INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More

Bose Olympiad Project Round is Live now. Learn More

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

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

JOIN TRIAL
Google