Grades 11-12 Video Solutions 2010-11&12 Video Solutions 2010 problem26

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Question number 26. Numbers 1 to 10 were listed in the table. Students picked two of these numbers at random and erased them, but then added the sum of these  numbers decreased by one to the table. Proceeding in this manner, eventually just one number was left in the table. This number was which of the  following? So let's begin by writing down numbers 1 through 10. We can just write them down in the list. So we have 1, 2, all the way up to 10. And if we were to  sum them, we can do that. If I put pluses here between the numbers, we can use the formula for the sum of the first consecutive 10 integers. And that sum  would come out to 55. Remember the formula is n times quantity n plus 1 divided by 2. And here n is equal to 10. And then what each student is doing is  coming up to the board and removing two of these numbers. So I can do that over here. I can just remove them. And then if I were to add these up, for example, 8  plus 9 is 17. Then the student here takes that 17 and writes it back in, but decreased by 1. So there is now a 16 over here. And now instead of 10 numbers  in the sum, we have 9. So doing this again, I can remove two numbers, take their sum, in this case 11, and then put that back into the list, decreased by 1.  So now I have a 10 in there. And now 8 numbers are being added together. So after two steps, what I have here on the right-hand side is no longer 55. The  first time we added back 17, decreased by 1. We decreased the right-hand side by 1. And the second time, also, we had replaced two numbers with their sum,  decreased by 1. So now we have another minus 1 on the right-hand side. And we do this. We continue thus until all the numbers except for 1 are gone. So  continue until one number remains. And that happens exactly in nine steps. So we have performed two steps. We need seven more steps. And so the final answer will  be 55. We have the minus 2 and minus 7, and that's equal to 46. And so that is our resulting number, and that gives us choice C, 46.

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