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# RMO 2019 Problems and Solutions | Previous Year Questions This post will provide you all the RMO (Regional Mathematics Olympiad) 2019 problems and solutions. You may find some solutions with hints too.

RMO 2019, Problem 1:

Suppose is a nonzero real number such that both and are rational numbers. Prove that is a rational number.

RMO 2019, Problem 2:

Let be a triangle with circumcircle and let be the centroid of triangle Extend  and to meet the circle again in and , respectively. Suppose  and Prove that and are equilateral triangles.

RMO 2019, Problem 3:

Let be positive real numbers such that . Prove that RMO 2019, Problem 4:

Consider the following array formed by using the numbers 1,2,3,4,5,6: RMO 2019, Problem 5:

In an acute angled triangle let be the orthocenter, and let be the feet of altitudes from to the opposite sides, respectively. Let be midpoints of segments respectively. Let be feet of altitudes from on to the line . Prove that is perpendicular to .

RMO 2019, Problem 6:

Suppose 91 distinct positive integers greater than 1 are given such that there are at least 456 pairs among them which are relatively prime. Show that one can find four integers among them such that .

This post will provide you all the RMO (Regional Mathematics Olympiad) 2019 problems and solutions. You may find some solutions with hints too.

RMO 2019, Problem 1:

Suppose is a nonzero real number such that both and are rational numbers. Prove that is a rational number.

RMO 2019, Problem 2:

Let be a triangle with circumcircle and let be the centroid of triangle Extend  and to meet the circle again in and , respectively. Suppose  and Prove that and are equilateral triangles.

RMO 2019, Problem 3:

Let be positive real numbers such that . Prove that RMO 2019, Problem 4:

Consider the following array formed by using the numbers 1,2,3,4,5,6: RMO 2019, Problem 5:

In an acute angled triangle let be the orthocenter, and let be the feet of altitudes from to the opposite sides, respectively. Let be midpoints of segments respectively. Let be feet of altitudes from on to the line . Prove that is perpendicular to .

RMO 2019, Problem 6:

Suppose 91 distinct positive integers greater than 1 are given such that there are at least 456 pairs among them which are relatively prime. Show that one can find four integers among them such that .

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