MathDB

3

Part of 1993 National High School Mathematics League

Problems(2)

The Number of Sets of Sets

Source: 1993 National High School Mathematics League, Exam One, Problem 3

2/27/2020
Sets A,BA,B satisfy that A∪B={a1,a2,a3}A\cup B=\{a_1,a_2,a_3\}. If A≠BA\neq B, then (A,B)(A,B) is different from (B,A)(B,A). The number of such sets (A,B)(A,B) is (A)8(B)9(C)26(D)27\text{(A)}8\qquad\text{(B)}9\qquad\text{(C)}26\qquad\text{(D)}27
Geometry

Source: 1993 National High School Mathematics League, Exam Two, Problem 3

2/28/2020
Horizontal line mm passes the center of circle ⊙O\odot O. Line l⊥ml\perp m, ll and mm intersect at MM, and MM is on the right side of OO. Three points A,B,CA,B,C (BB is in the middle) lie on line ll, which are outside the circle, above line mm. AP,BQ,CRAP,BQ,CR are tangent to ⊙O\odot O at P,Q,RP,Q,R. Prove: (a) If ll is tangent to ⊙O\odot O, then AB⋅CR+BC⋅AP=AC⋅BQAB\cdot CR+BC\cdot AP=AC\cdot BQ. (b) If ll and ⊙O\odot O intersect, then AB⋅CR+BC⋅AP<AC⋅BQAB\cdot CR+BC\cdot AP<AC\cdot BQ. (c) If ll and ⊙O\odot O are apart, then AB⋅CR+BC⋅AP>AC⋅BQAB\cdot CR+BC\cdot AP>AC\cdot BQ.
geometry