rmo 2015-16

i am not able to solve some problems of rmo ???? but i want to crack it ??? is it be possible , can anyone suggests me the bokks ???

#Geometry #RMO

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

@Rajdeep Dhingra

A Former Brilliant Member - 6 years, 2 months ago

You could tell me those problems and I may be able to help you ? See this

Rajdeep Dhingra - 6 years, 2 months ago

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

Rudraksh Sisodia - 6 years, 2 months ago

Method 1:

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 - 6 years, 2 months ago

@Rajdeep Dhingra – heyy thnks a lot ,,, can u suggest me the books ???? are a aspirant of jee ??

Rudraksh Sisodia - 6 years, 2 months ago

@Rajdeep Dhingra – after all this question meant for which class ???

Rudraksh Sisodia - 6 years, 2 months ago

add me also in ur hangouts ... I am also preparing for RMO! I'll give you a hand in solving the problems !!

Hrithik Nambiar - 5 years, 8 months ago

Give me your Email ID.

Rajdeep Dhingra - 5 years, 8 months ago

[email protected]

Hrithik Nambiar - 5 years, 8 months ago

thanks!

Hrithik Nambiar - 5 years, 8 months ago

Which class do ya study in ?

Rajdeep Dhingra - 5 years, 8 months ago

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 - 5 years, 7 months ago

Can you guys help me in solving my note named :"Primes filled with primes".

Arihant Samar - 5 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$