MathDB

2

Part of 1986 National High School Mathematics League

Problems(2)

Complex Number Problem

Source: 1986 National High School Mathematics League, Exam One, Problem 2

2/24/2020
Set M={z∈C∣(z−1)2=∣z−1∣2}M=\{z\in\mathbb{C}|(z-1)^2=|z-1|^2\}, then (A)M={pure imaginary number}\text{(A)}M=\{\text{pure imaginary number}\} (B)M={real number}\text{(B)}M=\{\text{real number}\} (C)M={real number}⊂M⊂{complex number}\text{(C)}M=\{\text{real number}\}\subset M\subset\{\text{complex number}\} (D)M={complex number}\text{(D)}M=\{\text{complex number}\}
complex numbers
Area of the Triangle

Source: 1986 National High School Mathematics League, Exam Two, Problem 2

2/24/2020
In acute triangle ABCABC, D∈BC,E∈CA,F∈ABD\in BC,E\in CA,F\in AB. Prove that the necessary and sufficient condition of AD,BE,CFAD,BE,CF are heights of △ABC\triangle ABC is that S=R2(EF+FD+DE)S=\frac{R}{2}(EF+FD+DE). Note: SS is the area of △ABC\triangle ABC, RR is the circumradius of △ABC\triangle ABC.
geometrycircumcircle