MathDB

10

Part of 2021 Harvard-MIT Mathematics Tournament.

Problems(4)

2021 Algebra/NT #10: Set's natural density

Source:

5/30/2021
Let SS be a set of positive integers satisfying the following two conditions: • For each positive integer nn, at least one of n,2n,…,100nn, 2n, \dots, 100n is in SS. • If a1,a2,b1,b2a_1, a_2, b_1, b_2 are positive integers such that gcd⁡(a1a2,b1b2)=1\gcd(a_1a_2, b_1b_2) = 1 and a1b1,a2b2∈S,a_1b_1, a_2b_2 \in S, then a2b1,a1b2∈S.a_2b_1, a_1b_2 \in S. Suppose that SS has natural density rr. Compute the minimum possible value of ⌊105r⌋\lfloor 10^5r\rfloor. Note: SS has natural density rr if 1n∣S∩1,…,n∣\tfrac{1}{n}|S \cap {1, \dots, n}| approaches rr as nn approaches ∞\infty.
algebranumber theory
2021 Combo #10: Dependent chance of coin

Source:

5/30/2021
Jude repeatedly flips a coin. If he has already flipped nn heads, the coin lands heads with probability 1n+2\tfrac{1}{n+2} and tails with probability n+1n+2.\tfrac{n+1}{n+2}. If Jude continues flipping forever, let pp be the probability that he flips 33 heads in a row at some point. Compute ⌊180p⌋.\lfloor 180p \rfloor.
Combo
2021 Geo #10: radius of circumcircle

Source:

5/30/2021
Acute triangle ABCABC has circumcircle Γ\Gamma. Let MM be the midpoint of BC.BC. Points PP and QQ lie on Γ\Gamma so that ∠APM=90∘\angle APM = 90^{\circ} and Q≠AQ \neq A lies on line AM.AM. Segments PQPQ and BCBC intersect at SS. Suppose that BS=1,CS=3,PQ=8737,BS = 1, CS = 3, PQ = 8\sqrt{\tfrac{7}{37}}, and the radius of Γ\Gamma is rr. If the sum of all possible values of r2r^2 can be expressed as ab\tfrac ab for relatively prime positive integers aa and b,b, compute 100a+b100a + b.
geometrycircumcircle
2021Team #10

Source:

6/27/2021
Let n>1n>1 be a positive integer. Each unit square in an n×nn\times n grid of squares is colored either black or white, such that the following conditions hold:
∙\bullet Any two black squares can be connected by a sequence of black squares where every two consecutive squares in the sequence share an edge; ∙\bullet Any two white squares can be connected by a sequence of white squares where every two consecutive squares in the sequence share an edge; ∙\bullet Any 2×22\times 2 subgrid contains at least one square of each color.
Determine, with proof, the maximum possible difference between the number of black squares and white squares in this grid (in terms of nn).
Chessboardcombinatorics