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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:

equation

it must be a perfect square:

equation

equation

Use the sum and difference identity:

equation

For the number to be positive:

equation

However, p cannot be very large since the left hand side cannot exceed 1099.

Thus:

equation

Already for p=106 we obtain too large a number, while for p=104 we obtain too small a number:

equation

equation

Therefore p must lie between 105 and 105. Indeed, we obtain:

equation

equation

and there is only one solution to the problem.