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Difficulty: HardMean, Median, and Mode

A data set consists of 99 positive integers. The set has a unique mode of 1414, a median of 1212, and an arithmetic mean of 1111. If the range of the data set is 1010, what is the maximum possible value of the largest integer in the set?

  1. A
    1414
  2. B
    1515
  3. 1616Answer
  4. D
    1717
  5. E
    1818

Answer

The maximum possible value of the largest integer in the set is 1616.
The correct answer states that the maximum possible value is 1616. By ordering the 99 terms x1≤x2≤⋯≤x9x_1 \le x_2 \le \dots \le x_9, the sum of all terms must equal 9999, with median x5=12x_5 = 12 and x9=x1+10x_9 = x_1 + 10. If x9=16x_9 = 16, then x1=6x_1 = 6, and we can construct a valid set {6,6,8,9,12,14,14,14,16}\{6, 6, 8, 9, 12, 14, 14, 14, 16\} where 1414 is the unique mode appearing 33 times. Trying a larger value such as 1717 forces x1=7x_1 = 7, which makes it impossible to maintain 1414 as the unique mode without violating the total sum of 9999.

Step-by-Step Solution

1
Express the total sum and set structure using ordered variables.
Let the 99 positive integers in non-decreasing order be x1≤x2≤x3≤x4≤x5≤x6≤x7≤x8≤x9x_1 \le x_2 \le x_3 \le x_4 \le x_5 \le x_6 \le x_7 \le x_8 \le x_9. The total sum is 9×11=999 \times 11 = 99. The median is x5=12x_5 = 12. The range is x9−x1=10  ⟹  x9=x1+10x_9 - x_1 = 10 \implies x_9 = x_1 + 10.
Establishing the position of the median and the exact sum provides structural bounds for maximizing x9x_9.
2
Analyze the frequency requirements for the unique mode.
Since 14>1214 > 12 (the median), the number 1414 must lie in the upper half of the set (x6,x7,x8,x9x_6, x_7, x_8, x_9). To be a unique mode, 1414 must appear at least twice. If x9=16x_9 = 16, then x1=6x_1 = 6.
Maximizing x9x_9 is equivalent to maximizing x1x_1, so we test the largest possible values for x1x_1 and verify the uniqueness of the mode 1414.
3
Evaluate x1=7x_1 = 7 (which would give x9=17x_9 = 17).
If x1=7x_1 = 7, then x9=17x_9 = 17. The sum equation becomes 7+x2+x3+x4+12+x6+x7+x8+17=99  ⟹  x2+x3+x4+x6+x7+x8=637 + x_2 + x_3 + x_4 + 12 + x_6 + x_7 + x_8 + 17 = 99 \implies x_2 + x_3 + x_4 + x_6 + x_7 + x_8 = 63. Since x2,x3,x4≥7x_2, x_3, x_4 \ge 7 and x6,x7,x8≥12x_6, x_7, x_8 \ge 12, achieving this sum while keeping 1414 as the unique mode is impossible (it would require 77 to appear 44 times, making 77 the mode instead of 1414). Thus, x9x_9 cannot be 1717 or greater.
Testing x9=17x_9 = 17 shows a violation of the unique mode condition.
4
Construct a valid set for x1=6x_1 = 6 (giving x9=16x_9 = 16).
If x1=6x_1 = 6 and x9=16x_9 = 16, consider the set {6,6,8,9,12,14,14,14,16}\{6, 6, 8, 9, 12, 14, 14, 14, 16\}. Sum = 6+6+8+9+12+14+14+14+16=996+6+8+9+12+14+14+14+16 = 99. Median = 1212. Unique mode = 1414 (frequency 33). Range = 16−6=1016 - 6 = 10. All conditions are satisfied.
Constructing an explicit valid data set proves that 1616 is attainable and is the maximum.

Key Concept

Maximizing elements in a constrained discrete data set using mean, median, mode, and range properties
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Topics

Mean, Median, and Mode

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