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Solution

Positive odd integers not exceeding $100$ are:

$1,3,5,\ldots,99$

Number of odd integers $=50$

Perfect squares not exceeding $100$ are:

$1^2,2^2,3^2,\ldots,10^2$

Number of perfect squares $=10$

Odd perfect squares are:

$1^2,3^2,5^2,7^2,9^2$

Number of odd perfect squares $=5$

Required number:

$50+10-5=55$