MathDB

6

Part of 2006 Harvard-MIT Mathematics Tournament

Problems(4)

2006 Algebra #6: Vieta's (Funky) Expressions

Source:

7/30/2012
Let a,b,ca,b,c be the roots of x3−9x2+11x−1=0x^3-9x^2+11x-1=0, and let s=a+b+cs=\sqrt{a}+\sqrt{b}+\sqrt{c}. Find s4−18s2−8ss^4-18s^2-8s.
Vieta
2006 Calculus #6: A Triangle's Revolution

Source:

7/30/2012
A triangle with vertices at (1003,0)(1003,0), (1004,3)(1004,3), and (1005,1)(1005,1) in the xyxy-plane is revolved all the way around the yy-axis. Find the volume of the solid thus obtained.
calculus
2006 Combinatorics #6: Product Divisible by 20

Source:

7/31/2012
For how many ordered triplets (a,b,c)(a,b,c) of positive integers less than 1010 is the product a×b×ca\times b\times c divisible by 2020?
2006 Geometry #6: Trigonometric Inradius

Source:

7/31/2012
A circle of radius tt is tangent to the hypotenuse, the incircle, and one leg of an isosceles right triangle with inradius r=1+sin⁡π8r=1+\sin \frac{\pi}{8}. Find rtrt.
geometryinradiustrigonometry