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# 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

## 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 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.

## Number Theory, South Africa 2019, Problem 6

This problem in number theory is an elegant applciations of the modulo technique used in the diophantine equations. Try with our sequential hints

## AMC 10A Year 2005 Problem 21 Sequential Hints

American Mathematical Olympiard 10A Problem 21 Solutuon. The main idea here in this problem is to use some formulae of induction and finding factors.

## Number Theory, Korea Junior MO 2015, Problem 7

This problem in number theory is an elegant application of the ideas of the proof of infinitude of primes from Korea. Try with our sequential hints.

# some testimonials.

## Jayanta Majumdar, Glasgow, UK

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## Shubhrangshu Das, Bangalore, India

"My son, Shuborno has been studying under Ashani from last one year. During this period, we have seen our son grow both intellectually and emotionally. His concepts and approach towards solving a problem has become more mature now. Not that he can solve each and every problem but he loves to think on the tough concepts. For this, all credit goes to Ashani, who is never in a hurry to solve a problem quickly. Rather he tries to slowly build the foundation of the students by discussing even minute concepts. His style of teaching is also unique combining different concepts and giving mathematics a more holistic approach. He is also very motivating and helpful. We are lucky that our son is under such good guidance. Rare to get such a dedicated teacher."

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I am impressed with their quality and professionalism. We are very thankful to Cheenta and hope to benefit from them in the coming years. I would strongly recommend them to any student who wants to learn Math by doing challenging problems, specially if they are looking for Math than what their school can offer.”

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