MathDB

6

Part of 2016 Harvard-MIT Mathematics Tournament

Problems(7)

2016 Combo #6

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12/30/2016
Define the sequence a1,a2…a_1, a_2 \dots as follows: a1=1a_1=1 and for every n≥2n\ge 2, an={n−2if an−1=0an−1−1if an−1≠0 a_n = \begin{cases} n-2 & \text{if } a_{n-1} =0 \\ a_{n-1} -1 & \text{if } a_{n-1} \neq 0 \end{cases} A non-negative integer dd is said to be {\em jet-lagged} if there are non-negative integers r,sr,s and a positive integer nn such that d=r+sd=r+s and that an+r=an+sa_{n+r} = a_n +s. How many integers in {1,2,…,2016}\{1,2,\dots, 2016\} are jet-lagged?
2016 Algebra #6

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12/24/2016
Call a positive integer N≥2N \ge 2 ``special'' if for every kk such that 2≤k≤N2 \leq k \leq N, NN can be expressed as a sum of kk positive integers that are relatively prime to NN (although not necessarily relatively prime to each other). How many special integers are there less than 100100?
2016 Geo #6

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12/30/2016
Let ABCABC be a triangle with incenter II, incircle γ\gamma and circumcircle Γ\Gamma. Let MM, NN, PP be the midpoints of sides BC‾\overline{BC}, CA‾\overline{CA}, AB‾\overline{AB} and let EE, FF be the tangency points of γ\gamma with CA‾\overline{CA} and AB‾\overline{AB}, respectively. Let UU, VV be the intersections of line EFEF with line MNMN and line MPMP, respectively, and let XX be the midpoint of arc BAC^\widehat{BAC} of Γ\Gamma. Given that AB=5AB = 5, AC=8AC = 8, and ∠A=60∘\angle A = 60^{\circ}, compute the area of triangle XUVXUV.
2016 Guts #6

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12/24/2016
Consider a 2×n2 \times n grid of points and a path consisting of 2n−12n-1 straight line segments connecting all these 2n2n points, starting from the bottom left corner and ending at the upper right corner. Such a path is called <spanclass=′latex−italic′>efficient</span><span class='latex-italic'>efficient</span> if each point is only passed through once and no two line segments intersect. How many efficient paths are there when n=2016n = 2016?
2016 Team #6

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12/30/2016
A nonempty set SS is called \emph{well-filled} if for every m∈Sm \in S, there are fewer than 12m\frac 12 m elements of SS which are less than mm. Determine the number of well-filled subsets of {1,2,…,42}\left\{ 1,2,\dots,42 \right\}.
2016 General #6: Arranging Numbers

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11/15/2016
The numbers 1,2…111, 2\ldots11 are arranged in a line from left to right in a random order. It is observed that the middle number is larger than exactly one number to its left. Find the probability that it is larger than exactly one number to its right.
HMMT
2016 Theme #6: Complex points

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11/22/2016
Let P1,P2,…,P6P_1, P_2, \ldots, P_6 be points in the complex plane, which are also roots of the equation x6+6x3−216=0x^6+6x^3-216=0. Given that P1P2P3P4P5P6P_1P_2P_3P_4P_5P_6 is a convex hexagon, determine the area of this hexagon.
HMMTgeometry