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| Note the function, get the value(s) (Posted on 2007-04-28) |
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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
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Submitted by K Sengupta
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Rating: 4.0000 (1 votes)
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Solution:
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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.
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An alternate methodology is provided by Charlie in this location.
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Subject |
Author |
Date |
 | Solution | Dej Mar | 2007-04-28 21:38:46 |
| solution | Charlie | 2007-04-28 10:10:39 |
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