MathDB

3

Part of 1994 National High School Mathematics League

Problems(2)

Sum of Sequence

Source: 1994 National High School Mathematics League, Exam One, Problem 3

2/28/2020
Sequence (an)(a_n) satisfies that 3an+1+an=4(n≥1),a1=93a_{n+1}+a_n=4(n\geq1),a_1=9, let Sn=∑i=1naiS_n=\sum_{i=1}^{n}a_i, then the minumum value of nn such that ∣Sn−n−6∣<1125|S_n-n-6|<\frac{1}{125} is (A)5(B)6(C)7(D)8\text{(A)}5\qquad\text{(B)}6\qquad\text{(C)}7\qquad\text{(D)}8
Geometry

Source: 1994 National High School Mathematics League, Exam Two, Problem 3

3/2/2020
Circumcircle of △ABC\triangle ABC is ⊙O\odot O, incentre of △ABC\triangle ABC is II. ∠B=60∘.∠A<∠C\angle B=60^{\circ}.\angle A<\angle C. Bisector of outer angle ∠A\angle A intersects ⊙O\odot O at EE. Prove: (a) IO=AEIO=AE. (b) The radius of ⊙O\odot O is RR, then 2R<IO+IA+IC<(1+3)R2R<IO+IA+IC<(1+\sqrt3)R.
geometrycircumcircleincenter