MathDB

3

Part of 1989 Iran MO (2nd round)

Problems(2)

Prove that the limit is 2 - [Iran Second Round 1989]

Source:

12/6/2010
Let {an}n≥1\{a_n\}_{n \geq 1} be a sequence in which a1=1a_1=1 and a2=2a_2=2 and an+1=1+a1a2a3⋯an−1+(a1a2a3⋯an−1)2∀n≥2.a_{n+1}=1+a_1a_2a_3 \cdots a_{n-1}+(a_1a_2a_3 \cdots a_{n-1} )^2 \qquad \forall n \geq 2. Prove that lim⁡n→∞(1a1+1a2+1a3+⋯+1an)=2\lim_{n \to \infty} \biggl( \frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\cdots + \frac{1}{a_n} \biggr) =2
limitinductionalgebra proposedalgebra
Faithful Lines - [Iran Second Round 1989]

Source:

12/6/2010
A line dd is called faithful to triangle ABCABC if dd be in plane of triangle ABCABC and the reflections of dd over the sides of ABCABC be concurrent. Prove that for any two triangles with acute angles lying in the same plane, either there exists exactly one faithful line to both of them, or there exist infinitely faithful lines to them.
geometrygeometric transformationreflectiongeometry proposed