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This problem trains for: SAT-I, AMC-8.
How many 5-digit perfect squares are of the form: 110xy where x and y are digits?
Writing the given number in a somewhat expanded form:
it must be a perfect square:
Use the sum and difference identity:
For the number to be positive:
However, p cannot be very large since the left hand side cannot exceed 1099.
Already for p=106 we obtain too large a number, while for p=104 we obtain too small a number:
Therefore p must lie between 105 and 105. Indeed, we obtain:
and there is only one solution to the problem.