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The visitors reached 151 for one at lunch after India had declared their first innings closed at 613 for seven after tea on the second day.
Hayden was 59 not out after raising a fluent 123-run opening partnership with fellow left-hander Katich, who made 64 before he was dismissed. Skipper Ricky Ponting was 14 not out at the interval.
Australia were still 462 runs behind India's total, having resumed on their overnight score of 50 without loss.
Katich, who hit two centuries on his test comeback in the West Indies this year, looked good for another until spinner Amit Mishra bowled him after he missed a sharp leg break.
He cracked 10 fours, including fluent drives past mid-on against paceman Zaheer Khan and leg spinner Anil Kumble, to calm any dressing room nerves in the morning.
Hayden, searching for success in the series, got a fiery welcome from Zaheer, who struck him on his helmet with the day's first delivery and then on his body two balls later.
He settled down to play a range of powerful shots, flicking the bowler to square leg for four and Mishra over midwicket for six after the youngster had replaced skipper Kumble.
Hayden, weighed down by a foot injury before the series, showed good footwork and drove fast bowler Ishant Sharma through cover for one of his 10 fours to reach his 28th test fifty.
Indian opener Gautam Gambhir had top-scored with 206 and Vangipurappu Laxman struck 200 not out in the Indian innings.
Gambhir was handed a one-test ban for a code breach after being found guilty of deliberately elbowing Australian all rounder Shane Watson while taking a run on the first day.
India lead the four-match series 1-0 after winning the second test in Mohali by 320 runs. The fourth and final test starts in Nagpur on Nov. 6.
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