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Explore the Back-StoryThis 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 ${gcd}(a, b)={gcd}(b, c)={gcd}(c, d)={gcd}(d, a)=1$.

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 ${gcd}(a, b)={gcd}(b, c)={gcd}(c, d)={gcd}(d, a)=1$.

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