2004 Indonesia MO Problems/Problem 3
Problem
In how many ways can we change the sign with or , such that the following equation is true?
Solution
The sum of the numbers from to is , which is more than . That means the sum of the numbers that are being subtracted is .
To find out the possible ways to pick distinct numbers from to that add up to , we will use casework.
- If the largest number being subtracted is , then the other number being subtracted is , for a total of possibility.
- If the largest number being subtracted is , then the other number being subtracted is , for a total of possibility.
- If the largest number being subtracted is , then the possible sets of numbers that are also being subtracted are and , for a total of possibilities.
- If the largest number being subtracted is , then the possible sets of numbers that are also being subtracted are and , for a total of possibilities.
- If the largest number being subtracted is , then the possible sets of numbers that are also being subtracted are and , for a total of possibilities.
- If the largest number being subtracted is (or lower), there are no possibilities.
In total, there are ways to put plus and minus signs such that the equation would be true.
See Also
2004 Indonesia MO (Problems) | ||
Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by Problem 4 |
All Indonesia MO Problems and Solutions |