MathDB

2

Part of 1999 National High School Mathematics League

Problems(2)

Integral Points

Source: 1999 National High School Mathematics League, Exam One, Problem 2

3/9/2020
The number of intengral points (x,y)(x,y) that fit (∣x∣−1)2+(∣y∣−1)2<2(|x|-1)^2+(|y|-1)^2<2 is (A)16(B)17(C)18(D)25\text{(A)}16\qquad\text{(B)}17\qquad\text{(C)}18\qquad\text{(D)}25
calculusintegration
Complex Number

Source: 1999 National High School Mathematics League, Exam Two, Problem 2

3/10/2020
Let a,b,ca,b,c be real numbers, z1,z2,z3z_{1},z_{2},z_{3} be complex numbers such that {∣z1∣=∣z2∣=∣z3∣=1z1z2+z2z3+z3z1=1\begin{cases} \displaystyle|z_1|=|z_2|=|z_3|=1\\ \displaystyle\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{3}}+\frac{z_{3}}{z_{1}}=1\\ \end{cases} Find ∣az1+bz2+cz3∣|az_{1}+bz_{2}+cz_{3}|.
complex numbers