6. I said to my classmates, "I won a race last year." I said to my classmates that I'd won a race year.

В прямой речи глагол стоит в Past Simple (won), значит, в косвенной речи нужно использовать Past Perfect (had won).

I said to my classmates, "I won a race last year." I said to my classmates that I'd won a race the previous year.

Ответ: the previous