MathDB

10

Part of 2021 BMT

Problems(5)

BMT Discrete #10 - Graphs Without Triangles

Source:

11/29/2021
Let NN be the number of ways to draw 22 straight edges between 10 labeled points, of which no three are collinear, such that no triangle with vertices among these 10 points is created, and there is at most one edge between any two labeled points. Compute N9!\dfrac{N}{9!}.
graph theorycountingBmt
BMT 2021 Guts Round p10

Source:

10/7/2022
Compute the number of nonempty subsets SS of {1,2,3,4,5,6,7,8,9,10}\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} such that max⁡  S+min⁡  S2\frac{\max \,\, S + \min \,\,S}{2} is an element of SS.
combinatoricsalgebra
BMT 2021 Geometry #10

Source:

8/12/2023
Consider △ABC\vartriangle ABC such that CA+AB=3BCCA + AB = 3BC. Let the incircle ω\omega touch segments CA‾\overline{CA} and AB‾\overline{AB} at EE and FF, respectively, and define PP and QQ such that segments PE‾\overline{P E} and QF‾\overline{QF} are diameters of ω\omega. Define the function DD of a point KK to be the sum of the distances from KK to PP and QQ (i.e. D(K)=KP+KQD(K) = KP + KQ). Let W,X,YW, X, Y , and ZZ be points chosen on lines BC↔\overleftrightarrow {BC}, CE↔\overleftrightarrow {CE}, EF↔\overleftrightarrow {EF}, and FB↔\overleftrightarrow {F B}, respectively. Given that BC=133BC =\sqrt{133} and the inradius of △ABC\vartriangle ABC is 14\sqrt{14}, compute the minimum value of D(W)+D(X)+D(Y)+D(Z)D(W) + D(X) + D(Y ) + D(Z).
geometry
BMT 2021 General p10

Source:

9/27/2023
Triangle △ABC\vartriangle ABC has side lengths AB=AC=27AB = AC = 27 and BC=18BC = 18. Point DD is on AB‾\overline{AB} and point EE is on AC‾\overline{AC} such that ∠BCD=∠CBE=∠BAC\angle BCD = \angle CBE = \angle BAC. Compute DEDE.
geometry
2021 BMT Algebra #10

Source:

3/10/2024
Given a positive integer nn, define fn(x)f_n(x) to be the number of square-free positive integers kk such that kx≤nkx \le n. Then, define v(n)v_(n) as v(n)=∑i=1n∑j=1nfn(i2)−6fn(ij)+fn(j2).v(n) =\sum^n_{i=1}\sum^n_{j=1}f_n(i^2)- 6f_n (ij) + f_n(j^2). Compute the largest positive integer 2≤n≤1002 \le n \le 100 for which v(n)−v(n−1)v(n)-v(n-1) is negative. (Note: A square-free positive integer is a positive integer that is not divisible by the square of any prime.)
algebra