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# RMO 2016 Delhi Region

Here is the paper of RMO 2016 Delhi.Pls post answers and solutions to all questions. Also tell ur marks and estimated cut off Thanks!

Note by Kaustubh Miglani
3 months, 1 week ago

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Problem 4

$$f\left( a,b,c \right) =\quad \sum { \frac { a }{ { a }^{ 3 }+{ b }^{ 2 }+{ c } } } =\quad \sum { \frac { 1 }{ { a }^{ 2 }+\frac { { b }^{ 2 } }{ a } +\frac { c }{ a } } } \\ \\ Now,\quad \frac { { a }^{ 2 }+{ \frac { { b }^{ 2 } }{ a } + }{ \frac { c }{ a } } }{ 3 } \ge \sqrt [ 3 ]{ { b }^{ 2 }{ c } } \quad \Rightarrow \quad \frac { 1 }{ { a }^{ 2 }+\frac { { b }^{ 2 } }{ a } +\frac { c }{ a } } \le \quad \frac { 1 }{ 3\sqrt [ 3 ]{ { b }^{ 2 }c } } \left\{ A.M-G.M \right\} \\ \sum { \frac { 1 }{ { a }^{ 2 }+\frac { { b }^{ 2 } }{ a } +\frac { c }{ a } } } \le \quad \sum { \frac { 1 }{ 3\sqrt [ 3 ]{ { b }^{ 2 }c } } } \le \sqrt [ 3 ]{ abc } \le \frac { a+b+c }{ 3 } =1\quad \\ \\ \Rightarrow f\left( a,b,c \right) \le 1,\quad with\quad equality\quad at\quad a=b=c=1$$ · 3 months, 1 week ago

Will my answer be correct? · 3 months, 1 week ago

I USED Titus lemma. Got same answer · 3 months, 1 week ago

THEN U GOTTA PROVE IT COZ IT'S NOT IN STANDARD TEXT BOOKS · 1 week, 2 days ago

It is in standard books.Atleast cauchy is · 1 week, 2 days ago

Problem 1

We can see that as we move the point $$P$$ on the circumference of the circle$$[$$excluding $$X$$ and $$Y],$$the $$\angle XPY=\angle XP_1Y$$ remains constant.So this shows that $$AB=A_1B_1.$$Now we use extended sin rule to complete the problem.
Let the circum-radius of $$\triangle PAB$$ be $$R$$ and $$\triangle P_1A_1B_1$$ be $$R_1.$$
In $$\triangle PAB, \frac{AB}{sin\angle P}=2R$$ and in $$\triangle P_1A_1B_1,\frac{A_1B_1}{sin\angle P_1}=\frac{AB}{sin\angle P}=2R_1.$$ Therefore $$2R=2R_1\Rightarrow\boxed {R=R_1}.$$Hence Proved. · 3 months, 1 week ago

nice solution · 3 months, 1 week ago

Hey can u explain in detail why A1B1=AB · 1 week, 6 days ago

Because angle subtended by both of these chords at centre are equal. angle A1 X A =P X P1=P Y P1=B Y B1 · 1 week, 2 days ago

Which class are u in and which school? · 1 week, 2 days ago

I m in class 10 And from bbps dw. But ur profile says u live in noida · 1 week ago

I am graduating from IIT Bombay. What is ur favourite college bro? · 1 week, 1 day ago

IIT Is the one Bombay ,Perhaps · 1 week, 1 day ago