MathDB

2

Part of 2019 Harvard-MIT Mathematics Tournament

Problems(4)

HMMT Team 2019/2: "Weird" bijections on N

Source:

2/17/2019
Let N={1,2,3,… }\mathbb{N} = \{1, 2, 3, \dots\} be the set of all positive integers, and let ff be a bijection from N\mathbb{N} to N\mathbb{N}. Must there exist some positive integer nn such that (f(1),f(2),…,f(n))(f(1), f(2), \dots, f(n)) is a permutation of (1,2,…,n)(1, 2, \dots, n)?
HMMTcombinatoricsPrinceton
HMMT Algebra/NT 2019/2: Simple exponent manipulation

Source:

2/17/2019
Let N=2(22)N = 2^{\left(2^2\right)} and xx be a real number such that N(NN)=2(2x)N^{\left(N^N\right)} = 2^{(2^x)}. Find xx.
HMMTalgebra
HMMT Combinatorics 2019/2: Tricky Dice

Source:

2/17/2019
Your math friend Steven rolls five fair icosahedral dice (each of which is labelled 1,2,…,201,2, \dots,20 on its sides). He conceals the results but tells you that at least half the rolls are 2020. Suspicious, you examine the first two dice and find that they show 2020 and 1919 in that order. Assuming that Steven is truthful, what is the probability that all three remaining concealed dice show 2020?
HMMTprobability
HMMT Geometry 2019/2: Simple rectangle geometry with equal areas

Source:

2/17/2019
In rectangle ABCDABCD, points EE and FF lie on sides ABAB and CDCD respectively such that both AFAF and CECE are perpendicular to diagonal BDBD. Given that BFBF and DEDE separate ABCDABCD into three polygons with equal area, and that EF=1EF = 1, find the length of BDBD.
HMMTgeometryrectangle