# 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
2 years, 1 month 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.

- 1 year, 9 months ago

- 1 year, 9 months ago

Where are the new problems.

- 1 year, 10 months ago

Try the new problems

- 2 years ago

i HAve added newproblems. TRy them.

- 2 years ago

Any more problems?

- 2 years ago

- 2 years, 1 month ago

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

- 2 years, 1 month ago

Is RMO DELHI results out?

At which website?

- 2 years ago

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

- 2 years ago

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

- 2 years ago

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

- 2 years ago

@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.

- 2 years, 1 month ago