MathDB

3

Part of 2003 National High School Mathematics League

Problems(2)

Parabola

Source: 2003 National High School Mathematics League, Exam One, Problem 3

3/16/2020
Line passes the focal point FF of parabola y2=8(x+2)y^2=8(x+2) with bank angle of 60∘60^{\circ} intersects the parabola at A,BA,B. Perpendicular bisector of ABAB intersects xx-axis at PP, then the length of PFPF is (A)163(B)83(C)1633(D)83\text{(A)}\frac{16}{3}\qquad\text{(B)}\frac{8}{3}\qquad\text{(C)}\frac{16}{3}\sqrt3\qquad\text{(D)}8\sqrt3
conicsparabolageometryperpendicular bisector
Graph Theory

Source: 2003 National High School Mathematics League, Exam Two, Problem 3

3/16/2020
A space figure is consisted of nn vertexes and ll lines connecting these vertices, where n=q2+q+1,l≥12q(q+1)2+1,q≥2,q∈Z+n=q^2+q+1, l\geq\frac{1}{2}q(q+1)^2+1,q\geq2,q\in\mathbb{Z}_+. The figure satisfies: every four vertices are not coplane, every vertex is connected by at least one line, and there is a vertex connected by at least q+2q+2 lines. Prove that there exists a space quadrilateral in the figure. Note: a space quadrilateral is figure with four vertices A,B,C,DA, B, C, D and four lines AB,BC,CD,DA AB, BC, CD, DA.
graph theory