The average of 10 distinct nonnegative integers is 14.5. The average of the maximum and the minimum of these numbers is less than 15. Which of the following is the maximum possible value of the minimum?

Correct option is C

Given:

• Total numbers = 10 (distinct, nonnegative integers)
• Average of numbers = 14.5
• So, total sum = 14.5 × 10 = 145
• Let the minimum = m and maximum = M
• Also given: (m + M)/2 < 15 → m + M < 30

We are to find the maximum possible value of m (the minimum), under these conditions.

Formula:

1. Average = Total Sum / Number of terms
2. Condition: (Minimum + Maximum) / 2 < 15

Solution:

Step 1: Let’s denote the 10 numbers as:
m, a₂, a₃, ..., a₉, M (all distinct and nonnegative, with m < a₂ < ... < a₉ < M)

Step 2: Sum of all numbers = 145
We want to maximize m, given that m + M < 30
→ So, M < 30 - m

We now try values of m starting from high to low, and check feasibility.

Try m = 10

Then M < 30 - 10 → M < 20
Let’s choose M = 19 (highest possible under constraint)

Now, pick 8 other distinct numbers between 11 and 18 (since m = 10 and M = 19):
→ Possible: 11,12,13,14,15,16,17,18

Sum = 10 (m) + 11+12+13+14+15+16+17+18 (a₂ to a₉) + 19 (M)
→ Middle 8 sum = 116
→ Total = 10 + 116 + 19 = 145

So, this fits all conditions.

Final Answer: (c) 10