## 14 Oct Regional Mathematical Olympiad India 2017 Questions

The Regional Mathematical Olympiad (RMO) is organized every year in India throughout the country as a preliminary screening for the Indian National Mathematical Olympiad (INMO). This year it was held on 8th October, 2017. The questions of this year's RMO can be found below.

(All questions carry equal marks, maximum possible score is 102.)

- Let be a given angle less than and let be an interior point of the angular region determined by the angle . Show, with proof, how to construct using only ruler and compass, a line segment passing through such that lies on the ray and lies on the , and .
- Show that the equation has no solutions in integers for .
- Let and be two polynomials with real coefficients such that for all real . Find all the real roots of .
- Consider unit squares in the -plane centered at point with integer coordinates, . It is required to colour each unit square in such a way that whenever and , the three squares with centres at have distinct colours. What is the least possible number of colours needed?
- Let be a circle with chord which is not a diameter. Let be a circle on one side of such that it is tangent to at and internally tangent to at . Likewise, let be a circle on the other side of such that it is tangent to at and internally tangent to at $. Suppose the line intersects at and the line intersects at . Prove that is a diameter of .
- Let be real numbers, each greater than . Prove that .

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#### Manjil Saikia

Managing Editor of Gonit Sora and Research Fellow, Faculty of Mathematics, University of Vienna, Austria.

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