2004 Indonesia MO Problems/Problem 3

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.

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

Page

Toolbox

Search

2004 Indonesia MO Problems/Problem 3

Problem

Solution

See Also