Home > Numbers
| Note the function, get the value(s) (Posted on 2007-04-28) |
 |
Let f(x) be a function defined on set of non negative integers and taking values in the same set such that f satisfies:
x – f(x) = 19[x/19] – 90[f(x)/90] and, 1900< f(1990) < 2000.
Determine the possible value(s) that f(1990) can take.
NOTE: [n] denotes the greatest integer < = n
|
|
Submitted by K Sengupta
|
|
Rating: 4.0000 (1 votes)
|
|
|
Solution:
|
(Hide)
|
We know that 1900< f(1990)< 2000
Or, 1900/90 < f(1990)/90< 2000/90
Or, [1900/90] <[f(1990)/90]< [2000/90] + 1
Or, 21< = [f(1990)/90] < 23
Accordingly, [f(1990)/90] = 21 or 22.
Let, [f(1990)/90] = 21.
Then, substituting x = 1990 in the relation:
x – g(x) = 19[x/19] – 90[f(x)/90], we have:
1990 – f(1990) = 19[1990/19] – 90[g(1990)/90]
Or, 1990 – f(1990) = 19*104 – 90*21 = 86
Or, f(1990) = 1904
Let, [f(1990)/90] = 21.
Then, substituting x = 1990 in the relation:
x – g(x) = 19[x/19] – 90[f(x)/90], we have:
1990 – f(1990) = 19[1990/19] – 90[g(1990)/90]
Or, 1990 – f(1990) = 19*104 – 90*22= -4
Or, f(1990) = 1994
Hence, f(1990) can assume only two values and these are 1904 and 1994.
*******************************************
An alternate methodology is provided by Charlie in this location.
|
Comments: (
You must be logged in to post comments.)
| |
Subject |
Author |
Date |
 | Solution | Dej Mar | 2007-04-28 21:38:46 |
| solution | Charlie | 2007-04-28 10:10:39 |
|
|
Please log in:
Forums (0)
Newest Problems
Random Problem
FAQ |
About This Site
Site Statistics
New Comments (3)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On
Chatterbox:
|
