MathDB

6

Part of 2019 Harvard-MIT Mathematics Tournament

Problems(4)

HMMT Team 2019/6: A real geometry problem

Source:

2/17/2019
Scalene triangle ABCABC satisfies ∠A=60∘\angle A = 60^{\circ}. Let the circumcenter of ABCABC be OO, the orthocenter be HH, and the incenter be II. Let DD, TT be the points where line BCBC intersects the internal and external angle bisectors of ∠A\angle A, respectively. Choose point XX on the circumcircle of △IHO\triangle IHO such that HX∥AIHX \parallel AI. Prove that OD⊥TXOD \perp TX.
HMMTgeometry
HMMT Algebra/NT 2019/6: (a√2 + b√3) mod √2 + (a√2 + b√3) mod √3 = √2

Source:

2/17/2019
For positive reals pp and qq, define the remainder when pp and qq as the smallest nonnegative real rr such that p−rq\tfrac{p-r}{q} is an integer. For an ordered pair (a,b)(a, b) of positive integers, let r1r_1 and r2r_2 be the remainder when a2+b3a\sqrt{2} + b\sqrt{3} is divided by 2\sqrt{2} and 3\sqrt{3} respectively. Find the number of pairs (a,b)(a, b) such that a,b≤20a, b \le 20 and r1+r2=2r_1 + r_2 = \sqrt{2}.
HMMTalgebra
HMMT Combinatorics 2019/6: Sequence of eight reflections preserving center

Source:

2/17/2019
A point PP lies at the center of square ABCDABCD. A sequence of points {Pn}\{P_n\} is determined by P0=PP_0 = P, and given point PiP_i, point Pi+1P_{i+1} is obtained by reflecting PiP_i over one of the four lines ABAB, BCBC, CDCD, DADA, chosen uniformly at random and independently for each ii. What is the probability that P8=PP_8 = P?
HMMTcombinatorics
HMMT Geometry 2019/6: This is not an olympiad

Source:

2/17/2019
Six unit disks C1C_1, C2C_2, C3C_3, C4C_4, C5C_5, C6C_6 are in the plane such that they don't intersect each other and CiC_i is tangent to Ci+1C_{i+1} for 1≤i≤61 \le i \le 6 (where C7=C1C_7 = C_1). Let CC be the smallest circle that contains all six disks. Let rr be the smallest possible radius of CC, and RR the largest possible radius. Find R−rR - r.
HMMTgeometry