Grades 11-12 Video Solutions 2026-2026_11-12

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Question 21. The integers 1, 2, through 40 are written on a blackboard. David performs 39 operations on these numbers. On the k-th operation, where k equals 1, 2, 3, through 39, if k is not a multiple of 7, he picks any two numbers a, b, erases them, and writes a plus b minus 1. If k is a multiple of 7, he picks any two numbers a, b, erases them, and writes a plus b plus 5. What number remains at the end? a, 781, b, 801, c, 802, c, 803, Consider the total sum of the numbers, s. In the beginning, s is equal to the sum of the first 40 positive integers, which is 820. On the kth operation, David either subtracts 1 from the total sum, or adds 5 to the total sum. Since there are 39 operations total, he will add 5 to the total 5 times, on operations 7, 14, 21, 28, and 35. The remaining 34 operations all subtract 1 from the total. Thus, the remaining sum of the numbers is 820-34, plus 5 times 5, which equals 811. Since we have one number remaining, this number must equal the sum, so our answer is c, 811.

Keywords

integer operations

sum manipulation

blackboard puzzle

number transformation

contest mathematics