MathDB

8

Part of 2013 Stanford Mathematics Tournament

Problems(6)

2013 General Problem 8

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2/4/2013
According to Moor's Law, the number of shoes in Moor's room doubles every year. In 2013, Moor's room starts out having exactly one pair of shoes. If shoes always come in unique, matching pairs, what is the earliest year when Moor has the ability to wear at least 500 mismatches pairs of shoes? Note that left and right shoes are distinct, and Moor must always wear one of each.
logarithms
SMT 2013 Calculus #8

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2/3/2013
The function f(x)f(x) is defined for all x≥0x\ge 0 and is always nonnegative. It has the additional property that if any line is drawn from the origin with any positive slope mm, it intersects the graph y=f(x)y=f(x) at precisely one point, which is 1m\frac{1}{\sqrt{m}} units from the origin. Let aa be the unique real number for which ff takes on its maximum value at x=ax=a (you may assume that such an aa exists). Find ∫0af(x) dx\int_{0}^{a}f(x) \, dx.
calculusfunctionanalytic geometrygraphing linesslopeintegration
2013 SMT Geometry #8

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11/2/2014
Let equilateral triangle ABCABC with side length 66 be inscribed in a circle and let PP be on arc ACAC such that AP⋅PC=10AP \cdot P C = 10. Find the length of BPBP.
geometry
SMT 2013 Algebra #8

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2/3/2013
Find the sum of all real xx such that 4x2+15x+17x2+4x+12=5x2+16x+182x2+5x+13.\frac{4x^2 + 15x + 17}{x^2 + 4x + 12}=\frac{5x^2 + 16x + 18}{2x^2 + 5x + 13}.
SMT 2013 Team #8

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2/4/2013
Rational Man and Irrational Man both buy new cars, and they decide to drive around two racetracks from time t=0t=0 to time t=∞t=\infty. Rational Man drives along the path parametrized by \begin{align*}x&=\cos(t)\\y&=\sin(t)\end{align*} and Irrational Man drives along the path parametrized by \begin{align*}x&=1+4\cos\frac{t}{\sqrt{2}}\\ y&=2\sin\frac{t}{\sqrt{2}}.\end{align*} Find the largest real number dd such that at any time tt, the distance between Rational Man and Irrational Man is not less than dd.
trigonometry
SMT 2013 Advanced Topics #8

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12/31/2016
Farmer John owns 2013 cows. Some cows are enemies of each other, and Farmer John wishes to divide them into as few groups as possible such that each cow has at most 3 enemies in her group. Each cow has at most 61 enemies. Compute the smallest integer GG such that, no matter which enemies they have, the cows can always be divided into at most GG such groups?
2013Advanced Topics