MathDB

1

Part of 1996 National High School Mathematics League

Problems(2)

Common Points

Source: 1996 National High School Mathematics League, Exam One, Problem 1

3/3/2020
Connect the commom points of circlex2+(y−1)2=1x^2+(y-1)^2=1 and ellipse 9x2+(y+1)2=99x^2+(y+1)^2=9 with line segments, the figure is a (A)\text{(A)} line segment (B)\text{(B)} scalene triangle (C)\text{(C)} equilateral triangle (D)\text{(D)} quadrilateral
conicsellipse
Two Sequences

Source: 1996 National High School Mathematics League, Exam Two, Problem 1

3/4/2020
The sum of first nn items of squence (an)(a_n) : SnS_n satisfies that Sn=2an−1S_n=2a_n-1, squence (bn)(b_n) satisfies that bk+1=ak+bkb_{k+1}=a_k+b_k for all k=1,2,⋯k=1,2,\cdots. Find the sum of first nn items of (bn)(b_n).