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INMO 2017 Board

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

\(\huge\ \frac { 1 }{ { x }_{ 1 }+2017 } +\frac { 1 }{ { x }_{ 2 }+2017 } +...+\frac { 1 }{ { x }_{ 2017
}+2017 } =\frac { 1 }{ 2017 }\)

Prove that

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


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.

These are sample problems.

Note by Priyanshu Mishra
1 year, 2 months ago

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Problem 1 was just an AM-GM.

Let \(y_i = \dfrac {1}{x_i + 2017}\), so \(x_i = \dfrac {1 - 2017y_i}{y_i}\)

We have

\[\displaystyle \sum_{j=1}^{2017} y_j = \dfrac{1}{2017}\]

Thus,

\[\begin{align} \dfrac{1}{2017} - y_i &= \displaystyle \sum_{\substack{j=1\\ j \neq i}}^{2017} y_j\\ 1 - 2017 y_i &= 2017 \displaystyle \sum_{\substack{j=1\\ j \neq i}}^{2017} y_j\\ \end{align}\]

However, we also have

\[\displaystyle \sum_{\substack{j=1\\ j \neq i}}^{2017} y_j \geq 2016 \left ( \displaystyle \prod_{\substack{j=1\\ j \neq i}}^{2017} y_j \right ) ^{\dfrac {1}{2016}}\]

Therefore,

\[\begin{align} \displaystyle \prod_{i=1}^{2017} x_i &= \displaystyle \prod_{i=1}^{2017} \dfrac {1 - 2017y_i}{y_i}\\ & = \dfrac {\displaystyle 2017^{2017} \prod_{i=1}^{2017} \left (\displaystyle \sum_{\substack{j=1\\ j \neq i}}^{2017} y_j \right )}{\displaystyle \prod_{i=1}^{2017} y_i}\\ &\geq \dfrac {(2016 \times 2017)^{2017} \displaystyle \prod_{i=1}^{2017} \left ( \displaystyle \prod_{\substack{j=1\\ j \neq i}}^{2017} y_j \right )^{\dfrac{1}{2016}}}{\displaystyle \prod_{i=1}^{2017} y_i}\\ &= 2016^{2017} \times 2017^{2017}\\ \end{align}\]

This just rearranges to give us the desired expression.

Sharky Kesa - 10 months, 2 weeks ago

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

Zeal Academy - 11 months ago

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Where are the new problems.

Aaron Jerry Ninan - 11 months, 2 weeks ago

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@Sharky Kesa,

Try the new problems

Priyanshu Mishra - 1 year, 1 month ago

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@Sharky Kesa, @Harsh Shrivastava,

i HAve added newproblems. TRy them.

Priyanshu Mishra - 1 year, 1 month ago

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Any more problems?

Sharky Kesa - 1 year, 1 month 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 - 1 year, 2 months ago

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@Racchit Jain

Is RMO DELHI results out?

At which website?

Priyanshu Mishra - 1 year, 1 month ago

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

Racchit Jain - 1 year, 1 month ago

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

Priyanshu Mishra - 1 year, 1 month ago

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

Racchit Jain - 1 year, 1 month 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.

Come and enjoy solving problems here.

Priyanshu Mishra - 1 year, 2 months ago

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check this one

Shivam Jadhav - 1 year, 2 months ago

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