1966 IMO Problems/Problem 4: Difference between revisions
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{{IMO box|year=1966|num-b=3|num-a=5}} | |||
Revision as of 11:53, 29 January 2021
Problem
Prove that for every natural number 
, and for every real number 
(
; 
any integer)
![\[\frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx}\]](http://latex.artofproblemsolving.com/4/a/3/4a3adede6c471d5c65e37bdfb9f4961dc8f6b823.png)
Solution
Assume that 
is true, then we use 
and get 
.
First, we prove
LHS=
Using the above formula, we can rewrite the original series as
Which gives us the desired answer of 
See Also
| 1966 IMO (Problems) • Resources | ||
| Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
| All IMO Problems and Solutions | ||
