# RMO 2019 solutions with sequential hints

Regional Mathematics Olympiad, India (RMO) 2019… try the problems. We give sequential hints leading up to complete solution. ## 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

# Rest of india

## What is coming up?

Seminar on how to solve Math Olympiad Problems by Anushka Aggarwal

# Faculty panel for Math Olympiad #### SRIJIT MUKHERJEE

Director, Faculty Cheenta

Srijit Mukherjee is a B.Stat from Indian Statistical Institute. He is pursuing M.Stat from I.S.I. He is a director and faculty at Cheenta. Sankhadip Chakraborty is an INMO awardee. He has a B.Sc. in Mathematics from CMI and is pursuing Ph.D. at IMPA, Brazil. #### Ishan Sengupta

Faculty at Cheenta

Ishan Sengupta is pursuing B.Stat from Indian Statistical Institute, Kolkata. He is a faculty at Cheenta. #### A.R. Sricharan

Faculty, Cheenta

A.R. Sricharan is a B.Sc. in Mathematics from Chennai Mathematical Institute. He is pursuing M.Sc. from CMI and is a faculty at Cheenta

# Try some sequential hints

## Geometry of AM GM Inequality

AM GM Inequality has a geometric interpretation. Watch the video discussion on it and try some hint problems to sharpen your skills.

## Geometry of Cauchy Schwarz Inequality

Cauchy Schwarz Inequality is a powerful tool in Algebra. However it also has a geometric meaning. We provide video and problem sequence to explore that.

## AMC 10A Year 2014 Problem 20 Sequential Hints

A challenging number theory problem. Here the main idea is the visualization of a pattern of which appeared in the multiplication.

## RMO 2019 Maharashtra and Goa Problem 2 Geometry

Understand the problemGiven a circle $latex \Gamma$, let $latex P$ be a point in its interior, and let $latex l$ be a line passing through $latex P$. Construct with proof using a ruler and compass, all circles which pass through $latex P$, are tangent to \$latex...

## RMO 2019 (Maharashtra Goa) Adding GCDs

Can you add GCDs? This problem from RMO 2019 (Maharashtra region) has a beautiful solution. We also give some bonus questions for you to try.

## Number Theory, Ireland MO 2018, Problem 9

This problem in number theory is an elegant applications of the ideas of quadratic and cubic residues of a number. Try with our sequential hints.

## Number Theory, France IMO TST 2012, Problem 3

This problem is an advanced number theory problem using the ideas of lifting the exponents. Try with our sequential hints.

## Combinatorics – AMC 10A 2008 Problem 23 Sequential Hints

AMC 10A 2008, Problem 23 needed a clever trick of set theory and combinations. See the solution with sequential hints for a subset theory-based problem

## Algebra, Austria MO 2016, Problem 4

This algebra problem is an elegant application of culminating the ideas of polynomials to give a simple proof of an inequality. Try with our sequential hints.

## Number Theory, Cyprus IMO TST 2018, Problem 1

This problem is a beautiful and simple application of the ideas of inequality and bounds in number theory. Try with our sequential hints.

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