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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 x is a nonzero real number such that both x^{5} and 20 x+\frac{19}{x} are rational numbers. Prove that x is a rational number.

RMO 2019, Problem 2:

Let A B C be a triangle with circumcircle \Omega and let G be the centroid of triangle A B C . Extend A G B G and C G to meet the circle \Omega again in A_{1}, B_{1} and C_{1}, respectively. Suppose \angle B A C=\angle A_{1} B_{1} C_{1} \angle A B C=\angle A_{1} C_{1} B_{1} and \angle A C B=\angle B_{1} A_{1} C_{1} . Prove that A B C and A_{1} B_{1} C_{1} are equilateral triangles.

RMO 2019, Problem 3:

Let a, b, c be positive real numbers such that a+b+c=1. Prove that

    \[\frac{a}{a^{2}+b^{3}+c^{3}}+\frac{b}{b^{2}+c^{3}+a^{3}}+\frac{c}{c^{2}+a^{3}+b^{3}} \leq \frac{1}{5 a b c}\]

RMO 2019, Problem 4:

Consider the following 3 \times 2 array formed by using the numbers 1,2,3,4,5,6:

    \[\left(\begin{array}{ll}a_{11} & a_{12} \a_{21} & a_{22} \a_{31} & a_{32}\end{array}\right)=\left(\begin{array}{ll}1 & 6 \2 & 5 \3 & 4\end{array}\right)\]

RMO 2019, Problem 5:

In an acute angled triangle A B C, let H be the orthocenter, and let D, E, F be the feet of altitudes from A, B, C to the opposite sides, respectively. Let L, M, N be midpoints of segments A H, E F, B C, respectively. Let X, Y be feet of altitudes from L, N on to the line D F. Prove that X M is perpendicular to M Y.

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 a, b, c, d among them such that {gcd}(a, b)={gcd}(b, c)={gcd}(c, d)={gcd}(d, a)=1.

Some Useful Links:

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 x is a nonzero real number such that both x^{5} and 20 x+\frac{19}{x} are rational numbers. Prove that x is a rational number.

RMO 2019, Problem 2:

Let A B C be a triangle with circumcircle \Omega and let G be the centroid of triangle A B C . Extend A G B G and C G to meet the circle \Omega again in A_{1}, B_{1} and C_{1}, respectively. Suppose \angle B A C=\angle A_{1} B_{1} C_{1} \angle A B C=\angle A_{1} C_{1} B_{1} and \angle A C B=\angle B_{1} A_{1} C_{1} . Prove that A B C and A_{1} B_{1} C_{1} are equilateral triangles.

RMO 2019, Problem 3:

Let a, b, c be positive real numbers such that a+b+c=1. Prove that

    \[\frac{a}{a^{2}+b^{3}+c^{3}}+\frac{b}{b^{2}+c^{3}+a^{3}}+\frac{c}{c^{2}+a^{3}+b^{3}} \leq \frac{1}{5 a b c}\]

RMO 2019, Problem 4:

Consider the following 3 \times 2 array formed by using the numbers 1,2,3,4,5,6:

    \[\left(\begin{array}{ll}a_{11} & a_{12} \a_{21} & a_{22} \a_{31} & a_{32}\end{array}\right)=\left(\begin{array}{ll}1 & 6 \2 & 5 \3 & 4\end{array}\right)\]

RMO 2019, Problem 5:

In an acute angled triangle A B C, let H be the orthocenter, and let D, E, F be the feet of altitudes from A, B, C to the opposite sides, respectively. Let L, M, N be midpoints of segments A H, E F, B C, respectively. Let X, Y be feet of altitudes from L, N on to the line D F. Prove that X M is perpendicular to M Y.

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 a, b, c, d among them such that {gcd}(a, b)={gcd}(b, c)={gcd}(c, d)={gcd}(d, a)=1.

Some Useful Links:

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