Difference between revisions of "1976 IMO Problems/Problem 6"

Revision as of 10:47, 26 February 2008

A sequence $(u_{n})$ is defined by

$u_{0} = 2 \quad u_{1} = \frac {5}{2}, u_{n + 1} = u_{n}(u_{n - 1}^{2} - 2) - u_{1} \quad \textnormal{for} n = 1,\ldots$

Prove that for any positive integer $n$ we have

$\lfloor u_{n} \rfloor = 2^{\frac {(2^{n} - ( - 1)^{n})}{3}}$

(where $\lfloor x\rfloor$ denotes the smallest integer $\leq$ x) $.$

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@@ Line 6: / Line 6: @@
 Prove that for any positive integer <math>n</math> we have
-<cmath>[u_{n}] = 2^{\frac {(2^{n} - ( - 1)^{n})}{3}}</cmath>
+<cmath>\lfloor u_{n} \rfloor = 2^{\frac {(2^{n} - ( - 1)^{n})}{3}}</cmath>
-(where [x] denotes the smallest integer <math>\leq</math> x)<math>.</math>
+(where <math>\lfloor x\rfloor</math> denotes the smallest integer <math>\leq</math> x)<math>.</math>
 == Solution ==

1976 IMO (Problems) • Resources
Preceded by Problem 5	1 • 2 • 3 • 4 • 5 • 6	Followed by Final Question
All IMO Problems and Solutions