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Zorluk: OrtaMean, Median, and Mode

A set SS consists of 66 distinct positive integers. The arithmetic mean of the integers in SS is 1515, and the median of SS is 1414. If the largest integer in SS is 2828, what is the greatest possible value of the second-largest integer in SS?

  1. A
    2424
  2. B
    2525
  3. 2727Cevap
  4. D
    2828
  5. E
    3131

Cevap

The greatest possible value of the second-largest integer in the set is 2727.
The correct option is 2727. With 66 distinct positive integers x1<x2<x3<x4<x5<x6x_1 < x_2 < x_3 < x_4 < x_5 < x_6, the total sum is 6×15=906 \times 15 = 90. The median gives x3+x4=28x_3 + x_4 = 28. Given x6=28x_6 = 28, we have x1+x2+x5=34x_1 + x_2 + x_5 = 34. Since all integers are distinct and 2828 is the largest, x5x_5 must be strictly less than 2828, making 2727 the maximum integer bound. Setting x5=27x_5 = 27 allows x1=1,x2=6,x3=13,x4=15,x5=27,x6=28x_1 = 1, x_2 = 6, x_3 = 13, x_4 = 15, x_5 = 27, x_6 = 28, which satisfies every requirement of the problem.

Adım Adım Çözüm

1
Determine the total sum of the 66 integers.
Sum = 6×15=906 \times 15 = 90.
Since the mean of 66 integers is 1515, the sum of all elements equals the count multiplied by the mean.
2
Express the median condition algebraically.
Let the ordered set be x1<x2<x3<x4<x5<x6x_1 < x_2 < x_3 < x_4 < x_5 < x_6. Then x3+x42=14  ⟹  x3+x4=28\frac{x_3 + x_4}{2} = 14 \implies x_3 + x_4 = 28.
For an even number of terms (n=6n=6), the median is the average of the 3rd and 4th terms.
3
Substitute known values into the sum equation.
x1+x2+(x3+x4)+x5+x6=90  ⟹  x1+x2+28+x5+28=90  ⟹  x1+x2+x5=34x_1 + x_2 + (x_3 + x_4) + x_5 + x_6 = 90 \implies x_1 + x_2 + 28 + x_5 + 28 = 90 \implies x_1 + x_2 + x_5 = 34.
We know x3+x4=28x_3 + x_4 = 28 and x6=28x_6 = 28 (the largest integer).
4
Apply constraints to maximize x5x_5.
Since x6=28x_6 = 28 is the largest element and all integers are distinct, x5<28x_5 < 28, so x5≤27x_5 \le 27. Testing x5=27x_5 = 27 yields x1+x2=34−27=7x_1 + x_2 = 34 - 27 = 7.
To verify x5=27x_5 = 27 is achievable, choose distinct positive integers x1=1,x2=6,x3=13,x4=15,x5=27,x6=28x_1 = 1, x_2 = 6, x_3 = 13, x_4 = 15, x_5 = 27, x_6 = 28, which satisfies all conditions.

Anahtar Kavram

Mean and Median Properties in Constrained Sets
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Konular

Mean, Median, and Mode

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