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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

## Squares and Triangles | AIME I, 2008 | Question 2

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2008 based on Squares and Triangles.

## Percentage Problem | AIME I, 2008 | Question 1

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2008 based on Percentage. you may use sequential hints.

## Smallest Positive Integer | PRMO 2019 | Question 14

Try this beautiful problem from the Pre-RMO, 2019 based on Smallest Positive Integer. You may use sequential hints to solve the problem.

## Triangles and Internal bisectors | PRMO 2019 | Question 10

Try this beautiful problem from the Pre-RMO, 2019 based on Triangles and Internal bisectors. You may use sequential hints to solve the problem.

## Angles in a circle | PRMO-2018 | Problem 80

Try this beautiful problem from PRMO, 2018 based on Angles in a circle. You may use sequential hints to solve the problem.

## Circles and Triangles | AIME I, 2012 | Question 13

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2012 based on Circles and triangles.

## Complex Numbers and Triangles | AIME I, 2012 | Question 14

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2012 based on Complex Numbers and Triangles.

## Digit Problem from SMO, 2012 | Problem 14

Try this beautiful problem from Singapore Mathematics Olympiad, SMO, 2012 based on digit. You may use sequential hints to solve the problem.

## Problem on Semicircle | AMC 8, 2013 | Problem 20

Try this beautiful problem from AMC-8, 2013, (Problem-20) based on area of semi circle.You may use sequential hints to solve the problem.

## Radius of semicircle | AMC-8, 2013 | Problem 23

Try this beautiful problem from Geometry: Radius of semicircle from AMC-8, 2013, Problem-23. You may use sequential hints to solve the problem.

# some testimonials.

## Jayanta Majumdar, Glasgow, UK

"We contacted Cheenta because our son, Sambuddha (a.k.a. Sam), seemed to have something of a gift in mathematical/logical thinking, and his school curriculum math was way too easy and boring for him. We were overjoyed when Mr Ashani Dasgupta administered an admission test and accepted Sam as a one-to-one student at Cheenta. Ever since it has been an excellent experience and we have nothing but praise for Mr Dasgupta. His enthusiasm for mathematics is infectious, and admirable is the amount of energy and thought he puts into each lesson. He covers a wide range of mathematical topics, and every lesson is packed with insights and methods. We are extremely pleased with the difference he has been making. Under his tutelage Sam has secured several gold awards from the UK Mathematics Trust (UKMT) and Scottish Mathematical Council (SMC). Recently Sam received a book award from the UKMT and got invited to masterclass sessions also organised by the UKMT. Mr Dasgupta's tutoring was crucial for these achievements. We think Cheenta is rendering an excellent service to humanity by identifying young mathematical minds and nurturing them towards becoming inspired mathematicians of the future."

## 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."