Organisation : Indian Statistical Institute Bangalore
Exam : Regional Mathematical Olympiad, Karnataka
Document Type : Sample Question Paper
Category or Subject : Mathematics
Year : 2015
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Website : http://www.isibang.ac.in/~statmath/olympiad/
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Paper – 2 : https://www.pdfquestion.in/uploads/9550-Paper-2.pdf
Paper – 3 : https://www.pdfquestion.in/uploads/9550-Paper-3.pdf
Paper – 4 : https://www.pdfquestion.in/uploads/9550-Paper-4.pdf
Indian Statistical Institute RMO Question Paper
Time: 3 hours
Maximum marks: 102.
Related / Similar Question Paper : MTA PRMO 2019 Question Paper
a. Calculators (in any form) and protractors are not allowed.
b. Rulers and compasses are allowed.
c. Answer all the questions.
d. All questions carry equal marks.
e. Answer to each question should start on a new page. Clearly indicate the question number.
Paper – 1
1. In a cyclic quadrilateral ABCD, let the diagonals AC and BD intersect at X. Let the circumcircles of triangles AXD and BXC intersect again at Y . If X is the incentre of triangle ABY , show that \CAD = 90.
2. Let P1(x) = x2 +a1x+b1 and P2(x) = x2 +a2x+b2 be two quadratic poly- nomials with integer coecients. Suppose a1 6= a2 and there exist integers m 6= n such that P1(m) = P2(n), P2(m) = P1(n). Prove that a1??a2 is even.
3. Find all fractions which can be written simultaneously in the forms 7k – 5/5k – 3 and 6l – 1/4l – 3 , for some integers k; l.
4. Suppose 28 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?
Paper – 2
1. Let ABC be a triangle. Let B0 and C0 denote respectively the re ction of B and C in the internal angle bisector of \A. Show that the triangles ABC and AB0C0 have the same incentre.
2. Let P(x) = x2 + ax + b be a quadratic polynomial with real coefficents. Suppose there are real numbers s 6= t such that P(s) = t and P(t) = s. Prove that b st is a root of the equation x2 + ax + b – st = 0.
3. Find all integers a; b; c such that a2 = bc + 1; b2 = ca + 1:
4. Suppose 32 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?
Paper – 3
1. Suppose 36 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?
2. Show that there are infinitely many positive real numbers a which are not integers such that a(a??3fag) is an integer. (Here fag denotes the fractional part of a. For example f1:5g = 0:5; f??3:4g = 0:6.)
3. Show that there are infinitely many triples (x; y; z) of integers such that x3 + y4 = z31.
Paper – 4
1. Let ABC be a triangle. Let B0 denote the reaction of B in the internal angle bisector ` of \A. Show that the circumference of the triangle CB0I lies on the line `, where I is the in centre of ABC.
2. Let P(x) = x2 + ax + b be a quadratic polynomial where a is real and b is rational. Suppose P(0)2, P(1)2, P(2)2 are integers. Prove that a and b are integers.
3. Suppose 40 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?
4. Two circles and intersect at two distinct points A and B. A line through B intersects and again at C and D, respectively. Suppose that CA = CD. Show that the centre of lies on .
Important Information concerning the Mathematics Olympiad Programme :
** The National Board for Higher Mathematics (NBHM) which is in charge of the mathematical olympiad activity, has entrusted its implementation to the Homi Bhabha Centre for Science Education (HBCSE) to be carried out in coordination with the MO-Cell and the National Coordinator appointed by NBHM, supervised by a Committee constituted by NBHM.
** All major policy decisions with regard to the Mathematical Olympiads programme are made with the concurrence of NBHM. The following MAJOR changes are announced this year.
** A pre-Regional Mathematical Olympiad (PRMO) exam will be held on August 20, 2017 at different centres all over India. In order to be eligible for the RMO it is mandatory that students qualify through this centrally administered PRMO. No other independently administered examinations will be recognized from this year onwards.
** 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.
** The PRMO will be a machine-correctable test of 30 questions. Each question has an answer which is a number with one or two digits. Sample PRMO questions and Sample OMR sheet showing marked answers can be downloaded from here.
** The PRMO exam will be organized this year by IAPT (the Indian Association of Physics Teachers), the same association that also organizes the National Standards Examination, which is the first step for participation in the International Physics, Chemistry, Biology, Astronomy and Junior Science olympiads. The website for PRMO is available at iapt.org.in/
** A link to a portal is available at the IAPT webpage where schools can register as centres; this portal will be open from May 20 to June 20, 2017. Any school with at least 5 registrations can register on the portal as a “registered centre”. However, the list of approved exam centres will be issued by IAPT. Students who register through a given school which is a registered centre may be re-assigned to another nearby exam centre.
** All Kendriya Vidyalaya, Jawahar Navodaya Vidyalaya and Atomic Energy Central Schools may register as registered centres regardless of the number of students.
** The list of registered centres will be available on the IAPT website from June 20, 2017.
** Students can approach one of the approved registered centres and through the centre register for the PRMO exam between June 20 and July 25, 2017 for a fee of Rs. 200. Kendriya Vidyalaya Sangathan has made its schools available as centres free of cost; for KV students there is a provision to register with payment of a fee of Rs. 100.
** A carbon copy of the answer sheet (OMR) will be given to the student after the exam; students can check their answers as against the answers which will be uploaded on the HBCSE website