Grades 11-12 Video Solutions 2021-video 2021 11-12/22

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Problem number 22 states, the numbers 1, 2, 7, 9, 10, 15, and 19 are written on a blackboard. Two players alternately delete one number, each until only one number remains on the  blackboard. The sum of the numbers deleted by one of the players is twice the sum of the numbers deleted by the other player. What is the number that remains?  So here we have our seven numbers that are on the blackboard. And we'll say that player A removes values that sum up to A, and player B removes values  that sum up to B. And we know that one of these scores is twice the other. And we can say that after they choose all their numbers, the number C will remain on the blackboard.  So let's calculate A plus B plus C. This would just be the sum of all the numbers, since all of these numbers are included in either A, B, or C. So adding that up, we get a value  of 63. So A plus B plus C equals 63. So now let's try and solve for C. Using our identity that A equals 2B, we can substitute that into our original sum equation.  We get 3B plus C equals 63. B is an integer, and 63 is divisible by 3. So subtracting 3B from both sides, we get the value of C to be 63 minus 3B.  Since we know 63 is divisible by 3, and 3 times B will also be divisible by 3, we know that C will also be divisible by 3.  Out of the values on the blackboard, only 2 are divisible by 3. 9 and 15. So now let's assume that C equals 15.  So let's substitute and subtract, and we get that 3B is equal to 48, or that B is equal to 16. And since we know that A is equal to 2B, then we know that A is equal to 32.  And here's the problem. No matter what combination of numbers we have, it is impossible for the sum of three of the original numbers to be equal to 16 or 32.  This means that our original assumption that C equals 15 must be wrong. So this means that C must be 9, and B must be 18, and A must be 36. So let's try making the sum.  We can see that 19 plus 10 plus 7 is going to make a sum of 36, and 1, 2, and 15 is going to make a sum of 18.  The number that remains is 9, which means that 9 is the remaining number on the blackboard. So the question asked us, what is the number that remains? The answer is 9, letter B.

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