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Name of the Centre : Homi Bhabha Centre For Science Education
Name Of The Exam : Pre-Regional Mathematical Olympiad
Name Of The Subject : Mathematics
Document type : Sample Questions Papers
Year : 2017
Website : http://olympiads.hbcse.tifr.res.in/how-to-prepare/past-papers/
Download Sample Question Paper : https://www.pdfquestion.in/uploads/22651-PRMO.pdf
Download Answer Key : https://www.pdfquestion.in/uploads/22651-Keyprmo.pdf

HBCSE PRMO Olympiad Question Paper

Time : 09:00-12:00 (3 hours)
Maximum Marks: 75

Related : Homi Bhabha Centre For Science Education INChO Indian National Chemistry Olympiad Question Paper 2017 : www.pdfquestion.in/13653.html

Instructions

1. This booklet consists of 6 pages (excluding this sheet) and total of 6 questions.
2. This booklet is divided in two parts: Questions with Summary Answer Sheet and Detailed Answer Sheet. Write roll number at the top wherever asked.
3. The final answer to each sub-question should be neatly written in the box provided below each sub-question in the Questions & Summary Answer Sheet.

4. You are also required to show your detailed work for each question in a reasonably neat and coherent way in the Detailed Answer Sheet. You must write the relevant Question Number on each of these pages.

5. Marks will be awarded on the basis of what you write on both the Summary Answer Sheet and the Detailed Answer Sheet. Simple short answers and plots may be directly entered in the Summary Answer Sheet. Marks may be deducted for absence of detailed work in questions involving loner calculations. Strike out any rough work that you do not want to be evaluated.

6. Adequate space has been provided in the answer sheet for you to write/calculate your answers. In case you need extra space to write, you may request for additional blank sheets from the invigilator. Write your roll number on the extra sheets and get them attached to your answer sheet and indicate number of extra sheets attached at the top of this page.

7. Non-programmable scientific calculators are allowed. Mobile phones cannot be used as calculators.
8. Use blue or black pen to write answers. Pencil may be used for diagrams/graphs/sketches.
9. This entire booklet must be returned at the end of the examination.

Sample Questions

1. How many positive integers less than 1000 have the property that the sum of the digits of each such number is divisible by 7 and the number itself is divisible by 3?

2. Suppose a, b are positive real numbers such that apa + bpb = 183, apb + bpa = 182. Find 9/5 (a + b).

3. A contractor has two teams of workers: team A and team B. Team A can complete a job in 12 days and team B can do the same job in 36 days. Team A starts working on the job and team B joins team A after four days. The team A withdraws after two more days. For how many more days should team B work to complete the job?

4. Let a, b be integers such that all the roots of the equation (x2+ax+20)(x2+17x+b) = 0 are negative integers. What is the smallest possible value of a + b ?
5. Let u, v,w be real numbers in geometric progression such that u > v > w. Suppose u40 = vn = w60. Find the value of n.

6. Find the number of positive integers n, such that pn + pn + 1 < 11.
7. A pen costs 11 and a notebook costs 13. Find the number of ways in which a person can spend exactly 1000 to buy pens and notebooks.

8. There are five cities A,B,C,D,E on a certain island. Each city is connected to every other city by road. In how many ways can a person starting from city A come back to A after visiting some cities without visiting a city more than once and without taking the same road more than once? (The order in which he visits the cities also matters: e.g., the routes A ! B ! C ! A and A ! C ! B ! A are different.)

9. There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated (that is each room is exactly opposite to one other room). Four guests have to be accommodated in four of the eight rooms (that is, one in each) such that no two guests are in adjacent rooms or in opposite rooms. In how many ways can the guests be accommodated?

10. In a class, the total numbers of boys and girls are in the ratio 4 : 3. On one day it was found that 8 boys and 14 girls were absent from the class, and that the number of boys was the square of the number of girls. What is the total number of students in the class?

11. In a rectangle ABCD, E is the midpoint of AB; F is a point on AC such that BF is perpendicular to AC; and FE perpendicular to BD. Suppose BC = 8p3. Find AB. 14. Suppose x is a positive real number such that {x}, [x] and x are in a geometric progression. Find the least positive integer n such that xn > 100. (Here [x] denotes the integer part of x and {x} = x − [x].)

12. Integers 1, 2, 3, . . . ,n, where n > 2, are written on a board. Two numbers m, k such that 1 < m < n, 1 < k < n are removed and the average of the remaining numbers is found to be 17. What is the maximum sum of the two removed numbers?

13. Five distinct 2-digit numbers are in a geometric progression. Find the middle term.
14. Suppose the altitudes of a triangle are 10, 12 and 15. What is its semi-perimeter?
15. If the real numbers x, y, z are such that x2 + 4y2 + 16z2 = 48 and xy + 4yz + 2zx = 24, what is the value of x2 + y2 + z2?

16. Suppose 1, 2, 3 are the roots of the equation x4 + ax2 + bx = c. Find the value of c.
17. What is the number of triples (a, b, c) of positive integers such that (i) a < b < c < 10 and (ii) a, b, c, 10 form the sides of a quadrilateral?

18. Find the number of ordered triples (a, b, c) of positive integers such that abc = 108.

19. Suppose in the plane 10 pairwise nonparallel lines intersect one another. What is the maximum possible number of polygons (with finite areas) that can be formed?

20. Suppose an integer x, a natural number n and a prime number p satisfy the equation 7×2 − 44x + 12 = pn. Find the largest value of p.

21. Let P be an interior point of a triangle ABC whose sidelengths are 26, 65, 78. The line through P parallel to BC meets AB in K and AC in L. The line through P parallel to CA meets BC in M and BA in N. The line through P parallel to AB meets CA in S and CB in T. If KL, MN, ST are of equal lengths, find this common length.

22. Let ABCD be a rectangle and let E and F be points on CD and BC respectively such that area(ADE) = 16, area(CEF) = 9 and area(ABF) = 25. What is the area of triangle AEF?

23. Let AB and CD be two parallel chords in a circle with radius 5 such that the centre O lies between these chords. Suppose AB = 6, CD = 8. Suppose further that the area of the part of the circle lying between the chords AB and CD is (m⇡ +n)/k, where m, n, k are positive integers with gcd(m, n, k) = 1. What is the value of m + n + k?

24. Let p, q be prime numbers such that n3pq − n is a multiple of 3pq for all positive integers n. Find the least possible value of p + q.
25. For each positive integer n, consider the highest common factor hn of the two numbers n! + 1 and (n + 1)!. For n < 100, find the largest value of hn.

26. Consider the areas of the four triangles obtained by drawing the diagonals AC and BD of a trapezium ABCD. The product of these areas, taken two at time, are computed. If among the six products so obtained, two products are 1296 and 576, determine the square root of the maximum possible area of the trapezium to the nearest integer.