These problems are my submissions to Xuming's Geometry group.These problems are taken from previous RMO papers so that some of my friends who are preparing for RMO will be benefited by this discussion and thereby prepare and improve themselves :)

Q1) Let \(AL\) and \(BK\) be the angle bisectors in a non-isosceles triangle \(ABC,\) where \(L\) lies on \(BC\) and \(K\) lies on \(AC.\) The perpendicular bisector of \(BK\) intersects the line \(AL\) at \(M\). Point \(N\) lies on the line \(BK\) such that \(LN\) is parallel to \(MK.\) Prove that \(LN=NA.\)

Q2) Let \(ABC\) be a triangle and let \(BB_1,CC_1\) be respectively the bisectors of \(\angle{B},\angle{C}\) with \(B_1\) on \(AC\) and \(C_1\) on \(AB\), Let \(E,F\) be the feet of perpendiculars drawn from \(A\) onto \(BB_1,CC_1\) respectively. Suppose \(D\) is the point at which the incircle of \(ABC\) touches \(AB\). Prove that \(AD=EF\)

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Since we know that in a triangle the perpendicular bisector of a side and angular bisector of the angle opposite to this side meet at a point which is concyclic with the vertices of the triangle. Here, angular bisector of \(\angle BAK\) and perpendicular bisector of \( BK\) meet at \(M\). So, \(A\), \(B\), \(M\) and \(K\) are concyclic. So, \(\angle AMK = \angle ABK\).

But, \(LN || MK\). So, \(\angle ALN = \angle AMK = \angle ABK = \angle ABN\). So, \(A\), \(B\), \(L\) and \(N\) are cyclic. It implies that \(\angle NAL = \angle NBL = \angle NBA = \angle ALN\). So, \(\angle ALN = \angle ANL\), which implies that \(NA = NL\). – Surya Prakash · 1 year, 10 months ago

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– Nihar Mahajan · 1 year, 10 months ago

Yeah , this is the standard solution. At first the problem looks tough. But as you decipher the configuration , you crack it like a left hand's play (provided that you are right handed :P)Log in to reply

– Brilliant Member · 11 months ago

what rank did you get in jee 2016 ?Log in to reply

@Surya Prakash @Mehul Arora @Agnishom Chattopadhyay @Alan Yan @Shivam Jadhav @Ambuj Shrivastava @Swapnil Das @Sharky Kesa @Saarthak Marathe @Kushagra Sahni @naitik sanghavi If you are interested in this , please make your proposals soon and await the geometry challenges,discussion!

@Calvin Lin Sorry , for mass tagging. But I think this will encourage the participation :) – Nihar Mahajan · 1 year, 10 months ago

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– Shivam Jadhav · 1 year, 10 months ago

I made mineLog in to reply

@Trevor Arashiro @Satyajit Mohanty @Vishnu Bhagyanath @Karthik Venkata and others too. – Nihar Mahajan · 1 year, 10 months ago

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– Mehul Arora · 1 year, 10 months ago

:3 :3 :3 :3 :3 :3 :3 :3 :3Log in to reply

modern algebra text – Gebretsadkan Gebereyohannes · 1 year, 8 months ago

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Here's mine @Nihar Mahajan - Surya Prakash's Proposals – Surya Prakash · 1 year, 10 months ago

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@Calvin Lin @Xuming Liang Here's my submission! I hope that this marvelous group forms and accelerates soon :) – Nihar Mahajan · 1 year, 10 months ago

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