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Find the number of integers $n$ that satisfy $n^2 < 64 - 20n - n^2.$

 Nov 25, 2023
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The given inequality is equivalent to n2<64−20n, or 2n2+20n−64<0. Factoring the left side, we get 2(n−4)(n+8)<0. Since n−4 n>−8. Thus, the possible values of n are −7,−6,−5,…,3,2.

There are 3+(−7)+1=15​ integers in this range.

 Nov 25, 2023

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