MathDB

2

Part of 1988 National High School Mathematics League

Problems(2)

Ellipse Problem

Source: 1988 National High School Mathematics League, Exam One, Problem 2

2/25/2020
If the coordinate origin is inside the ellipse k2x2+y2−4kx+2ky+k2−1=0k^2x^2+y^2-4kx+2ky+k^2-1=0, then the range value of kk is (A)∣k∣>1(B)∣k∣≠1(C)−1<k<1(D)0<∣k∣<1\text{(A)}|k|>1\qquad\text{(B)}|k|\neq1\qquad\text{(C)}-1<k<1\qquad\text{(D)}0<|k|<1
conicsellipse
Three Equal Division

Source: 1988 National High School Mathematics League, Exam Two, Problem 2

2/25/2020
In △ABC\triangle ABC, P,Q,RP,Q,R divides the perimeter of △ABC\triangle ABC into three equal parts. P,Q∈ABP,Q\in AB. Prove that S△PQRS△ABC>29\frac{S_{\triangle PQR}}{S_{\triangle ABC}}>\frac{2}{9}.
geometryperimeter