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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

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

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@Aditya Dhawan Will my answer be correct? Kaustubh Miglani · 3 months, 1 week ago

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@Aditya Dhawan I USED Titus lemma. Got same answer Kaustubh Miglani · 3 months, 1 week ago

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

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@Yuvraj Singh It is in standard books.Atleast cauchy is Kaustubh Miglani · 1 week, 2 days ago

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Problem 1

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. Ayush Rai · 3 months, 1 week ago

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@Ayush Rai nice solution Abhishek Alva · 3 months, 1 week ago

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@Ayush Rai Hey can u explain in detail why A1B1=AB Kaustubh Miglani · 1 week, 6 days ago

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@Kaustubh Miglani 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 Yuvraj Singh · 1 week, 2 days ago

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@Yuvraj Singh Which class are u in and which school? Kaustubh Miglani · 1 week, 2 days ago

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

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

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@Yuvraj Singh IIT Is the one Bombay ,Perhaps Kaustubh Miglani · 1 week, 1 day ago

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@Kaustubh Miglani Prefered Jee rank?mine was 17 Yuvraj Singh · 1 week, 1 day ago

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