MathDB

8

Part of 2008 Harvard-MIT Mathematics Tournament

Problems(5)

Arctan Expression

Source:

3/2/2008
Compute \arctan\left(\tan65^\circ \minus{} 2\tan40^\circ\right). (Express your answer in degrees.)
trigonometry
Integral of Exponent "Rational" Function

Source: HMMT 2008 Calculus Problem 8

3/2/2008
Let T \equal{} \int_0^{\ln2} \frac {2e^{3x} \plus{} e^{2x} \minus{} 1} {e^{3x} \plus{} e^{2x} \minus{} e^x \plus{} 1}dx. Evaluate eT e^T.
calculusintegrationfunctionlogarithmscalculus computations
Sequence of Sets

Source:

3/3/2008
Determine the number of ways to select a sequence of 8 8 sets A1,A2,…,A8 A_1,A_2,\ldots,A_8, such that each is a subset (possibly empty) of {1,2} \{1,2\} and Am A_m contains An A_n if m m divides n n.
Equilateral Triangle Path

Source:

3/9/2008
Let ABC ABC be an equilateral triangle with side length 2, and let Γ \Gamma be a circle with radius 12 \frac {1}{2} centered at the center of the equilateral triangle. Determine the length of the shortest path that starts somewhere on Γ \Gamma, visits all three sides of ABC ABC, and ends somewhere on Γ \Gamma (not necessarily at the starting point). Express your answer in the form of \sqrt p \minus{} q, where p p and q q are rational numbers written as reduced fractions.
inequalitiesgeometrygeometric transformationreflectiontriangle inequality
"Trodgor" the Burninator

Source:

3/23/2008
Trodgor the dragon is burning down a village consisting of 90 90 cottages. At time t \equal{} 0 an angry peasant arises from each cottage, and every 8 8 minutes (480 480 seconds) thereafter another angry peasant spontaneously generates from each non-burned cottage. It takes Trodgor 5 5 seconds to either burn a peasant or to burn a cottage, but Trodgor cannot begin burning cottages until all the peasants around him have been burned. How many seconds does it take Trodgor to burn down the entire village?
LaTeX