Cheenta
How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
Learn More

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

Check the Answer


Answer: is 4.

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.

Subscribe to Cheenta at Youtube


Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com