MathDB

3

Part of 2014 Harvard-MIT Mathematics Tournament

Problems(5)

2014 HMMT #3: Power of logarithmic expression

Source:

2/23/2014
Let A=16((log⁡2(3))3−(log⁡2(6))3−(log⁡2(12))3+(log⁡2(24))3) A = \frac{1}{6}((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3) .
Compute 2A2^A.
HMMTlogarithms
2014 Combinatorics #3: Strings of As

Source:

2/23/2014
Bob writes a random string of 55 letters, where each letter is either A,B,C,A, B, C, or DD. The letter in each position is independently chosen, and each of the letters A,B,C,DA, B, C, D is chosen with equal probability. Given that there are at least two A′sA's in the string, find the probability that there are at least three A′sA's in the string.
probability
2014 Geometry #3: External Angle Bisector

Source:

2/25/2014
ABCABC is a triangle such that BC=10BC = 10, CA=12CA = 12. Let MM be the midpoint of side ACAC. Given that BMBM is parallel to the external bisector of ∠A\angle A, find area of triangle ABCABC. (Lines ABAB and ACAC form two angles, one of which is ∠BAC\angle BAC. The external angle bisector of ∠A\angle A is the line that bisects the other angle.
geometryangle bisectorarea of a triangleHeron's formula
2014 Guts #3: Hexagon, Circle, and Rectangle

Source:

2/25/2014
[4] Let ABCDEFABCDEF be a regular hexagon. Let PP be the circle inscribed in △BDF\triangle{BDF}. Find the ratio of the area of circle PP to the area of rectangle ABDEABDE.
geometryrectangleratio
2014 Team #3: Kinda Sorta the Marriage Lemma?

Source:

3/2/2014
There are nn girls G1,…,GnG_1,\ldots, G_n and nn boys B1,…,BnB_1,\ldots,B_n. A pair (Gi,Bj)(G_i,B_j) is called <spanclass=′latex−italic′>suitable</span><span class='latex-italic'>suitable</span> if and only if girl GiG_i is willing to marry boy BjB_j. Given that there is exactly one way to pair each girl with a distinct boy that she is willing to marry, what is the maximal possible number of suitable pairs?