RMO2018 Tamil Nadu Solutions and Problems

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

Also see
Advanced Math Olympiad Program

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 $
- Sequential Hints and Discussion
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.
- Sequential Hints and Discussion
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 $.
- Sequential Hints and Discussion
Suppose 100 points in the plane are colored using two colors, red and white, such that each red point is the center of a circle passing through at least three white points. What is the least possible number of white points?
In a cyclic quadrilateral ABCD with circumcenter O, the diagonals AC and BD intersect at X. Let the circumcircles of triangles AXD and BXC intersect at Y. Let the circumcircles of triangle AXB and CXD intersect at Z. If O lies inside ABCD and if the points O, X, Y, Z are all distinct, prove that O, X, Y, Z lie in a circle.
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.

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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 $
- Sequential Hints and Discussion
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.
- Sequential Hints and Discussion
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 $.
- Sequential Hints and Discussion
Suppose 100 points in the plane are colored using two colors, red and white, such that each red point is the center of a circle passing through at least three white points. What is the least possible number of white points?
In a cyclic quadrilateral ABCD with circumcenter O, the diagonals AC and BD intersect at X. Let the circumcircles of triangles AXD and BXC intersect at Y. Let the circumcircles of triangle AXB and CXD intersect at Z. If O lies inside ABCD and if the points O, X, Y, Z are all distinct, prove that O, X, Y, Z lie in a circle.
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.