2005 PMWC Problems/Problem I5
Problem
Consider the following conditions on the positive integer (natural number) :
1.
2.
3.
4.
5.
If only three of these conditions are true, what is the value of ?
Solution
We can solve each of these conditions for , and also truncate the right-hand side of the inequality since we know is an integer.
1.
2.
3.
4.
5.
Exactly two of these statements are false.
Note that statement 4 implies statement 1, which implies statement 5, which implies statement 2. This means that 2 and 5 must both be true, or else 1, 4, and 5 are all false, which is in opposition to the fact that only two are false.
Note further that if 3 is false, , which implies that all four of the other statements are true. This also disagrees with the stipulation that exactly three statements are true, and two are false. Therefore statement 3 is true.
This leaves 1 and 4 to be the only statements that are allowed to be false. If 1 and 4 are false, and the rest are true, we have (the inverse of statement 1) and (statement 5). The only natural number that satisfies this is .
See also
2005 PMWC (Problems) | ||
Preceded by Problem I4 |
Followed by Problem I6 | |
I: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 T: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 |