MathDB

9

Part of 2018 Harvard-MIT Mathematics Tournament

Problems(7)

2018 Algebra / NT #9

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2/12/2018
Assume the quartic x4−ax3+bx2−ax+d=0x^4-ax^3+bx^2-ax+d=0 has four real roots 12≤x1,x2,x3,x4≤2.\frac{1}{2}\leq x_1,x_2,x_3,x_4\leq 2. Find the maximum possible value of (x1+x2)(x1+x3)x4(x4+x2)(x4+x3)x1.\frac{(x_1+x_2)(x_1+x_3)x_4}{(x_4+x_2)(x_4+x_3)x_1}.
inequalitiesalgebra
2018 Combinatorics #9

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2/12/2018
How many ordered sequences of 3636 digits have the property that summing the digits to get a number and taking the last digit of the sum results in a digit which is not in our original sequence? (Digits range from 00 to 99.)
2018 Geometry #9

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2/12/2018
Po picks 100100 points P1,P2,⋯ ,P100P_1,P_2,\cdots, P_{100} on a circle independently and uniformly at random. He then draws the line segments connecting P1P2,P2P3,…,P100P1.P_1P_2,P_2P_3,\ldots,P_{100}P_1. Find the expected number of regions that have all sides bounded by straight lines.
geometry
2018 Team #9

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2/12/2018
Evan has a simple graph with vv vertices and ee edges. Show that he can delete at least e−v+12\frac{e-v+1}{2} edges so that each vertex still has at least half of its original degree.
2018 General #9

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11/12/2018
2020 players are playing in a Super Mario Smash Bros. Melee tournament. They are ranked 1−201-20, and player nn will always beat player mm if n<mn<m. Out of all possible tournaments where each player plays 1818 distinct other players exactly once, one is chosen uniformly at random. Find the expected number of pairs of players that win the same number of games.
probabilityexpected value
2018 Theme #9

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11/13/2018
Circle ω1\omega_1 of radius 11 and circle ω2\omega_2 of radius 22 are concentric. Godzilla inscribes square CASHCASH in ω1\omega_1 and regular pentagon MONEYMONEY in ω2\omega_2. It then writes down all 20 (not necessarily distinct) distances between a vertex of CASHCASH and a vertex of MONEYMONEY and multiplies them all together. What is the maximum possible value of his result?
geometry
2018 November Team #9

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11/13/2018
Let A,B,CA,B,C be points in that order along a line, such that AB=20AB=20 and BC=18BC=18. Let ω\omega be a circle of nonzero radius centered at BB, and let ℓ1\ell_1 and ℓ2\ell_2 be tangents to ω\omega through AA and CC, respectively. Let KK be the intersection of ℓ1\ell_1 and ℓ2\ell_2. Let XX lie on segment KA‾\overline{KA} and YY lie on segment KC‾\overline{KC} such that XY∥BCXY\|BC and XYXY is tangent to ω\omega. What is the largest possible integer length for XYXY?