MathDB

6

Part of 2018 Harvard-MIT Mathematics Tournament

Problems(7)

2018 Algebra / NT #6

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2/12/2018
Let α,β,\alpha,\beta, and γ\gamma be three real numbers. Suppose that cos⁡α+cos⁡β+cos⁡γ=1\cos\alpha+\cos\beta+\cos\gamma=1 sin⁡α+sin⁡β+sin⁡γ=1.\sin\alpha+\sin\beta+\sin\gamma=1. Find the smallest possible value of cos⁡α.\cos \alpha.
2018 Combinatorics #6

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2/12/2018
Sarah stands at (0,0)(0,0) and Rachel stands at (6,8)(6,8) in the Euclidena plane. Sarah can only move 11 unit in the positive xx or yy direction, and Rachel can only move 11 unit in the negative xx or yy direction. Each second, Sarah and Rachel see each other, independently pick a direction to move, and move to their new position. Sarah catches Rachel if Sarah and Rachel are every at the same point. Rachel wins if she is able to get (0,0)(0,0) without being caught; otherwise, Sarah wins. Given that both of them play optimally to maximize their probability of winning, what is the probability that Rachel wins?
2018 Geometry #6

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2/12/2018
Let ABCABC be an equilateral triangle of side length 1.1. For a real number 0<x<0.5,0<x<0.5, let A1A_1 and A2A_2 be the points on side BCBC such that A1B=A2C=x,A_1B=A_2C=x, and let TA=△AA1A2.T_A=\triangle AA_1A_2. Construct triangles TB=△BB1B2T_B=\triangle BB_1B_2 and TC=△CC1C2T_C=\triangle CC_1C_2 similarly.
There exist positive rational numbers b,cb,c such that the region of points inside all three triangles TA,TB,TCT_A,T_B,T_C is a hexagon with area 8x2−bx+c(2−x)(x+1)⋅34.\dfrac{8x^2-bx+c}{(2-x)(x+1)}\cdot \dfrac{\sqrt 3}{4}. Find (b,c).(b,c).
geometry
2018 Team #6

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2/12/2018
Let n≥2n \geq 2 be a positive integer. A subset of positive integers SS is said to be comprehensive if for every integer 0≤x<n0 \leq x < n, there is a subset of SS whose sum has remainder xx when divided by nn. Note that the empty set has sum 0. Show that if a set SS is comprehensive, then there is some (not necessarily proper) subset of SS with at most n−1n-1 elements which is also comprehensive.
2018 General #6

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11/12/2018
Call a polygon normal if it can be inscribed in a unit circle. How many non-congruent normal polygons are there such that the square of each side length is a positive integer?
HMMTgeometrycombinatorics
2018 Theme #6

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11/13/2018
Farmer James invents a new currency, such that for every positive integer n≤6n\le 6, there exists an nn-coin worth n!n! cents. Furthermore, he has exactly nn copies of each nn-coin. An integer kk is said to be nice if Farmer James can make kk cents using at least one copy of each type of coin. How many positive integers less than 2018 are nice?
combinatorics
2018 November Team #6

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11/13/2018
Triangle △PQR\triangle PQR, with PQ=PR=5PQ=PR=5 and QR=6QR=6, is inscribed in circle ω\omega. Compute the radius of the circle with center on QR‾\overline{QR} which is tangent to both ω\omega and PQ‾\overline{PQ}.
geometry