MathDB

2

Part of 2018 Harvard-MIT Mathematics Tournament

Problems(7)

2018 Algebra / NT #2

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2/12/2018
Compute the positive real number xx satisfying x(2x6)=3.x^{(2x^6)}=3.
2018 Combinatorics #2

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2/12/2018
Let a<ba<b be five-digit palindromes (without leading zeroes) such that there is no other five-digit palindrome strictly between aa and bb. What are all possible values of b−ab-a? (A number is a palindrome if it reads the same forwards and backwards in base 1010.)
2018 Geometry #2

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2/12/2018
Points A,B,C,DA,B,C,D are chosen in the plane such that segments AB,BC,CD,DAAB,BC,CD,DA have lengths 2,7,5,12,2,7,5,12, respectively. Let mm be the minimum possible value of the length of segment ACAC and let MM be the maximum possible value of the length of segment AC.AC. What is the ordered pair (m,M)(m,M)?
geometry
2018 General #2

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11/12/2018
Twenty-seven players are randomly split into three teams of nine. Given that Zack is on a different team from Mihir and Mihir is on a different team from Andrew, what is the probability that Zack and Andrew are on the same team?
probability
2018 Team #2

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2/12/2018
Is the number (1+12)(1+14)(1+16)⋯(1+12018) \left( 1 + \frac12 \right) \left( 1 + \frac14 \right) \left( 1 + \frac16 \right)\cdots\left( 1 + \frac{1}{2018} \right) greater than, less than, or equal to 5050?
2018 Theme #2

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11/16/2018
Consider the addition problem: \begin{tabular}{ccccc} &C&A&S&H\\ +&&&M&E\\ \hline O&S&I&D&E \end{tabular} where each letter represents a base-ten digit, and C,M,O≠0.C,M,O \ne 0. (Distinct letters are allowed to represent the same digit.) How many ways are there to assign values to the letters so that the addition problem is true?
algebra
2018 November Team #2

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11/13/2018
Alice starts with the number 0. She can apply 100 operations on her number. In each operation, she can either add 1 to her number, or square her number. After applying all operations, her score is the minimum distance from her number to any perfect square. What is the maximum score she can attain?