Organisation : Homi Bhabha Centre For Science Education
Exam : Pre-Regional Mathematical Olympiad
Document Type : Previous Question Paper
Year : 2015
Subject : Mathematics
Website : https://olympiads.hbcse.tifr.res.in/
Download Model Question Paper :
Set A : https://www.pdfquestion.in/uploads/9136-seta.pdf
Set B : https://www.pdfquestion.in/uploads/9136-setb.pdf
Pre-Regional Mathematical Olympiad Previous Question Paper :
Date : October 4, 2015
Question Paper Set: A
1. There are 20 questions in this question paper. Each question carries 5 marks.
2. Answer all questions.
3. Time allotted: 2.5 hours.
Related : Homi Bhabha Centre For Science Education Regional Mathematical Olympiad Previous Question Paper : www.pdfquestion.in/9132.html
Questions :
1. A man walks a certain distance and rides back in 33/4 hours; he could ride both ways in 2 1/2hours. How many hours would it take him to walk both ways? [5]
2. Positive integers a and b are such that a + b = a=b + b=a. What is the value of a2 + b2? [2]
3. The equations x2 – 4x + k = 0 and x2 + kx – 4 = 0, where k is a real number, have exactly one common root. What is the value of k? [3]
4. Let P(x) be a non-zero polynomial with integer coecients. If P(n) is divisible by n for each positive integer n, what is the value of P(0)? [0]
5. How many line segments have both their endpoints located at the vertices of a given cube? [28]
6. Let E(n) denote the sum of the even digits of n. For example, E(1243) = 2 + 4 = 6. What is the value of E(1) + E(2) + E(3) + + E(100)? [400]
7. How many two-digit positive integers N have the property that the sum of N and the number obtained by reversing the order of the digits of N is a perfect square? [8]
8. The gure below shows a broken piece of a circular plate made of glass.C is the midpoint of AB, and D is the midpoint of arc AB. Given that AB = 24 cm and CD = 6 cm, what is the radius of the plate in centimetres? (The gure is not drawn to scale.) [15]
9. A 2 3 rectangle and a 3 4 rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square? [25]
10. What is the greatest possible perimeter of a right-angled triangle with integer side lengths if one of the sides has length 12 ? [84]
11. Let a, b, and c be real numbers such that a – 7b + 8c = 4 and 8a + 4b – c = 7. What is the value of a2 – b2 + c2? [1]
12. In rectangle ABCD, AB = 8 and BC = 20. Let P be a point on AD such that 6 BPC = 90. If r1, r2, r3 are the radii of the incircles of triangles APB, BPC and CPD, what is the value of r1 + r2 + r3? [8]
13. At a party, each man danced with exactly four women and each woman danced with exactly three men. Nine men attended the party. How many women attended the party? [12]
14. If 3x + 2y = 985 and 3x – 2y = 473, what is the value of xy? [48]
15. Let n be the largest integer that is the product of exactly 3 distinct prime numbers, x, y and 10x + y, where x and y are digits. What is the sum of the digits of n? [12]
Question Paper Set : B
** There are 20 questions in this question paper.
** Each question carries 5 marks.
** Answer all questions.
** Time allotted: 2.5 hours.
QUESTIONS :
1. A man walks a certain distance and rides back in 3 /3 4hours; he could ride both ways in 2 1/2 hours. How many hours would it take him to walk both ways? [5]
2. The equations x2 – 4x + k = 0 and x2 + kx – 4 = 0, where k is a real number, have exactly one common root. What is the value of k? [3]
3. Positive integers a and b are such that a + b = a=b + b=a. What is the value of a2 + b2 [2]
4. How many line segments have both their endpoints located at the vertices of a given cube? [28]
5. Let P(x) be a non-zero polynomial with integer coecients. If P(n) is divisible by n for each positive integer n, what is the value of P(n)[0]
6. How many two-digit positive integers N have the property that the sum of N and the number obtained by reversing the order of the digits of N is a perfect square? [8]
7. Let E(n) denote the sum of the even digits of n. For example, E(1243) = 2 + 4 = 6. What is the value of E(1) + E(2) + E(3) + + E(100)? [400]