MathDB

8

Part of 2019 Harvard-MIT Mathematics Tournament

Problems(4)

HMMT Team 2019/8: Variant on old USA TST

Source:

2/17/2019
Can the set of lattice points {(x,y)∣x,y∈Z,1≤x,y≤252,x≠y}\{(x, y) \mid x, y \in \mathbb{Z}, 1 \le x, y \le 252, x \neq y\} be colored using 10 distinct colors such that for all a≠ba \neq b, b≠cb \neq c, the colors of (a,b)(a, b) and (b,c)(b, c) are distinct?
HMMTcombinatoricsgraph theory
HMMT Algebra/NT 2019/8: Dirichlet square root of all-ones

Source:

2/17/2019
There is a unique function f:N→Rf: \mathbb{N} \to \mathbb{R} such that f(1)>0f(1) > 0 and such that ∑d∣nf(d)f(nd)=1\sum_{d \mid n} f(d) f\left(\frac{n}{d}\right) = 1 for all n≥1n \ge 1. What is f(20182019)f(2018^{2019})?
HMMTalgebranumber theory
HMMT Combinatorics 2019/8: a, b, gcd(a, b) different colors if all different

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2/17/2019
For a positive integer NN, we color the positive divisors of NN (including 1 and NN) with four colors. A coloring is called multichromatic if whenever aa, bb and gcd⁡(a,b)\gcd(a, b) are pairwise distinct divisors of NN, then they have pairwise distinct colors. What is the maximum possible number of multichromatic colorings a positive integer can have if it is not the power of any prime?
HMMTcombinatorics
HMMT Geometry 2019/8: Nine-point center on BC

Source:

2/17/2019
In triangle ABCABC with AB<ACAB < AC, let HH be the orthocenter and OO be the circumcenter. Given that the midpoint of OHOH lies on BCBC, BC=1BC = 1, and the perimeter of ABCABC is 6, find the area of ABCABC.
HMMTgeometry