2020 AMC 10A Problems/Problem 17
DefineHow many integers are there such that ?
Notice that is a product of many integers. We either need one factor to be 0 or an odd number of negative factors.
Case 1: There are 100 integers for which
Case 2: For there to be an odd number of negative factors, must be between an odd number squared and an even number squared. This means that there are total possible values of . Simplifying, there are possible numbers.
Summing, there are total possible values of . ~PCChess
Notice that is nonpositive when is between and , and , and (inclusive), which means that the amount of values equals .
This reduces to
Solution 3 (end behavior)
We know that is a -degree function with a positive leading coefficient. That is, .
Since the degree of is even, its end behaviors match. And since the leading coefficient is positive, we know that both ends approach as goes in either direction.
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