MathDB

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Part of 2007 Harvard-MIT Mathematics Tournament

Problems(4)

2007 Algebra #1: Floor of Factorials

Source:

6/21/2012
Compute ⌊2007!+2004!2006!+2005!⌋.\left\lfloor \dfrac{2007!+2004!}{2006!+2005!}\right\rfloor. (Note that ⌊x⌋\lfloor x \rfloor denotes the greatest integer less than or equal to xx.)
factorialfloor function
2007 Geometry #1: A Cube and its Properties

Source:

6/22/2012
A cube of edge length s>0s>0 has the property that its surface area is equal to the sum of its volume and five times its edge length. Compute all possible values of ss.
geometry3D geometry
2007 Calculus #1: Computation of a Limit

Source:

6/21/2012
Compute: lim⁡x→0 x21−cos⁡(x)\lim_{x\to 0}\text{ }\dfrac{x^2}{1-\cos(x)}
calculuslimit
2007 Guts #1: Recursive Sequence

Source:

6/22/2012
Define the sequence of positive integers ana_n recursively by a1=7a_1=7 and an=7an−1a_n=7^{a_{n-1}} for all n≥2n\geq 2. Determine the last two digits of a2007a_{2007}.