelve

Difficulty: easy-ish. Does your student remember his trig identities?

Problem 30

Difficulty: Medium.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

answer 12 and answer 30.


  1. I don't know if my solution for 12 was standard: (s^2 + c^2) = 1
    so (s^4 + 2s^2c^2 + c^4) = 1
    so s^4 + c^4 = 1 - 2s^2c^2

    since sin2x = 2sinxcosx
    sin2x/2 = sinxcosx,
    and (sin2x)^2/4 = s^2c^2, which is 1/28. Double and subtract.

    More interesting for me (and easier) was 30, where I offer that 3, 5, and 7 must be included, in that order, so I ignore them.

    Instead, I will Select 2 numbers from the remaining 6 C(6,2) = 15 ways, x Select two places from the five in the number C(5,2) = 10 x And place the digits in the chosen spaces in two ways.


  2. #12: pretty standard. It's exactly this kind of algebraic manipulation that the test creators are famous for.

    #30. Rewording the question to your advantage, eh?

  3. Those "keep the digits in order" problems can often be reworded to our advantage.

    Funny, I scratched my head more for #12.