MathDB

10.1

Part of VMEO IV 2015

Problems(2)

rationals wanted, a\alpha^2 + b\alpha+ c = \sqrt{d} if \alpha^3 = \alpha + 1

Source: 2015 VMEO IV Juniors 10.1 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2463156_2015_vmeo_iv

9/15/2021
Given a real number α\alpha satisfying α3=α+1\alpha^3 = \alpha + 1. Determine all 44-tuples of rational numbers (a,b,c,d)(a, b, c, d) satisfying: aα2+bα+c=d.a\alpha^2 + b\alpha+ c = \sqrt{d}.
algebrarational
limit of S_n where a_{n+1}=\frac{1}{a_1 + a_2 +... + a_n} -\sqrt2

Source: 2015 VMEO IV Seniors 10.1 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2463156_2015_vmeo_iv

9/19/2021
Where nn is a positive integer, the sequence ana_n is determined by the formula an+1=1a1+a2+...+an−2, a1=1.a_{n+1}=\frac{1}{a_1 + a_2 +... + a_n} -\sqrt2, \,a_1 = 1. Find the limit of the sequence SnS_n defined by Sn=a1+a2+...+anS_n=a_1 + a_2 +... + a_n.
algebraSequenceRecurrencerecurrence relationLimit of sequence