Organisation : Homi Bhabha Centre For Science Education
Exam : Regional Mathematical Olympiad
Document Type : Previous Question Paper
Year : 2015
Website : https://olympiads.hbcse.tifr.res.in/
Download Model Question Paper :
CRMO 2015 Paper 1 : https://www.pdfquestion.in/uploads/9132-crmo-paper1.pdf
CRMO 2015 Paper 2 : https://www.pdfquestion.in/uploads/9132-crmo-paper-2.pdf
CRMO 2015 Paper 3 : https://www.pdfquestion.in/uploads/9132-crmo-paper-3.pdf
CRMO 2015 Paper 4 : https://www.pdfquestion.in/uploads/9132-crmo-paper-4.pdf
RMO 2015 (Mumbai region) : https://www.pdfquestion.in/uploads/9132-crmo-paper-5.pdf
Regional Mathematical Olympiad Previous Question Paper :
Time: 3 hours
Year : 2015 Paper – I
Date : December 06, 2015
Related : Homi Bhabha Centre For Science Education Indian National Mathematical Olympiad INMO Previous Question Paper : www.pdfquestion.in/9127.html
Instructions:
** Calculators (in any form) and protractors are not allowed.
** Rulers and compasses are allowed.
** Answer all the questions.
** All questions carry equal marks. Maximum marks: 102.
** Answer to each question should start on a new page. Clearly indicate the question number.
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 forms7k – 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?
5. Let ABC be a right triangle with \B = 90. Let E and F be respectively the mid-points of AB and AC. Suppose the incentre I of triangle ABC lies on the circumcircle of triangle AEF. Find the ratio BC=AB.
6. Find all real numbers a such that 3 < a < 4 and a(a -3fag) is an integer. (Here fag denotes the fractional part of a. For example 1.5 = 0:5; -3.4 = 0:6.)
Regional Mathematical Olympiad-2015 :
Time: 3 hours
Instructions:
** Calculators (in any form) and protractors are not allowed.
** Rulers and compasses are allowed.
** Answer all the questions.
** All questions carry equal marks. Maximum marks: 102.
** Answer to each question should start on a new page.
** Clearly indicate the question number.
1. Let ABC be a triangle. Let B and C denote respectively the re ection 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 coecients. 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?
5. Two circles T and M in the plane intersect at two distinct points A and B, and the centre of M lies on T. Let points C and D be on T and M, respectively, such that C;B and D are collinear. Let point E on be such that DE is parallel to AC. Show that AE = AB.
6. Find all real numbers a such that 4 < a < 5 and a(a -3 {a} is an integer. (Here {a} denotes the fractional part of a. For example {1:5} = 0:5; {-3,4}= 0:6.)
Indian National Mathematical Olympiad-(INMO) :
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.