MathDB

7

Part of 2003 IMO Shortlist

Problems(2)

A beautiful sequence with a nice property!

Source: IMO ShortList 2003, number theory problem 7

8/17/2004
The sequence a0a_0, a1a_1, a2,a_2, …\ldots is defined as follows: a_0=2, \qquad a_{k+1}=2a_k^2-1  \text{for }k \geq 0. Prove that if an odd prime pp divides ana_n, then 2n+32^{n+3} divides p2−1p^2-1.
[hide="comment"] Hi guys ,
Here is a nice problem:
Let be given a sequence ana_n such that a0=2a_0=2 and an+1=2an2−1a_{n+1}=2a_n^2-1 . Show that if pp is an odd prime such that p∣anp|a_n then we have p2≡1(mod2n+3)p^2\equiv 1\pmod{2^{n+3}}
Here are some futher question proposed by me :Prove or disprove that : 1) gcd(n,an)=1gcd(n,a_n)=1 2) for every odd prime number pp we have am≡±1(modp)a_m\equiv \pm 1\pmod{p} where m=p2−12km=\frac{p^2-1}{2^k} where k=1k=1 or 22
Thanks kiu si u
Edited by Orl.
modular arithmeticpolynomialRecurrenceSequenceDivisibilityprimeIMO Shortlist
IMO ShortList 2003, geometry problem 7

Source: IMO ShortList 2003, geometry problem 7

6/4/2004
Let ABCABC be a triangle with semiperimeter ss and inradius rr. The semicircles with diameters BCBC, CACA, ABAB are drawn on the outside of the triangle ABCABC. The circle tangent to all of these three semicircles has radius tt. Prove that s2<t≤s2+(1−32)r.\frac{s}{2}<t\le\frac{s}{2}+\left(1-\frac{\sqrt{3}}{2}\right)r. Alternative formulation. In a triangle ABCABC, construct circles with diameters BCBC, CACA, and ABAB, respectively. Construct a circle ww externally tangent to these three circles. Let the radius of this circle ww be tt. Prove: s2<t≤s2+12(2−3)r\frac{s}{2}<t\le\frac{s}{2}+\frac12\left(2-\sqrt3\right)r, where rr is the inradius and ss is the semiperimeter of triangle ABCABC.
Proposed by Dirk Laurie, South Africa
geometryperimetercircumcirclegeometric inequalityTriangleIMO Shortlist