| Title: | Mathematics at DEC |
| Moderator: | RUSURE::EDP |
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
We need more problems in this file. Here is one:
Find all rational numbers q such that:
n
-----
\ n i
/ ( )(q-1) = 1/1024, where n is an integer.
----- i
i=0
(The above equation says the sum for all integer values of i from 0 to n,
inclusive, of the number of combinations of n things taken i at a time
multiplied by (q-1) to the power of i is 1/1024.)
This was a problem in a high school competition. The time limit is six
minutes (although it was one of the hardest problems, so more time is
reasonable).
-- edp
| T.R | Title | User | Personal Name | Date | Lines |
|---|---|---|---|---|---|
| 333.1 | BEING::POSTPISCHIL | Sat Sep 21 1985 22:43 | 12 | ||
No takers yet? A hint follows the form-feed. n ----- \ n i n-i n / ( )p q = (p+q) ----- i i=0 -- edp | |||||
| 333.2 | TAV02::NITSAN | Sun Sep 22 1985 10:34 | 15 | ||
n n
----- -----
\ n i \ n i n-i n n
1/1024 = / ( )(q-1) = / ( )(q-1) 1 = (q-1+1) = q
----- i ----- i
i=0 i=0
n _____ n ______
/ / 10
So, q = \/1/1024 = \/(1/2) which is rational for n=1,2,5,10 and yields:
q = 1/1024, 1/32, 1/4, 1/2
Nitsan.
| |||||
| 333.3 | ALIEN::POSTPISCHIL | Mon Sep 23 1985 13:19 | 7 | ||
Re .2: Very good, but you forgot (as I did, originally, as well as the creator of the problem) -1/32 and -1/2. -- edp | |||||