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Inequality | HANOI 2018

Try this beautiful problem from American Invitational Mathematics Examination, HANOI, 2018 based on Inequality.

Inequality - HANOI 2018

Find the number of integers that satisfy the inequality $n^{4}-n^{3}-3n^{2}-3n-17 \lt 0$.

• is 4
• is 6
• is 8
• cannot be determined from the given information

Key Concepts

Algebra

Theory of Equations

Inequality

HANOI, 2018

Inequalities (Little Mathematical Library) by Korovkin

Try with Hints

First hint

We have $(n+1)^{3}+16 \gt n^{4} \geq 0$ which implies $n \geq -3$.

Second Hint

For $n \geq 4$ we have $n^{4}-(n+1)^{3}$ $\geq 3n^{3}-3n^{2}-3n-1$ $\geq 12n^{2}-3n^{2}-3n-1$ $=n(n-3)+8n^{2}-1 \gt 16$.

Final Step

Then $-3 \leq n \leq 3$. By directly calculation we obtain n=-1,0,1,2 that is 4 such integers.