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2021 AMC 10B Problem 19

Problem 19 of 25HarderAlgebraProbability & Statistics

Suppose that SS is a finite set of positive integers. If the greatest integer in SS is removed from S,S, then the average value (arithmetic mean) of the integers remaining is 32.32. If the least integer in SS is also removed, then the average value of the integers remaining is 35.35. If the greatest integer is then returned to the set, the average value of the integers rises to 40.40. The greatest integer in the original set SS is 7272 greater than the least integer in S.S. What is the average value of all the integers in the set S?S?

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Solution

Let the sum of all the integers be s,s, the greatest number be g,g, the least number be l,l, and the size of SS be n.n. The two averages of sets with n−1n-1 elements give s−l=40(n−1),s−g=32(n−1). \begin{aligned} s-l&=40(n-1),\\ s-g&=32(n-1). \end{aligned} Subtracting and using g−l=72g-l=72 gives 72=8(n−1),72=8(n-1), so n=10.n=10. Removing both extremes leaves 88 numbers with sum 8⋅35=280.8\cdot35=280. On the other hand, removing only the least number leaves a sum of 9⋅40=360,9\cdot40=360, so g=360−280=80.g=360-280=80. Hence l=80−72=8,l=80-72=8, and the original sum is s=280+80+8=368.s=280+80+8=368. The required average is sn=36810=36.8.\frac{s}{n}=\frac{368}{10}=36.8. Thus, the answer is D .
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Tagged: mean · system of equations

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