MathDB

3

Part of 2009 Harvard-MIT Mathematics Tournament

Problems(5)

2009 Algebra #3 - Sum of Tangents

Source:

1/2/2012
If tan⁡x+tan⁡y=4\tan x + \tan y = 4 and cot⁡x+cot⁡y=5\cot x + \cot y = 5, compute tan⁡(x+y)\tan(x + y).
trigonometryAMCAIME
2009 Calculus #3: Rational Function Integral

Source:

6/23/2012
Compute eAe^A where AA is defined as ∫3/44/32x2+x+1x3+x2+x+1dx.\int_{3/4}^{4/3}\dfrac{2x^2+x+1}{x^3+x^2+x+1}dx.
calculusfunctionintegrationlogarithms
2009 Combinatorics #3 - Rearrangements with Restrictions

Source:

1/7/2012
How many rearrangements of the letters of "HMMTHMMTHMMTHMMT" do not contain the substring "HMMTHMMT"? (For instance, one such arrangement is HMMHMTMTHMMHMTMT.)
HMMT
2009 Geometry #1: Folding a Piece of Paper

Source:

6/23/2012
A rectangular piece of paper with side lengths 5 by 8 is folded along the dashed lines shown below, so that the folded flaps just touch at the corners as shown by the dotted lines. Find the area of the resulting trapezoid.
[asy] size(150); defaultpen(linewidth(0.8)); draw(origin--(8,0)--(8,5)--(0,5)--cycle,linewidth(1)); draw(origin--(8/3,5)^^(16/3,5)--(8,0),linetype("4 4")); draw(origin--(4,3)--(8,0)^^(8/3,5)--(4,3)--(16/3,5),linetype("0 4")); label("55",(0,5/2),W); label("88",(4,0),S); [/asy]
geometrygeometric transformationreflectiontrapezoidPythagorean Theorem
2009 Geometry #3: Awesome Points

Source:

6/23/2012
Let TT be a right triangle with sides having lengths 33, 44, and 55. A point PP is called awesome if P is the center of a parallelogram whose vertices all lie on the boundary of TT. What is the area of the set of awesome points?
geometryparallelogram