MathDB

2

Part of 1995 National High School Mathematics League

Problems(2)

Complex Plane

Source: 1995 National High School Mathematics League, Exam One, Problem 2

3/2/2020
Complex numbers of apexes of 20-regular polygon inscribed to unit circle refer to are Z1,Z2,⋯ ,Z20Z_1,Z_2,\cdots,Z_{20} on complex plane. Then the number of points in Z11995,Z21995,⋯ ,Z201995Z_1^{1995},Z_2^{1995},\cdots,Z_{20}^{1995} refer to is (A)4(B)5(C)10(D)20\text{(A)}4\qquad\text{(B)}5\qquad\text{(C)}10\qquad\text{(D)}20
complex numbers
Three Roots

Source: 1995 National High School Mathematics League, Exam Two, Problem 2

3/3/2020
Find all real number pp, such that the three roots of the equation 5x3−5(p+1)x2+(71p−1)x+1=66p5x^3-5(p+1)x^2+(71p-1)x+1=66p are all positive integers.
number theory