It was a maiden century for Broom as New Zealand struggled to 251.
It was a good toss to win for Bangladesh skipper Mushrafe Mortaza who needs a victory to keep the three-match series alive after New Zealand won the first ODI by 77 runs.
He put New Zealand into bat to make the most of the heavy overcast conditions. When the clouds cleared and the pitched flattened out Bangladesh maintained the pressure.
His 109 off 107 deliveries included three sixes and eight fours. He offered only one chance when dropped by debutant wicketkeeper Nurul Hasan in the penultimate over.
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Mashrafe had Bangladesh in charge from the start when removed Martin Guptill with the third ball of the match.
He also dismissed the big-hitting Colin Munro for three in the middle of the innings and came back to remove Lockie Ferguson at the tail.
Taskin Ahmed dismissed Kane Williamson (14) for the second successive match and first ODI century-maker Tom Latham struggled to 22 before he fell lbw to Shakib Al Hasan leaving New Zealand 47-3 in the 14th over.
For Bangladesh, Mashrafe took three for 49 while Taskin Ahmed and Shakib Al Hasan both took two for 45.
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