MathDB

2009 Stanford Mathematics Tournament

Part of Stanford Mathematics Tournament

Subcontests

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SMT 2009 Algebra 6

The dollar is now worth 1980\frac{1}{980} ounce of gold. After the nthn^{th} 7001 billion dollars bailout package passed by congress, the dollar gains 122n−1\frac{1}{2{}^2{}^{n-1}} of its (n−1)th(n-1)^{th} value in gold. After four bank bailouts, the dollar is worth 1b(1−12c)\frac{1}{b}(1-\frac{1}{2^c}) in gold, where b,cb, c are positive integers. Find b+cb + c.

SMT 2009 Geometry 6

Equilateral triangle ABCABC has side lengths of 2424. Points DD, EE, and FF lies on sides BCBC, CACA, ABAB such that AD⊥BC{AD}\perp{BC}, DE⊥AC{DE}\perp{AC}, and EF⊥AB{EF}\perp{AB}. GG is the intersection of ADAD and EFEF. Find the area of quadrilateral BFGDBFGD

SMT 2009 Team #6

Rhombus ABCDABCD has side length 1 1. The size of ∠A\angle A (in degrees) is randomly selected from all real numbers between 00 and 9090. Find the expected value of the area of ABCDABCD.
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SMT 2009 Algebra 2

King Midas spent 100x%\frac{100}{x}\% of his gold deposit yesterday. He is set to earn gold today. What percentage of the amount of gold King Midas currently has would he need to earn today to end up with as much gold as he started?

SMT 2009 Team 2

Factor completely the expression (a−b)3+(b−c)3+(c−a)3(a-b)^3+(b-c)^3+(c-a)^3

SMT 2009 Geometry 2

The pattern in the fi gure below continues inward in finitely. The base of the biggest triangle is 1. All triangles are equilateral. Find the shaded area.
[asy] defaultpen(linewidth(0.8)); pen blu = rgb(0,112,191); real r=sqrt(3); fill((8,0)--(0,8r)--(-8,0)--cycle, blu); fill(origin--(4,4r)--(-4,4r)--cycle, white); fill((2,2r)--(0,4r)--(-2,2r)--cycle, blu); fill((0,2r)--(1,3r)--(-1,3r)--cycle, white);[/asy]
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