INMO 2017 Board

(1)(1) Let x1,...,x2017{ x }_{ 1 },...,{ x }_{ 2017 } be positive reals such that

 1x1+2017+1x2+2017+...+1x2017+2017=12017\huge\ \frac { 1 }{ { x }_{ 1 }+2017 } +\frac { 1 }{ { x }_{ 2 }+2017 } +...+\frac { 1 }{ { x }_{ 2017 }+2017 } =\frac { 1 }{ 2017 }

Prove that

 x1x2...x2017201720162017\huge\ \frac { \sqrt [ 2017 ]{ { x }_{ 1 }{ x }_{ 2 }...{ x }_{ 2017 } } }{ 2016 } \ge 2017


Two circles enclose non-intersecting areas. Common tangent lines to the two circles, one external and one internal, are drawn. Consider two straight lines each of which passes through the tangent points on one of the circles. Prove that the intersection point of the lines lies on the straight line that connects the centers of the circles.


Hello everybody. Please post solutions of these problems and post problems on your own also.

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Note by Priyanshu Mishra
3 years ago

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@Harsh Shrivastava, @Sharky Kesa, @Svatejas Shivakumar, @rajdeep das, @Racchit Jain,@Vaibhav Prasad @Kalash Verma @Nihar Mahajan @Adarsh Kumar @Akshat Sharda @AkshayYadav @Swapnil Das @Rajdeep Dhingra @Anik Mandal @Lakshya Sinha @Abhay Kumar @Dev Sharma and everyone.

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Priyanshu Mishra - 3 years ago

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For problem 1 apply titus lemma in the constraint given and then apply AM GM and it's done.

Racchit Jain - 3 years ago

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@Racchit Jain

Is RMO DELHI results out?

At which website?

Priyanshu Mishra - 2 years, 11 months ago

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They send you your marks by mail. Only marks though, the cutoff hasn't been decided yet.

Racchit Jain - 2 years, 11 months ago

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@Racchit Jain At which website is the name of list of selected students declared?

Priyanshu Mishra - 2 years, 11 months ago

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@Priyanshu Mishra It's not declared yet but I think it will be on hbcse

Racchit Jain - 2 years, 11 months ago

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Any more problems?

Sharky Kesa - 2 years, 11 months ago

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@Sharky Kesa, @Harsh Shrivastava,

i HAve added newproblems. TRy them.

Priyanshu Mishra - 2 years, 11 months ago

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@Sharky Kesa,

Try the new problems

Priyanshu Mishra - 2 years, 11 months ago

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Where are the new problems.

Aaron Jerry Ninan - 2 years, 9 months ago

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INMO 2017 Solution- www.zeal.academy http://www.zeal.academy/INMO%202017%20solutions(HM).pdf

zeal academy - 2 years, 9 months ago

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Problem 1 was just an AM-GM.

Let yi=1xi+2017y_i = \dfrac {1}{x_i + 2017}, so xi=12017yiyix_i = \dfrac {1 - 2017y_i}{y_i}

We have

j=12017yj=12017\displaystyle \sum_{j=1}^{2017} y_j = \dfrac{1}{2017}

Thus,

12017yi=jij=12017yj12017yi=2017jij=12017yj\begin{aligned} \dfrac{1}{2017} - y_i &= \displaystyle \sum_{\stackrel{j=1}{j \neq i}}^{2017} y_j\\ 1 - 2017 y_i &= 2017 \displaystyle \sum_{\stackrel{j=1}{j \neq i}}^{2017} y_j\\ \end{aligned}

However, we also have

jij=12017yj2016(jij=12017yj)12016\displaystyle \sum_{\stackrel{j=1}{j \neq i}}^{2017} y_j \geq 2016 \left ( \displaystyle \prod_{\stackrel{j=1}{j \neq i}}^{2017} y_j \right ) ^{\dfrac {1}{2016}}

Therefore,

i=12017xi=i=1201712017yiyi=20172017i=12017(jij=12017yj)i=12017yi(2016×2017)2017i=12017(jij=12017yj)12016i=12017yi=20162017×20172017\begin{aligned} \displaystyle \prod_{i=1}^{2017} x_i &= \displaystyle \prod_{i=1}^{2017} \dfrac {1 - 2017y_i}{y_i}\\ & = \dfrac {\displaystyle 2017^{2017} \prod_{i=1}^{2017} \left (\displaystyle \sum_{\stackrel{j=1}{j \neq i}}^{2017} y_j \right )}{\displaystyle \prod_{i=1}^{2017} y_i}\\ &\geq \dfrac {(2016 \times 2017)^{2017} \displaystyle \prod_{i=1}^{2017} \left ( \displaystyle \prod_{\stackrel{j=1}{j \neq i}}^{2017} y_j \right )^{\dfrac{1}{2016}}}{\displaystyle \prod_{i=1}^{2017} y_i}\\ &= 2016^{2017} \times 2017^{2017}\\ \end{aligned}

This just rearranges to give us the desired expression.

Sharky Kesa - 2 years, 8 months ago

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