A finite set M of positive integers consists of distinct perfect squares and the number 92. The average of the numbers in M is 85. If we remove 92 from M, the average drops to 84. If N2 is the largest possible square in M, what is the value of N?

Step 1: = 85 …(i)

Step 2: If we remove 92, then

Step 3: a12 + a22 + a32 + \dots + an2

Step 4: = 84 …(ii)

Step 5: From (i) and (2)

Step 6: 84n + 92 × 85n + 85

Step 7: Now, a12 + a22 + a32 + \dots + a62 + a72 = 84 \times 7 = 588

Step 8: If a1 = 1, a2 = 2, a3 = 3, a4 = 4, a5 = 5, a6 = 6

Step 9: Then 12 + 22 + 32 + 42 + 52 + 62 + a72 = 588

Step 10: => a72 = 588 − 91

Step 11: => a72 = 497 (not possible)

Step 12: If a1 = 1, a2 = 2, a3 = 3, a4 = 4, a5 = 5, a6 = 7

Step 13: Then 12 + 22 + 32 + 42 + 52 + 72 + a72 = 588

Step 14: => a72 = 484

Step 15: => a7 = 22

Step 16: => N = 22