MathDB

4

Part of 1995 National High School Mathematics League

Problems(2)

Roots of the Equation

Source: 1995 National High School Mathematics League, Exam One, Problem 4

3/2/2020
Equation ∣x−2n∣=kx(n∈Z+)|x-2n|=k\sqrt{x}(n\in\mathbb{Z}_+) has two different real roots on (2n−1,2n+1](2n-1,2n+1], then the range value of kk is (A)k>0(B)0<k≤12n+1(C)12n+1<k≤12n+1\text{(A)}k>0\qquad\text{(B)}0<k\leq\frac{1}{\sqrt{2n+1}}\qquad\text{(C)}\frac{1}{2n+1}<k\leq\frac{1}{\sqrt{2n+1}} (D)\text{(D)} none above
Color all points

Source: 1995 National High School Mathematics League, Exam Two, Problem 4

3/3/2020
Color all points on a plane in red or blue. Prove that there exists two similar triangles, their similarity ratio is 19951995, and apexes of both triangles are in the same color.
geometrysimilar trianglesratio