Organisation : Homi Bhabha Centre For Science Education
Exam : Indian National Mathematical Olympiad-(INMO)
Document Type : Sample Question Paper
Year : 2016
Website : https://olympiads.hbcse.tifr.res.in/
Download Model Question Paper : https://www.pdfquestion.in/uploads/9127-INMO-2016.pdf
Indian National Mathematical Olympiad Sample Question Paper :
Time: 4 hours
Date : January 17, 2016
Related : Homi Bhabha Centre For Science Education Indian National Mathematical Olympiad INMO Previous Question Papers : www.pdfquestion.in/7250.html
Instructions :
1. Calculators (in any form) and protractors are not allowed.
2. Rulers and compasses are allowed.
3. Answer all the questions. Maximum marks: 100.
4. Answer to each question should start on a new page. Clearly indicate the question number.
1. Let ABC be triangle in which AB = AC. Suppose the orthocentre of the triangle lies on the incircle. Find the ratio AB=BC.
2. For positive real numbers a; b; c, which of the following statements necessarily implies a = b = c: (I) a(b3 + c3) = b(c3 + a3) = c(a3 + b3), (II) a(a3 + b3) = b(b3 + c3) = c(c3 + a3) ? Justify your answer.
3. Let N denote the set of all natural numbers. Dene a function T : N ! N by T(2k) = k and T(2k + 1) = 2k + 2. We write T2(n) = T(T(n)) and in general Tk(n) = Tk??1(T(n)) for any k > 1.
(i) Show that for each n 2 N, there exists k such that Tk(n) = 1.
(ii) For k 2 N, let ck denote the number of elements in the set fn : Tk(n) = 1g. Prove that ck+2 = ck+1 + ck, for k 1.
4. Suppose 2016 points of the circumference of a circle are coloured red and the remaining points are coloured blue. Given any natural number n 3, prove that there is a regular n-sided polygon all of whose vertices are blue.
5. Let ABC be a right-angled triangle with \B = 90. Let D be a point on AC such that the inradii of the triangles ABD and CBD are equal. If this common value is r0 and if r is the inradius of triangle ABC, prove that 1/r=1/r+1/BD
6. Consider a nonconstant arithmetic progression a1; a2; : : : ; an; : : :. Suppose there exist relatively prime positive integers p > 1 and q > 1 such that a21 , a2 p+1 and a2q +1 are also the terms of the same arithmetic progression. Prove that the terms of the arithmetic progression are all integers.
Mathematical Olympiad syllabus :
** The syllabus for Mathematical Olympiad (regional, national and international) is pre-degree college mathematics.
** The areas covered are arithmetic of integers, geometry, quadratic equations and expressions, trigonometry, co-ordinate geometry, system of linear equations, permutations and combination, factorization of polynomial, inequalities, elementary combinatorics, probability theory and number theory, finite series and complex numbers and elementary graph theory.
** The syllabus does not include calculus and statistics.
** The major areas from which problems are given are algebra, combinatorics, geometry and number theory .
** The syllabus is in a sense spread over Class XI to Class XII levels, but the problems under each topic involve high level of difficulty and sophistication.
** The difficulty level increases from RMO to INMO to IMO.
Eligibility of Mathematical Olympiad :
** All Indian students who are born on or after August 1, 1998 and, in addition, are in Class VIII, IX, X and XI are eligible to appear for the PRMO 2017.
Stages of Mathematical Olympiad :
Stage 1:
** The first stage examination, the pre-Regional Mathematical Olympiad (PRMO) is a two and half hour examination with 30 questions. The answer to each question is either a single digit number or a two digit number and will need to be marked on a machine readable OMR response sheet. The PRMO question paper will be in English and Hindi. The syllabus may be found here. Instructions for how to enroll for PRMO may be found here.
** Sample PRMO questions and Sample OMR sheet showing marked answers can be downloaded from here.
** Queries regarding PRMO may be sent by email to prmo AT hbcse.tifr.res.in. Queries will not be replied to individually, but via the FAQ section on this website.
Stage 2:
** The second stage examination, the Regional Mathematical Olympiad (RMO) is a three hour examination with six problems. The RMOs are offered in English, Hindi and other regional languages as deemed appropriate by the respective Regional Coordinators. The syllabus may be found here. The problems under each topic involve high level of difficulty and sophistication.
Stage 3:
** The best-perfroming students from the RMO (approximately 900) qualify for the third stage – the Indian National Mathematical Olympiad (INMO). The INMO is held on the third Sunday of January at 28 centres across the country.
Stage 4:
** The top students from the INMO (approximately 35) are invited for the fourth stage, the International Mathematical Olympiad Training Camp (IMOTC) held at HBCSE during April to May. At this camp orientation is provided to students for the International Mathematical Olympiad (IMO).
** Emphasis is laid on developing conceptual foundations and problem-solving skills. Several selection tests are held during this camp. On the basis of perfromance in these tests six students are selected to represent India at the IMO. Resource persons from different institutions across the country are invited to the training camps.
Stage 5:
** The selected team undergoes a rigorous training programme for about 8-10 days at HBCSE prior to its departure for the IMO.
Stage 6:
** The Olympiad programme culminates with the participation of the students in the IMO. The students are accompanied by 4 teachers or mentors.