MathDB

2

Part of 1991 National High School Mathematics League

Problems(2)

Complex Number

Source: 1991 National High School Mathematics League, Exam One, Problem 2

2/26/2020
a,b,ca,b,c are three non-zero-complex numbers, and ab=bc=ca\frac{a}{b}=\frac{b}{c}=\frac{c}{a}, then the value of a+b−ca−b+c\frac{a+b-c}{a-b+c} is (ω=−12+32i\omega=-\frac{1}{2}+\frac{\sqrt3}{2}\text{i}) (A)1(B)±ω(C)1,ω,ω2(D)1,−ω,−ω2\text{(A)}1\qquad\text{(B)}\pm\omega\qquad\text{(C)}1,\omega,\omega^2\qquad\text{(D)}1,-\omega,-\omega^2
complex numbers
Convex Quadrilateral

Source: 1991 National High School Mathematics League, Exam Two, Problem 2

2/26/2020
Area of convex quadrilateral ABCDABCD is 11. Prove that we can find four points on its side (vertex included) or inside, satisfying: area of triangles comprised of any three points of the four points is larger than 14\frac{1}{4}.
geometry