2004 Indonesia MO Problems/Problem 3

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Problem

In how many ways can we change the sign $\ast$ with $+$ or $-$, such that the following equation is true?

\[1 \ast 2 \ast 3 \ast 4 \ast 5 \ast 6 \ast 7 \ast 8 \ast 9 \ast 10 = 29\]

Solution

The sum of the numbers from $1$ to $10$ is $55$, which is $26$ more than $29$. That means the sum of the numbers that are being subtracted is $13$.

To find out the possible ways to pick distinct numbers from $2$ to $10$ that add up to $13$, we will use casework.

  • If the largest number being subtracted is $10$, then the other number being subtracted is $3$, for a total of $1$ possibility.
  • If the largest number being subtracted is $9$, then the other number being subtracted is $4$, for a total of $1$ possibility.
  • If the largest number being subtracted is $8$, then the possible sets of numbers that are also being subtracted are $[5]$ and $[2,3]$, for a total of $2$ possibilities.
  • If the largest number being subtracted is $7$, then the possible sets of numbers that are also being subtracted are $[6]$ and $[2,4]$, for a total of $2$ possibilities.
  • If the largest number being subtracted is $6$, then the possible sets of numbers that are also being subtracted are $[2,5]$ and $[3,4]$, for a total of $2$ possibilities.
  • If the largest number being subtracted is $5$ (or lower), there are no possibilities.

In total, there are $\boxed{8}$ 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
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