MathDB

9

Part of 2017 Harvard-MIT Mathematics Tournament

Problems(7)

2017 Team #9: Well-centered and decomposable polygons

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2/19/2017
Let nn be an odd positive integer greater than 22, and consider a regular nn-gon G\mathcal{G} in the plane centered at the origin. Let a subpolygon G′\mathcal{G}' be a polygon with at least 33 vertices whose vertex set is a subset of that of G\mathcal{G}. Say G′\mathcal{G}' is well-centered if its centroid is the origin. Also, say G′\mathcal{G}' is decomposable if its vertex set can be written as the disjoint union of regular polygons with at least 33 vertices. Show that all well-centered subpolygons are decomposable if and only if nn has at most two distinct prime divisors.
roots of unitynumber theory
2017 Algebra/NT #9: Fibonacci number divisible by 127

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2/19/2017
The Fibonacci sequence is defined as follows: F0=0F_0=0, F1=1F_1=1, and Fn=Fn−1+Fn−2F_n=F_{n-1}+F_{n-2} for all integers n≥2n\ge 2. Find the smallest positive integer mm such that Fm≡0(mod127)F_m\equiv 0 \pmod {127} and Fm+1≡1(mod127)F_{m+1}\equiv 1\pmod {127}.
modular arithmeticFibonacci sequencenumber theory
2017 Geometry #9: Squares on the sides of the triangle

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2/19/2017
Let ABCABC be a triangle, and let BCDEBCDE, CAFGCAFG, ABHIABHI be squares that do not overlap the triangle with centers XX, YY, ZZ respectively. Given that AX=6AX=6, BY=7BY=7, and CA=8CA=8, find the area of triangle XYZXYZ.
geometry
2017 Combinatorics #9: Good Sets

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2/20/2017
Let mm be a positive integer, and let TT denote the set of all subsets of {1,2,…,m}\{1, 2, \dots, m\}. Call a subset SS of TT δ\delta-[I]good[/I] if for all s1,s2∈Ss_1, s_2\in S, s1≠s2s_1\neq s_2, ∣Δ(s1,s2)∣≥δm|\Delta (s_1, s_2)|\ge \delta m, where Δ\Delta denotes the symmetric difference (the symmetric difference of two sets is the set of elements that is in exactly one of the two sets). Find the largest possible integer ss such that there exists an integer mm and 10242047 \frac{1024}{2047}-good set of size ss.
symmetry
2017 Theme #9

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5/8/2018
New this year at HMNT: the exciting game of RNG baseball! In RNG baseball, a team of infinitely many people play on a square field, with a base at each vertex; in particular, one of the bases is called the home base. Every turn, a new player stands at home base and chooses a number n uniformly at random from {0,1,2,3,4}\{0, 1, 2, 3, 4\}. Then, the following occurs: • If n>0n>0, then the player and everyone else currently on the field moves (counterclockwise) around the square by n bases. However, if in doing so a player returns to or moves past the home base, he/she leaves the field immediately and the team scores one point. • If n=0n=0 (a strikeout), then the game ends immediately; the team does not score any more points. What is the expected number of points that a given team will score in this game?
combinatorics
2017 General #9

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5/8/2018
Find the minimum value of 58−42x+149−1401−x2\sqrt{58-42x}+\sqrt{149-140\sqrt{1-x^2}} where −1≤x≤1-1 \le x \le 1.
algebra
2017 Guts #9: Weird process

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2/21/2017
Jeffrey writes the numbers 11 and 100000000=108100000000 = 10^8 on the blackboard. Every minute, if x,yx, y are on the board, Jeffery replaces them with \frac{x + y}{2}   \text{and}   2 \left(\frac{1}{x} + \frac{1}{y}\right)^{-1}. After 20172017 minutes the two numbers are aa and bb. Find min⁡(a,b)\min(a, b) to the nearest integer.
algebra