Organisation : Mathematics Teachers’ Association (India)
Exam Name : Pre-Regional Mathematical Olympiad PRMO 2019-20
Document Type : Question Paper
Year : 2019
Website : http://www.mtai.org.in/activities/exams/prmo/
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MTA PRMO Question Paper
Download the sample question paper of Pre-Regional Mathematical Olympiad PRMO 2019-20 from the official website of Mathematics Teachers’ Association (India).
Related : Unified Council UIMO International Mathematics Olympiad Sample Question Paper : www.pdfquestion.in/34925.html
1. Use of mobile phones, smartphones, ipads, calculators, programmable wrist watches is STRICTLY PROHIBITED. Only ordinary pens and pencils are allowed inside the examination hall.
2. The correction is done by machines through scanning. On the OMR Sheet, darken bubbles completely with a black pencil or a black or blue ball pen. Darken the bubbles completely, only after you are sure of your answer; else, erasing may lead to the OMR sheet getting damaged and the machine may not be able to read the answer.
3. The name, email address, and date of birth entered on the OMR sheet will be your login credentials for accessing your PRMO score.
4. Incompletely, incorrectly or carelessly filled information may disqualify your candidature.
5. Each question has a one or two digit number as answer. The first diagram below shows improper and proper way of darkening the bubbles with detailed instructions. The second diagram shows how to mark a 2-digit number and a 1-digit number.
6. The answer you write on OMR sheet is irrelevant. The darkened bubble will be cosidered as your final answer.
7. Questions 1 to 6 carry 2 marks each; questions 7 to 21 carry 3 marks each; questions 22 to 30 carry 5 marks each.
8. All questions are compulsory.
9. There are no negative marks.
10. Do all rough work in the space provided below for it. You also have blank pages at the end of the question paper to continue with rough work.
11. After the exam, you may take away the Candidate’s copy of the OMR sheet.
12. Preserve your copy of OMR sheet till the end of current olympiad season. You will need it later for verification purposes.
13. You may take away the question paper after the examination.
1. From a square with sides of length 5, triangular pieces from the four corners are removed to form a regular octagon. Find the area removed to the nearest integer?
2. Let f(x) = x2 + ax + b. If for all nonzero real x and the roots of f(x) = 0 are integers, what is the value of a2 + b2 ?
3. Let x1 be a positive real number and for every integer n 1 let xn+1 = 1 + x1x2 : : : xn??1xn. If x5 = 43, what is the sum of digits of the largest prime factor of x6?
4. An ant leaves the anthill for its morning exercise. It walks 4 feet east and then makes a 160 turn to the right and walks 4 more feet. It then makes another 160 turn to the right and walks 4 more feet. If the ant continues this pattern until it reaches the anthill again, what is the distance in feet it would have walked?
5. Five persons wearing badges with numbers 1; 2; 3; 4; 5 are seated on 5 chairs around a circular table. In how many ways can they be seated so that no two persons whose badges have consecutive numbers are seated next to each other? (Two arrangements obtained by rotation around the table are considered different.)
6. Let abc be a three digit number with nonzero digits such that a2 + b2 = c2. What is the largest possible prime factor of abc?
7. On a clock, there are two instants between 12 noon and 1 PM, when the hour hand and the minute hand are at right angles. The difference in minutes between these two instants is written as a + b c , where a; b; c are positive integers, with b < c and b=c in the reduced form. What is the value of a + b + c ?
8. How many positive integers n are there such that 3 n 100 and x2n + x + 1 is divisible by x2 + x + 1 ?
9. Let the rational number p=q be closest to but not equal to 22=7 among all rational numbers with denominator < 100. What is the value of p ?? 3q ?
10. Let ABC be a triangle and let be its circumcircle. The internal bisectors of angles A, B and C intersect at A1, B1, and C1, respectively, and the internal bisectors of angles A1, B1 and C1 of the triangle A1B1C1 intersect at A2, B2 and C2, respectively. If the smallest angle of triangle ABC is 40, what is the magnitude of the smallest angle of triangle A2B2C2 in degrees?
11. How many distinct triangles ABC are there, up to similarity, such that the magnitudes of angles A, B and C in degrees are positive integers and satisfy cosAcosB + sinAsinB sin kC = 1 for some positive integer k, where kC does not exceed 360?
14. Find the smallest positive integer n 10 such that n + 6 is a prime and 9n + 7 is a perfect square.
15. In how many ways can a pair of parallel diagonals of a regular polygon of 10 sides be selected?
16. A pen costs Rs. 13 and a note book costs Rs. 17. A school spends exactly Rs. 10000 in the year 2017-18 to buy x pens and y note books such that x and y are as close as possible (i.e., jx??yj is minimum). Next year, in 2018-19, the school spends a little more than Rs. 10000 and buys y pens and x note books. How much more did the school pay?
PRMO Question Paper :
Paper 1 :