MathDB

7

Part of 2007 Harvard-MIT Mathematics Tournament

Problems(4)

2007 Algebra #7: 2007th Term of Recursion

Source:

6/21/2012
An infinite sequence of positive real numbers is defined by a0=1a_0=1 and an+2=6an−an+1a_{n+2}=6a_n-a_{n+1} for n=0,1,2,⋯n=0,1,2,\cdots. Find the possible value(s) of a2007a_{2007}.
2007 Calculus #7: Infinite Sum of Reciprocals

Source:

6/21/2012
Compute ∑n=1∞1n⋅(n+1)⋅(n+1)!.\sum_{n=1}^\infty \dfrac{1}{n\cdot(n+1)\cdot(n+1)!}.
calculus
2007 Geometry #7: Convex Quadrilateral

Source:

6/22/2012
Convex quadrilateral ABCDABCD has sides AB=BC=7AB=BC=7, CD=5CD=5, and AD=3AD=3. Given additionally that m∠ABC=60∘m\angle ABC=60^\circ, find BDBD.
geometrytrigonometrytrig identitiesLaw of Cosines
2007 Guts #7: Kevin Miscounts

Source:

6/22/2012
A student at Harvard named Kevin Was counting his stones by 1111 He messed up nn times And instead counted 99s And wound up at 20072007.
How many values of nn could make this limerick true?
Harvardcollegemodular arithmetic