MathDB

1

Part of 1984 National High School Mathematics League

Problems(2)

Complex Plane

Source: 1984 National High School Mathematics League, Exam One, Problem 1

2/22/2020
On complex plane, what figure does the set {Z‾2∣arg⁡Z=α,α∈[0,2π)}\{ \overline{Z}^2|\arg Z=\alpha,\alpha\in[0,2\pi)\} stands for? (A)\text{(A)} half-line arg⁡Z=2α\arg Z=2\alpha (B)\text{(B)} half-line arg⁡Z=−2α\arg Z=-2\alpha (C)\text{(C)} half-line arg⁡Z=α\arg Z=\alpha (D)\text{(D)} None above
True Or False?

Source: 1984 National High School Mathematics League, Exam Two, Problem 1

2/22/2020
Judge the following statements. If true, prove it; If false, give a counter-example. (a) P,QP,Q are two different points on the sime side of line ll. There exists two different circles, passing P,QP,Q, and tangent to line ll. (b) If a>0,b>0,a≠1,b≠1a>0,b>0,a\neq1,b\neq1, thenlog⁡ab+log⁡ba≥2\log_ab+\log_ba\geq2. (c) A,BA,B are two sets of points on coordinate plane. Cr={(x,y)∣x2+y2≤r2}C_r=\{(x,y)|x^2+y^2\leq r^2\}. For any r≥0r\geq0, Cr∪A⊂Cr∪BC_r\cup A\subset C_r\cup B, then A⊂BA\subset B.