# RMO 2019 solutions with sequential hints

## Problems in Regional Math Olympiad 2019 (this is being updated continually… stay tuned)

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

**2.** Let ABC be a triangle with circumcircle \( \Omega \) and let G be the centroid of the triangle ABC. Extend AG, BG, and CG to meet \( \Omega \) again at \( A_1, B_1 \) and \(C_1\) respectively. Suppose \( \angle BAC = \angle A_1B_1C_1 , \angle ABC = \angle A_1 C_1 B_1 \) and \( \angle ACB = \angle B_1 A_1 C_1 \). Prove that ABC and \( A_1B_1C_1 \) are equilateral triangles.

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

**4.** Consider the following \( 3 \times 2 \) array formed by the numbers 1, 2, 3, 4, 5, 6:

\( \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{33} \end{bmatrix} = \begin{bmatrix} 1 & 6 \\ 2 & 5 \\ 3 & 4 \end{bmatrix} \)

Observe that all row sums are equal, but the sum of the squares in 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, …, 3k such that

$$ \sum_{j=1}^k a_{1j} = \sum_{j=1}^k a_{2j} = \sum_{j=1}^k a_{3j}, \sum_{j=1}^k (a_{1j})^2 = \sum_{j=1}^k (a_{2j})^2 = \sum_{j=1}^k (a_{3j})^2 $$

**5.** In an acute angled triangle ABC, let H be the orthocenter, and let D, E, F be the feet of the 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 feet of altitudes from L, N on to the line DF. Prove that XM is perpendicular to MY.

**6.** Suppose 91 distinct positive integers greater than 1 are given such that there are atleast 456 pairs among them which are relatively prime. So that one can find four integers a, b, c, d among them such that gcd (a, b) = gcd (b, c) = gcd (c, a) = 1

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# Faculty panel for Math Olympiad

#### SRIJIT MUKHERJEE

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Faculty, Admin at Cheenta

#### Ishan Sengupta

Faculty at Cheenta

#### A.R. Sricharan

Faculty, Cheenta