RMO 2019

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

Let $ABC$ be a triangle with circumcircle $\Omega$ and let $G$ be the centroid of triangle $ABC$ . Extend $AG, BG$ and $CG$ to meet the circle $\Omega$ again in $A_1, B_1$ and $C_1$ . Suppose $\angle BAC = \angle A_1 B_1 C_1, \angle ABC = \angle A_1 C_1 B_1$ and $\angle ACB = B_1 A_1 C_1$ . Prove that $ABC$ and $A_1 B_1 C_1$ are equilateral triangles.

Let $a,b,c$ be positive real numbers such that $a+b+c=1$ . Prove that $$\frac{a}{a^2+b^3+c^3}+\frac{b}{b^2+a^3+c^3}+\frac{c}{c^2+a^3+b^3}\leq\frac{1}{5abc}$$

Consider the following $3\times 2$ array formed by using the numbers $1,2,3,4,5,6$ , $$\begin{pmatrix} a_{11}& a_{12}a_{21}& a_{22} a_{31}& a_{32}\end{pmatrix}=\begin{pmatrix}1& 62& 5 3& 4\end{pmatrix}.$$ Observe that all row sums are equal, but the sum of the square of the squares is not the same for each row. Extend the above array to a $3\times k$ array $(a_{ij})_{3\times k}$ for a suitable $k$ , adding more columns, using the numbers $7,8,9,\dots ,3k$ such that $$\sum_{j=1}^k a_{1j}=\sum_{j=1}^k a_{2j}=\sum_{j=1}^k a_{3j}~~\text{and}~~\sum_{j=1}^k (a_{1j})^2=\sum_{j=1}^k (a_{2j})^2=\sum_{j=1}^k (a_{3j})^2$$

In an acute angled triangle $ABC$ , 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 the midpoints of the segments $AH, EF, BC$ respectively. Let $X,Y$ be the feet of altitudes from $L,N$ on to the line $DF$ respectively. Prove that $XM$ is perpendicular to $MY$ .

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