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