MathDB

1

Part of 1999 National High School Mathematics League

Problems(2)

Geometric Series

Source: 1999 National High School Mathematics League, Exam One, Problem 1

3/9/2020
Give a geometric series (an)(a_n) with common ratio of qq, let b1=a1+a2+a3,b2=a4+a5+a6,⋯ ,bn=a3n+a3n+1+a3n+2b_1=a_1+a_2+a_3,b_2=a_4+a_5+a_6,\cdots,b_n=a_{3n}+a_{3n+1}+a_{3n+2}, then sequence (bn)(b_n) (A)\text{(A)} is an arithmetic sequence (B)\text{(B)} is a geometric series with common ratio of qq (C)\text{(C)} is a geometric series with common ratio of q3q^3 (D)\text{(D)} is neither an arithmetic sequence nor a geometric series
geometric seriesratioarithmetic sequence
Easy Geometry

Source: 1999 National High School Mathematics League, Exam Two, Problem 1

3/10/2020
In convex quadrilateral ABCDABCD, ∠BAC=∠CAD\angle BAC=\angle CAD. EE lies on segment CDCD, BEBE and ACAC intersect at F,F, DFDF and BCBC intersect at G.G. Prove that ∠GAC=∠EAC\angle GAC=\angle EAC.
geometry