"Find the three consecutive odd intergers..."

Let x = 1st odd integer

So, x+2 = 2nd odd integer

And, x+4 = 3rd odd integer

"whose sum..."

x +(x+2) +(x+4) = 3x +6

"...is equal to five more than two times the second largest number among the three."

3x +6 = 2(x+2) +5

3x +6 = 2x +4 +5

3x -2x = 4 +5 -6

x = 3

Hence,

x+2 = 5

x+4 = 7

Therefore, the 3 cosecutive odd integers are 3,5,7. --------answer.

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Check

Sum = 3+5+7 = 15

15 =? 2(5) +5

15 =? 10 +5

15 =? 15

Yes, so, Ok.