MathDB

6

Part of 2000 Harvard-MIT Mathematics Tournament

Problems(7)

2000 Advanced Topics #6: Exactly One Returns

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6/21/2012
66 people each have a hat. If they shuffle their hats and redistribute them, what is the probability that exactly one person gets their own hat back?
probability
2000 Guts #6: Diophantine equation

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10/12/2014
If integers m,n,km,n,k satisfy m2+n2+1=kmnm^2+n^2+1=kmn, what values can kk have?
algebrapolynomialVieta
2000 Algebra #6: Days to write Test

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10/5/2014
Barbara, Edward, Abhinav, and Alex took turns writing this test. Working alone, they could finish it in 1010, 99, 1111, and 1212 days, respectively. If only one person works on the test per day, and nobody works on it unless everyone else has spent at least as many days working on it, how many days (an integer) did it take to write this test?
2000 HMMT RMT Geometry # 6

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3/3/2024
What is the area of the largest circle contained in an equilateral triangle of area 838\sqrt3?
geometry
2000 Oral #6: Sum of four cubes

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10/6/2014
Prove that every multiple of 33 can be written as a sum of four cubes (positive or negatives).
geometry3D geometry
2000 HMMT RMT Team #6

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3/8/2024
If aa is a root of x3−x−1=0x^3-x-1 = 0, compute the value of a10+2a8−a7−3a6−3a5+4a4+2a3−4a4−6a−17.a^{10 }+ 2a^8 -a^7 - 3a^6 - 3a^5 + 4a^4 + 2a^3 - 4a^4 - 6a - 17.
algebra
2000 HMMT RMT General #6

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3/8/2024
Three cards, only one of which is an ace, are placed face down on a table. You select one, but do not look at it. The dealer turns over one of the other cards, which is not the ace (if neither are, he picks one of them randomly to turn over). You get a chance to change your choice and pick either of the remaining two face-down cards. If you selected the cards so as to maximize the chance of finding the ace on the second try, what is the probability that you selected it on the (a) first try? (b) second try?
combinatorics