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

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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

Sankhadip Chakraborty

Faculty, Admin 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

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

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

Functional Equation Problem from SMO, 2018 – Question 35

Try this problem from Singapore Mathematics Olympiad, SMO, 2018 based on Functional Equation. You may use sequential hints if required.

Sequence and greatest integer | AIME I, 2000 | Question 11

Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2000 based on Sequence and the greatest integer.

Arithmetic sequence | AMC 10A, 2015 | Problem 7

Try this beautiful problem from Algebra: Arithmetic sequence from AMC 10A, 2015, Problem. You may use sequential hints to solve the problem.

Series and sum | AIME I, 1999 | Question 11

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1999 based on Series and sum.

Inscribed circle and perimeter | AIME I, 1999 | Question 12

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2011 based on Rectangles and sides.

Problem based on Cylinder | AMC 10A, 2015 | Question 9

Try this beautiful problem from Mensuration: Problem based on Cylinder from AMC 10A, 2015. You may use sequential hints to solve the problem.

Cubic Equation | AMC-10A, 2010 | Problem 21

Try this beautiful problem from Algebra, based on the Cubic Equation problem from AMC-10A, 2010. You may use sequential hints to solve the problem.

Median of numbers | AMC-10A, 2020 | Problem 11

Try this beautiful problem from Geometry based on Median of numbers from AMC 10A, 2020. You may use sequential hints to solve the problem.

LCM and Integers | AIME I, 1998 | Question 1

Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 1998, Problem 1, based on LCM and Integers.

Problem on Fraction | AMC 10A, 2015 | Question 15

Try this beautiful Problem on Fraction from Algebra from AMC 10A, 2015. You may use sequential hints to solve the problem.

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