MathDB

2

Part of 2009 Harvard-MIT Mathematics Tournament

Problems(4)

2009 Algebra #2 - Sum of Coefficients of Expansion

Source:

1/2/2012
Let SS be the sum of all the real coefficients of the expansion of (1+ix)2009(1+ix)^{2009}. What is log⁡2(S)\log_2(S)?
logarithmsHMMTcomplex numbers
2009 Calculus #2: Nested Sines and Cosines

Source:

6/23/2012
The differentiable function F:R→RF:\mathbb{R}\to\mathbb{R} satisfies F(0)=−1F(0)=-1 and ddxF(x)=sin⁡(sin⁡(sin⁡(sin⁡(x))))⋅cos⁡(sin⁡(sin⁡(x)))⋅cos⁡(sin⁡(x))⋅cos⁡(x).\dfrac{d}{dx}F(x)=\sin (\sin (\sin (\sin(x))))\cdot \cos( \sin (\sin (x))) \cdot \cos (\sin(x))\cdot\cos(x). Find F(x)F(x) as a function of xx.
calculustrigonometryfunctionintegration
2009 Combinatorics #2 - Cards between two Jokers in deck

Source:

1/7/2012
Two jokers are added to a 5252 card deck and the entire stack of 5454 cards is shuffled randomly. What is the expected number of cards that will be strictly between the two jokers?
probabilityexpected value
2009 General Part 1 #2: 6 Digit Integers

Source:

6/23/2012
Suppose N is a 66-digit number having base-1010 representation a‾ b‾ c‾ d‾ e‾ f‾\underline{a}\text{ }\underline{b}\text{ }\underline{c}\text{ }\underline{d}\text{ }\underline{e}\text{ }\underline{f}. If NN is 6/76/7 of the number having base-1010 representation d‾ e‾ f‾ a‾ b‾ c‾\underline{d}\text{ }\underline{e}\text{ }\underline{f}\text{ }\underline{a}\text{ }\underline{b}\text{ }\underline{c}, find NN.
algorithmnumber theoryEuclidean algorithmgreatest common divisor