u can use any of the following books
1 . arthur engel
2. pre college mathematics
3 . rmo and inmo prep booklet by rajeev manocha
–
Rishabh Agarwal
·
1 year, 7 months ago

By what is also known as Titu's form of Cauchy-Schwarz we have
\(\frac{a^2}{c-1}+\frac{b^2}{d-1}+\frac{c^2}{e-1}+\frac{d^2}{a-1}+\frac{e^2}{b-1} \ge \dfrac {(\sum a)^2} {\sum a - 5} \ge 20\), since it comes to \((\sum a-10)^2 \ge 0\) . Equality occurs for \(a=b=c=d=e=2\)

Method 2:

We know that \((a-2)^2 \ge 0\) so \(a^2 \ge 4(a-1)\). Similarly we get \( \frac{a^2}{c-1}+\frac{b^2}{d-1}+\frac{c^2}{e-1}+\frac{d^2}{a-1}+\frac{e^2}{b-1} \ge \frac{4(a-1)}{c-1}+\frac{4(b-1)}{d-1}+\frac{4(c-1)}{e-1}+\frac{4(d-1)}{a-1}+\frac{4(e-1)}{b-1} \\ \ge 5\sqrt[5]{\frac{4(a-1)}{c-1}\cdot \frac{4(b-1)}{d-1}\cdot \frac{4(c-1)}{e-1}\cdot \frac{4(d-1)}{a-1}\cdot \frac{4(e-1)}{b-1} }= 20 . \)
–
Rajdeep Dhingra
·
2 years, 2 months ago

## Comments

Sort by:

TopNewestu can use any of the following books 1 . arthur engel 2. pre college mathematics 3 . rmo and inmo prep booklet by rajeev manocha – Rishabh Agarwal · 1 year, 7 months ago

Log in to reply

thanks! – Hrithik Nambiar · 1 year, 8 months ago

Log in to reply

– Rajdeep Dhingra · 1 year, 8 months ago

Which class do ya study in ?Log in to reply

hrithik.nambiar2002@gmail.com – Hrithik Nambiar · 1 year, 8 months ago

Log in to reply

add me also in ur hangouts ... I am also preparing for RMO! I'll give you a hand in solving the problems !! – Hrithik Nambiar · 1 year, 8 months ago

Log in to reply

– Rajdeep Dhingra · 1 year, 8 months ago

Give me your Email ID.Log in to reply

You could tell me those problems and I may be able to help you ? See this – Rajdeep Dhingra · 2 years, 2 months ago

Log in to reply

– Rudraksh Sisodia · 2 years, 2 months ago

http://olympiads.hbcse.tifr.res.in/uploads/crmo-2013-paper-4 here question number 3Log in to reply

By what is also known as Titu's form of Cauchy-Schwarz we have

\(\frac{a^2}{c-1}+\frac{b^2}{d-1}+\frac{c^2}{e-1}+\frac{d^2}{a-1}+\frac{e^2}{b-1} \ge \dfrac {(\sum a)^2} {\sum a - 5} \ge 20\), since it comes to \((\sum a-10)^2 \ge 0\) . Equality occurs for \(a=b=c=d=e=2\)

Method 2:

We know that \((a-2)^2 \ge 0\) so \(a^2 \ge 4(a-1)\). Similarly we get \( \frac{a^2}{c-1}+\frac{b^2}{d-1}+\frac{c^2}{e-1}+\frac{d^2}{a-1}+\frac{e^2}{b-1} \ge \frac{4(a-1)}{c-1}+\frac{4(b-1)}{d-1}+\frac{4(c-1)}{e-1}+\frac{4(d-1)}{a-1}+\frac{4(e-1)}{b-1} \\ \ge 5\sqrt[5]{\frac{4(a-1)}{c-1}\cdot \frac{4(b-1)}{d-1}\cdot \frac{4(c-1)}{e-1}\cdot \frac{4(d-1)}{a-1}\cdot \frac{4(e-1)}{b-1} }= 20 . \) – Rajdeep Dhingra · 2 years, 2 months ago

Log in to reply

– Rudraksh Sisodia · 2 years, 2 months ago

after all this question meant for which class ???Log in to reply

– Rudraksh Sisodia · 2 years, 2 months ago

heyy thnks a lot ,,, can u suggest me the books ???? are a aspirant of jee ??Log in to reply

Log in to reply

– Rudraksh Sisodia · 2 years, 2 months ago

u r in which class ????Log in to reply

– Rajdeep Dhingra · 2 years, 2 months ago

In class 8thLog in to reply

@Rajdeep Dhingra ! You are in class 9 this year right? Can we appear for RMO in the 9th grade?? – Hrithik Nambiar · 1 year, 8 months ago

HeyLog in to reply

– Rajdeep Dhingra · 1 year, 8 months ago

Yes, I am in class 9. We can give RMO but school should approve ya first.Log in to reply

– Hrithik Nambiar · 1 year, 8 months ago

but then I saw that the eligibility was class 11 and 10!!Log in to reply

@Rajdeep Dhingra – Azhaghu Roopesh M · 2 years, 2 months ago

Log in to reply

Can you guys help me in solving my note named :"Primes filled with primes". – Arihant Samar · 1 year, 7 months ago

Log in to reply