MathDB

P2

Part of 2014 BMT Spring

Problems(3)

BMT 2014 Spring - Geometry P2

Source:

12/29/2021
Let ABCABC be a fixed scalene triangle. Suppose that X,YX, Y are variable points on segments ABAB, ACAC, respectively such that BX=CYBX = CY . Prove that the circumcircle of △AXY\vartriangle AXY passes through a fixed point other than AA.
geometry
BMT 2014 Spring - Analysis P2

Source:

1/6/2022
Define η(f)\eta(f) to be the number of roots that are repeated of the complex-valued polynomial ff (e.g. η((x−1)3⋅(x+1)4)=5\eta((x-1)^3\cdot(x+1)^4)=5). Prove that for nonconstant, relatively prime f,g∈C[x]f,g\in\mathbb C[x], η(f)+η(g)+η(f+g)<deg⁡f+deg⁡g\eta(f)+\eta(g)+\eta(f+g)<\deg f+\deg g
Polynomials
BMT 2014 Spring - Discrete P2

Source:

1/6/2022
Given an integer n≥2n\ge2, the graph GG is defined by:
- Vertices of GG are represented by binary strings of length nn - Two vertices a,ba,b are connected by an edge if and only if they differ in exactly 22 places
Let SS be a subset of the vertices of GG, and let S′S' be the set of edges between vertices in SS and vertices not in SS. Show that if ∣S∣|S| (the size of SS) ≤2n−2\le2^{n-2}, then ∣S′∣≥∣S∣|S'|\ge|S|.
combinatoricsgraph theory