MathDB

10

Part of 2007 Harvard-MIT Mathematics Tournament

Problems(4)

2007 Algebra #10: 2007 Distinct Zeroes

Source:

6/21/2012
The polynomial f(x)=x2007+17x2006+1f(x)=x^{2007}+17x^{2006}+1 has distinct zeroes r1,…,r2007r_1,\ldots,r_{2007}. A polynomial PP of degree 20072007 has the property that P(rj+1rj)=0P\left(r_j+\dfrac{1}{r_j}\right)=0 for j=1,…,2007j=1,\ldots,2007. Determine the value of P(1)/P(−1)P(1)/P(-1).
algebrapolynomial
2007 Calculus #10: Integral from 0 to Infinity

Source:

6/21/2012
Compute ∫0∞e−xsin⁡(x)xdx\int_0^\infty \dfrac{e^{-x}\sin(x)}{x}dx
calculusintegration
2007 Geometry #10: Incircle of Quadrilateral

Source:

6/22/2012
ABCDABCD is a convex quadrilateral such that AB=2AB=2, BC=3BC=3, CD=7CD=7, and AD=6AD=6. It also has an incircle. Given that ∠ABC\angle ABC is right, determine the radius of this incircle.
geometry
2007 Guts #10: Difference of Base 10 Integers

Source:

6/22/2012
Let A12A_{12} denote the answer to problem 1212. There exists a unique triple of digits (B,C,D)(B,C,D) such that 10>A12>B>C>D>010>A_{12}>B>C>D>0 and A12BCD‾−DCBA12‾=BDA12C‾,\overline{A_{12}BCD}-\overline{DCBA_{12}}=\overline{BDA_{12}C}, where A12BCD‾\overline{A_{12}BCD} denotes the four digit base 1010 integer. Compute B+C+DB+C+D.