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True Or False?

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

February 22, 2020

Problem Statement

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.
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