MathDB

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Part of 2018 Harvard-MIT Mathematics Tournament

Problems(7)

2018 Algebra / NT #1

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2/12/2018
For some real number c,c, the graphs of the equation y=∣x−20∣+∣x+18∣y=|x-20|+|x+18| and the line y=x+cy=x+c intersect at exactly one point. What is cc?
2018 Combinatorics #1

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2/12/2018
Consider a 2×32\times 3 grid where each entry is either 00, 11, or 22. For how many such grids is the sum of the numbers in every row and in every column a multiple of 33? One valid grid is shown below: [120210]\begin{bmatrix} 1 & 2 & 0 \\ 2 & 1 & 0 \end{bmatrix}
2018 Geometry #1

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2/12/2018
Triangle GRTGRT has GR=5,GR=5, RT=12,RT=12, and GT=13.GT=13. The perpendicular bisector of GTGT intersects the extension of GRGR at O.O. Find TO.TO.
geometryperpendicular bisector
2018 Team #1

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2/12/2018
In an n×nn \times n square array of 1×11\times1 cells, at least one cell is colored pink. Show that you can always divide the square into rectangles along cell borders such that each rectangle contains exactly one pink cell.
geometryrectangleteam
2018 General #1

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11/12/2018
What is the largest factor of 130000130000 that does not contain the digit 00 or 55?
HMMTnumbe
2018 Theme #1

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11/13/2018
Square CASHCASH and regular pentagon MONEYMONEY are both inscribed in a circle. Given that they do not share a vertex, how many intersections do these two polygons have?
geometry
2018 November Team #1

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11/13/2018
Four standard six-sided dice are rolled. Find the probability that, for each pair of dice, the product of the two numbers rolled on those dice is a multiple of 4.
probability