MathDB

3

Part of 1983 National High School Mathematics League

Problems(2)

Rational Number Or Not?

Source: 1983 National High School Mathematics League, Exam One, Problem 3

2/21/2020
In triangle ABCABC, AB=ACAB=AC, ∣BC∣|BC| and d(A,BC)d(A,BC) are both integers. Then, sin⁡A\sin A and cos⁡A\cos A (A)\text{(A)} one is a rational number while the other is not (B)\text{(B)} both are rational numbers (C)\text{(C)} neither is rational number (D)\text{(D)} not sure
So many midpoints!

Source: 1983 National High School Mathematics League, Exam Two, Problem 3

2/22/2020
In quadrilateral ABCDABCD, S△ABD:S△BCD:S△ABC=3:4:1S_{\triangle ABD}:S_{\triangle BCD}:S_{\triangle ABC}=3:4:1. M∈AC,N∈CDM\in AC,N\in CD, satisfying that AMAC=CNCD\frac{AM}{AC}=\frac{CN}{CD}. If B,M,NB,M,N are collinear, prove that M,NM,N are mid points of AC,CDAC,CD.
geometry