RMO2018 Tamil Nadu Solutions (Sequential Hints) and related discussion follows. This is a work in progress.

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1. Let ABC be an acute-angled triangle and let D be an interior point of the line segment BC. Let the circumcircle of triangle ACD intersect AB at E (E between A and B) and let the circumcircle of triangle ABD intersect AC at F (F between A and C). Let O be the circumcenter triangle AEF. Prove that OD bisects $$\angle EDF$$
2. Find the set of all real values of a for which the real polynomial equation $$P(x) = x^2 – 2ax + b = 0$$ has real roots, given that $$P(0) \cdot P(1) \cdot P(2) \neq 0$$ and $$P(0), P(1), P(2)$$ form a geometric progression.
3. Show that there are infinitely many 4-tuples (a, b, c, d) of natural numbers such that $$a^3 + b^4 + c^5 = d^7$$.
6. Define a sequence $$< a_n >_{n \geq 1 }$$ of real numbers by $$a_1 = 2, a_{n+1} = \frac {a_n^2 +1 } {2} , \text{for} n \geq 1 .$$ Prove that $$\displaystyle {\sum_{j=1}^N \frac {1}{a_j +1} < 1 }$$ for every natural number N.