MathDB

4

Part of 2007 Harvard-MIT Mathematics Tournament

Problems(4)

2007 Algebra #4: Telescoping Product

Source:

6/21/2012
Compute 23−123+1⋅33−133+1⋅43−143+1⋅53−153+1⋅63−163+1.\dfrac{2^3-1}{2^3+1}\cdot\dfrac{3^3-1}{3^3+1}\cdot\dfrac{4^3-1}{4^3+1}\cdot\dfrac{5^3-1}{5^3+1}\cdot\dfrac{6^3-1}{6^3+1}.
2007 Calculus #4: Tangent Parabola and Exponential Curves

Source:

6/21/2012
Find the real number α\alpha such that the curve f(x)=exf(x)=e^x is tangent to the curve g(x)=αx2g(x)=\alpha x^2.
calculusconicsparabolafunction
2007 Geometry #4: Tangents to a Circle

Source:

6/22/2012
Circle ω\omega has radius 55 and is centered at OO. Point AA lies outside ω\omega such that OA=13OA=13. The two tangents to ω\omega passing through AA are drawn, and points BB and CC are chosen on them (one on each tangent), such that line BCBC is tangent to ω\omega and ω\omega lies outside triangle ABCABC. Compute AB+ACAB+AC given that BC=7BC=7.
geometry
2007 Guts #4: A Sequence of Digits

Source:

6/22/2012
A sequence consists of the digits 122333444455555…122333444455555\ldots such that the each positive integer nn is repeated nn times, in increasing order. Find the sum of the 45014501st and 40524052nd digits of this sequence.