MathDB

9

Part of 2016 Harvard-MIT Mathematics Tournament

Problems(7)

2016 Algebra #9

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12/24/2016
For any positive integer nn, SnS_{n} be the set of all permutations of {1,2,3,…,n}\{1,2,3,\dots,n\}. For each permutation π∈Sn\pi \in S_n, let f(π)f(\pi) be the number of ordered pairs (j,k)(j,k) for which π(j)>π(k)\pi(j)>\pi(k) and 1≤j<k≤n1\leq j<k \leq n. Further define g(π)g(\pi) to be the number of positive integers k≤nk \leq n such that π(k)≡k±1(modn)\pi(k)\equiv k \pm 1 \pmod{n}. Compute ∑π∈S999(−1)f(π)+g(π). \sum_{\pi \in S_{999}} (-1)^{f(\pi)+g(\pi)}.
linear algebraHMMT
2016 Combo #9

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12/30/2016
Let V={1,…,8}V = \left\{ 1, \dots, 8 \right\}. How many permutations σ:V→V\sigma : V \to V are automorphisms of some tree?
(A \emph{graph} consists of some set of vertices and some edges between pairs of distinct vertices. It is \emph{connected} if every two vertices in it are connected by some path of one or more edges. A \emph{tree} GG on VV is a connected graph with vertex set VV and exactly ∣V∣−1|V|-1 edges, and an \emph{automorphism} of GG is a permutation σ:V→V\sigma : V \to V such that vertices i,j∈Vi,j \in V are connected by an edge if and only if σ(i)\sigma(i) and σ(j)\sigma(j) are.)
2016 Geo #9

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12/30/2016
The incircle of a triangle ABCABC is tangent to BCBC at DD. Let HH and Γ\Gamma denote the orthocenter and circumcircle of △ABC\triangle ABC. The \emph{BB-mixtilinear incircle}, centered at OBO_B, is tangent to lines BABA and BCBC and internally tangent to Γ\Gamma. The \emph{CC-mixtilinear incircle}, centered at OCO_C, is defined similarly. Suppose that DH‾⊥OBOC‾\overline{DH} \perp \overline{O_BO_C}, AB=3AB = \sqrt3 and AC=2AC = 2. Find BCBC.
2016 Guts #9

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12/24/2016
Victor has a drawer with two red socks, two green socks, two blue socks, two magenta socks, two lavender socks, two neon socks, two mauve socks, two wisteria socks, and 20002000 copper socks, for a total of 20162016 socks. He repeatedly draws two socks at a time from the drawer at random, and stops if the socks are of the same color. However, Victor is red-green colorblind, so he also stops if he sees a red and green sock.
What is the probability that Victor stops with two socks of the same color? Assume Victor returns both socks to the drawer at each step.
2016 Team #9

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12/30/2016
Fix positive integers r>sr>s, and let FF be an infinite family of sets, each of size rr, no two of which share fewer than ss elements. Prove that there exists a set of size r−1r-1 that shares at least ss elements with each set in FF.
2016 General #9: Tetrations

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11/15/2016
Let the sequence aia_i be defined as ai+1=2aia_{i+1} = 2^{a_i}. Find the number of integers 1≤n≤10001 \le n \le 1000 such that if a0=na_0 = n, then 100100 divides a1000−a1a_{1000} - a_1.
HMMT
2016 Theme #9: Coloring a nonagon's points

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11/22/2016
The vertices of a regular nonagon are colored such that 1)1) adjacent vertices are different colors and 2)2) if 33 vertices form an equilateral triangle, they are all different colors. Let mm be the minimum number of colors needed for a valid coloring, and n be the total number of colorings using mm colors. Determine mnmn. (Assume each vertex is distinguishable.)
HMMT