https://artofproblemsolving.com/wiki/index.php?title=2004_Indonesia_MO_Problems/Problem_3&feed=atom&action=history 2004 Indonesia MO Problems/Problem 3 - Revision history 2022-07-01T08:11:54Z Revision history for this page on the wiki MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=2004_Indonesia_MO_Problems/Problem_3&diff=96551&oldid=prev Rockmanex3: Solution to Problem 3 — pluses and minuses 2018-07-28T20:53:28Z <p>Solution to Problem 3 — pluses and minuses</p> <p><b>New page</b></p><div>==Problem==<br /> <br /> In how many ways can we change the sign &lt;math&gt;\ast&lt;/math&gt; with &lt;math&gt;+&lt;/math&gt; or &lt;math&gt;-&lt;/math&gt;, such that the following equation is true?<br /> <br /> &lt;cmath&gt;1 \ast 2 \ast 3 \ast 4 \ast 5 \ast 6 \ast 7 \ast 8 \ast 9 \ast 10 = 29&lt;/cmath&gt;<br /> <br /> ==Solution==<br /> <br /> The sum of the numbers from &lt;math&gt;1&lt;/math&gt; to &lt;math&gt;10&lt;/math&gt; is &lt;math&gt;55&lt;/math&gt;, which is &lt;math&gt;26&lt;/math&gt; more than &lt;math&gt;29&lt;/math&gt;. That means the sum of the numbers that are being subtracted is &lt;math&gt;13&lt;/math&gt;.<br /> <br /> To find out the possible ways to pick distinct numbers from &lt;math&gt;2&lt;/math&gt; to &lt;math&gt;10&lt;/math&gt; that add up to &lt;math&gt;13&lt;/math&gt;, we will use [[casework]].<br /> <br /> * If the largest number being subtracted is &lt;math&gt;10&lt;/math&gt;, then the other number being subtracted is &lt;math&gt;3&lt;/math&gt;, for a total of &lt;math&gt;1&lt;/math&gt; possibility.<br /> * If the largest number being subtracted is &lt;math&gt;9&lt;/math&gt;, then the other number being subtracted is &lt;math&gt;4&lt;/math&gt;, for a total of &lt;math&gt;1&lt;/math&gt; possibility.<br /> * If the largest number being subtracted is &lt;math&gt;8&lt;/math&gt;, then the possible sets of numbers that are also being subtracted are &lt;math&gt;&lt;/math&gt; and &lt;math&gt;[2,3]&lt;/math&gt;, for a total of &lt;math&gt;2&lt;/math&gt; possibilities.<br /> * If the largest number being subtracted is &lt;math&gt;7&lt;/math&gt;, then the possible sets of numbers that are also being subtracted are &lt;math&gt;&lt;/math&gt; and &lt;math&gt;[2,4]&lt;/math&gt;, for a total of &lt;math&gt;2&lt;/math&gt; possibilities.<br /> * If the largest number being subtracted is &lt;math&gt;6&lt;/math&gt;, then the possible sets of numbers that are also being subtracted are &lt;math&gt;[2,5]&lt;/math&gt; and &lt;math&gt;[3,4]&lt;/math&gt;, for a total of &lt;math&gt;2&lt;/math&gt; possibilities.<br /> * If the largest number being subtracted is &lt;math&gt;5&lt;/math&gt; (or lower), there are no possibilities.<br /> <br /> In total, there are &lt;math&gt;\boxed{8}&lt;/math&gt; ways to put plus and minus signs such that the equation would be true.<br /> <br /> ==See Also==<br /> {{Indonesia MO box<br /> |year=2004<br /> |num-b=2<br /> |num-a=4<br /> }}<br /> <br /> [[Category:Intermediate Number Theory Problems]]</div> Rockmanex3