MathDB

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Part of 2014 Harvard-MIT Mathematics Tournament

Problems(5)

2014 Algebra #1: x and y

Source:

2/23/2014
Given that xx and yy are nonzero real numbers such that x+1y=10x+\frac{1}{y}=10 and y+1x=512y+\frac{1}{x}=\frac{5}{12}, find all possible values of xx.
quadratics
2014 Combinatorics #1: Probability Same Class

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2/23/2014
There are 100100 students who want to sign up for the class Introduction to Acting. There are three class sections for Introduction to Acting, each of which will fit exactly 2020 students. The 100100 students, including Alex and Zhu, are put in a lottery, and 60 of them are randomly selected to fill up the classes. What is the probability that Alex and Zhu end up getting into the same section for the class?
probability
2014 HMMT Geometry #1:

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2/24/2014
Let O1O_1 and O2O_2 be concentric circles with radii 4 and 6, respectively. A chord ABAB is drawn in O1O_1 with length 22. Extend ABAB to intersect O2O_2 in points CC and DD. Find CDCD.
HMMTgeometryanalytic geometrysymmetryPythagorean Theorempower of a point
2014 Guts #1: prime factorisation

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2/25/2014
[4] Compute the prime factorisation of 159999159999.
2014 Team #1: Circle in the Interior

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3/2/2014
Let ω\omega be a circle, and let AA and BB be two points in its interior. Prove that there exists a circle passing through AA and BB that is contained in the interior of ω\omega.