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
.