Cheenta is joining hands with Aditya Birla Education Academy for AMC Training.
Learn More

May 8, 2020

Problem from Inequality | PRMO-2018 | Problem 23

Try this beautiful Algebra problem from PRMO, 2018 based on Inequality.

Problem from Inequality | PRMO | Problem-23

What is the largest positive integer n such that \(\frac{a^2}{\frac{b}{29} +\frac{c}{31}} +\frac{b^2}{\frac{c}{29}+\frac{a}{31}} +\frac{c^2}{\frac{a}{29}+\frac{b}{31}}\)\(\geq n(a+b+c)\).

  • $20$
  • $24$
  • $13$

Key Concepts



Check the Answer


PRMO-2018, Problem 23

Pre College Mathematics

Try with Hints

we have to find out the largest value of \(n\).......

we know that ,

\(\frac{a^2}{x}+\frac{b^2}{y}+\frac{c^2}{z} \geq \frac{(a+b+c)^2}{x+y+z}\) .we may use this form to getting the largest positive integer \(n\)

Can you now finish the problem ..........


\(\frac{a^2}{\frac{b}{29} +\frac{c}{31}} +\frac{b^2}{\frac{c}{29}+\frac{a}{31}} +\frac{c^2}{\frac{a}{29}+\frac{b}{31}}\)\(\geq \frac{(a+b+c)^2}{a(\frac{1}{29}+\frac{1}{31}) +b(\frac{1}{29}+\frac{1}{31})+c(\frac{1}{29}+\frac{1}{31})}\)

\(\Rightarrow \frac{a^2}{\frac{b}{29} +\frac{c}{31}} +\frac{b^2}{\frac{c}{29}+\frac{a}{31}} +\frac{c^2}{\frac{a}{29}+\frac{b}{31}}\) \(\geq \frac{(a+b+c)}{(\frac{1}{29} +\frac{1}{31})}\)

\(\Rightarrow \frac{a^2}{\frac{b}{29} +\frac{c}{31}} +\frac{b^2}{\frac{c}{29}+\frac{a}{31}} +\frac{c^2}{\frac{a}{29}+\frac{b}{31}}\)\(\geq \frac{a+b+c}{\frac{60}{29 \times 31}}\)

\(\Rightarrow \frac{a^2}{\frac{b}{29} +\frac{c}{31}} +\frac{b^2}{\frac{c}{29}+\frac{a}{31}} +\frac{c^2}{\frac{a}{29}+\frac{b}{31}}\)\(\geq \frac{29\times 31}{60}(a+b+c)\)

\(\Rightarrow \frac{a^2}{\frac{b}{29} +\frac{c}{31}} +\frac{b^2}{\frac{c}{29}+\frac{a}{31}} +\frac{c^2}{\frac{a}{29}+\frac{b}{31}}\)\(\geq 14.98(a+b+c)\)

Therefore \(n=14\)

Subscribe to Cheenta at Youtube

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy

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.