MathDB

7

Part of 2016 Harvard-MIT Mathematics Tournament

Problems(7)

2016 Algebra #7

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12/24/2016
Determine the smallest positive integer n≥3n \ge 3 for which A≡210n(mod2170) A \equiv 2^{10n} \pmod{2^{170}} where AA denotes the result when the numbers 2102^{10}, 2202^{20}, …\dots, 210n2^{10n} are written in decimal notation and concatenated (for example, if n=2n=2 we have A=10241048576A = 10241048576).
2016 Combo #7

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12/30/2016
Kelvin the Frog has a pair of standard fair 88-sided dice (each labelled from 11 to 88). Alex the sketchy Kat also has a pair of fair 88-sided dice, but whose faces are labelled differently (the integers on each Alex's dice need not be distinct). To Alex's dismay, when both Kelvin and Alex roll their dice, the probability that they get any given sum is equal!
Suppose that Alex's two dice have aa and bb total dots on them, respectively. Assuming that a≠ba \neq b, find all possible values of min⁡{a,b}\min \{a,b\}.
2016 Geo #7

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12/30/2016
For i=0,1,…,5i=0,1,\dots,5 let lil_i be the ray on the Cartesian plane starting at the origin, an angle θ=iπ3\theta=i\frac{\pi}{3} counterclockwise from the positive xx-axis. For each ii, point PiP_i is chosen uniformly at random from the intersection of lil_i with the unit disk. Consider the convex hull of the points PiP_i, which will (with probability 1) be a convex polygon with nn vertices for some nn. What is the expected value of nn?
geometry
2016 Guts #7

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12/24/2016
A contest has six problems worth seven points each. On any given problem, a contestant can score either 00, 11, or 77 points. How many possible total scores can a contestant achieve over all six problems?
2016 Team #7

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12/30/2016
Let q(x)=q1(x)=2x2+2x−1q(x) = q^1(x) = 2x^2 + 2x - 1, and let qn(x)=q(qn−1(x))q^n(x) = q(q^{n-1}(x)) for n>1n > 1. How many negative real roots does q2016(x)q^{2016}(x) have?
2016 General #7: Incircle in a 13-14-15 triangle

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11/15/2016
Let ABC be a triangle with AB=13,BC=14,CA=15AB = 13, BC = 14, CA = 15. The altitude from AA intersects BCBC at DD. Let ω1\omega_1 and ω2\omega_2 be the incircles of ABDABD and ACDACD, and let the common external tangent of ω1\omega_1 and ω2\omega_2 (other than BCBC) intersect ADAD at EE. Compute the length of AEAE.
geometry
2016 Theme #7: Lattice points

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11/22/2016
Seven lattice points form a convex heptagon with all sides having distinct lengths. Find the minimum possible value of the sum of the squares of the sides of the heptagon.
HMMT