MathDB

4

Part of 2017 Harvard-MIT Mathematics Tournament

Problems(7)

2017 Team #4: Palindromic substrings

Source:

2/19/2017
Let w=w1w2…wnw = w_1 w_2 \dots w_n be a word. Define a substring of ww to be a word of the form wiwi+1…wj−1wjw_i w_{i + 1} \dots w_{j - 1} w_j, for some pair of positive integers 1≤i≤j≤n1 \le i \le j \le n. Show that ww has at most nn distinct palindromic substrings.
For example, aaaaaaaaaa has 55 distinct palindromic substrings, and abcataabcata has 55 (aa, bb, cc, tt, ataata).
combinatorics
2017 Algebra/NT #4: ab divides a^2017+b

Source:

2/19/2017
Find all pairs (a,b)(a,b) of positive integers such that a2017+ba^{2017}+b is a multiple of abab.
number theory
2017 Geometry #4: Quadrilateral Finding MN^2-PQ^2

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2/19/2017
Let ABCDABCD be a convex quadrilateral with AB=5AB=5, BC=6BC=6, CD=7CD=7, and DA=8DA=8. Let MM, PP, NN, QQ be the midpoints of sides ABAB, BCBC, CDCD, DADA respectively. Compute MN2−PQ2MN^2-PQ^2.
geometry
2017 Combinatorics #4: Walking Around a grid

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2/19/2017
Sam spends his days walking around the following 2×22\times 2 grid of squares. \begin{tabular}[t]{|c|c|}\hline 1&2\\ \hline 4&3 \\ \hline \end{tabular} Say that two squares are adjacent if they share a side. He starts at the square labeled 11 and every second walks to an adjacent square. How many paths can Sam take so that the sum of the numbers on every square he visits in his path is equal to 2020 (not counting the square he started on)?
2017 Theme #4

Source:

5/8/2018
Mary has a sequence m2,m3,m4,...m_2,m_3,m_4,... , such that for each b≥2b \ge 2, mbm_b is the least positive integer m for which none of the base-bb logarithms logb(m),logb(m+1),...,logb(m+2017)log_b(m),log_b(m+1),...,log_b(m+2017) are integers. Find the largest number in her sequence.
algebra
2017 General #4

Source:

5/8/2018
Triangle ABCABC has AB=10AB=10, BC=17BC=17, and CA=21CA=21. Point PP lies on the circle with diameter ABAB. What is the greatest possible area of APCAPC?
geometry
2017 Guts #4: Diophantine equation

Source:

2/21/2017
Find the number of ordered triples of nonnegative integers (a,b,c)(a, b, c) that satisfy (ab+1)(bc+1)(ca+1)=84.(ab + 1)(bc + 1)(ca + 1) = 84.
number theoryDiophantine equation