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# Indian Statistical Institute Bangalore Regional Mathematical Olympiad Model Question Paper : isibang.ac.in

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** Organisation **: Indian Statistical Institute Bangalore

**: Regional Mathematical Olympiad, Karnataka**

__Exam__**: Sample Question Paper**

__Document Type__**: Mathematics**

__Category or Subject__**: 2015**

__Year__** Website **: http://www.isibang.ac.in/~statmath/olympiad/

**:**

__Download Model Question Paper__**: http://www.pdfquestion.in/uploads/9550-Paper-1.pdf**

__Paper – 1__**: http://www.pdfquestion.in/uploads/9550-Paper-2.pdf**

__Paper – 2__**: http://www.pdfquestion.in/uploads/9550-Paper-3.pdf**

__Paper – 3__**: http://www.pdfquestion.in/uploads/9550-Paper-4.pdf**

__Paper – 4__## RMO Model Question Paper :

** Time**: 3 hours

**: 102.**

__Maximum marks__**:**

__Instructions__a. Calculators (in any form) and protractors are not allowed.

b. Rulers and compasses are allowed.

Related: RTSE Rationalist Talent Search Exam Class II Sample Question Paper : www.pdfquestion.in/9393.html

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 innitely 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 innitely many triples (x; y; z) of integers such that x3 + y4 = z31.

** Paper – 4 **:

1. Let ABC be a triangle. Let B0 denote the reection of B in the internal angle bisector ` of \A. Show that the circumcentre of the triangle CB0I lies on the line `, where I is the incentre 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 .