MathDB

7

Part of 2009 Harvard-MIT Mathematics Tournament

Problems(4)

2009 Algebra #7 - Find Product

Source:

1/2/2012
Simplify the product ∏m=1100∏n=1100xn+m+xn+m+2+x2n+1+x2m+1x2n+2xn+m+x2m \prod_{m=1}^{100}\prod_{n=1}^{100}\frac{x^{n+m}+x^{n+m+2}+x^{2n+1}+x^{2m+1}}{x^{2n}+2x^{n+m}+x^{2m}} Express your answer in terms of xx.
2009 Calculus #7: Strange Lines

Source:

6/23/2012
A line in the plane is called strange if it passes through (a,0)(a,0) and (0,10−a)(0,10-a) for some aa in the interval [0,10][0,10]. A point in the plane is called charming if it lies in the first quadrant and also lies below some strange line. What is the area of the set of all charming points?
calculusgeometry
2009 Combinatorics #7 - Filling out a 7x7 Grid

Source:

1/7/2012
Paul fi lls in a 7×77\times7 grid with the numbers 11 through 4949 in a random arrangement. He then erases his work and does the same thing again, to obtain two diff erent random arrangements of the numbers in the grid. What is the expected number of pairs of numbers that occur in either the same row as each other or the same column as each other in both of the two arrangements?
probabilityexpected value
2009 Geometry #7: Feet of Perpendiculars

Source:

6/23/2012
In triangle ABCABC, DD is the midpoint of BCBC, EE is the foot of the perpendicular from AA to BCBC, and FF is the foot of the perpendicular from DD to ACAC. Given that BE=5BE=5, EC=9EC=9, and the area of triangle ABCABC is 8484, compute ∣EF∣|EF|.
geometrysimilar triangles