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FIIT JEE RMO Regional Mathematical Olympiad Question Paper 2018 : fiitjee.com

Name of the Organisation :FIIT JEE Ltd
Name of the Paper : Regional Mathematical Olympiad-2018
Year : 2018
Document Type : Question Paper
Name of the Subject : Mathematical Olympiad
Website : http://www.fiitjee.com/DownLoads/Olympiad-Question-Papers-and-Solutions/17

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FIIT JEE Regional Mathematical Olympiad Question Paper

Sample Question Paper of FIIT JEE Regional Mathematical Olympiad Exam 2018 Mathematical Question Paper is now available in the official website of FIIT JEE Ltd

Related / Similar Question Paper :
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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
** Answers to each question should start on a new page. Clearly indicate the question number.

Model Questions

Time : 3 hours
Date : October 07, 2018

Download Question Paper :
RMO  : https://www.pdfquestion.in/uploads/rmo1.pdf
Pre RMO : https://www.pdfquestion.in/uploads/rmo2.pdf

1. Let ABC be a triangle with integer sides in which AB < AC. Let the tangent to the circumcircle of triangle ABC at A intersect the line BC at D. Suppose AD is also an integer. Prove that gcd(AB, AC) > 1.
Sol. Here DCA and ?DAB are similar
Let gcd of (b, c) = 1 so b will not divide b2 – c2 and also c will not divide b2 – c2 i.e. ? ? ?bc
a = (b2 – c2)? and given that b + c > a
b + c > (b2 – c2)?
(b – c)? < 1 which is not possible
So, gcd = b, c will be > 1

2. Let n be a natural number. Find all real numbers x satisfying the equation

3. For a rational number r, its period is the length of the smallest repeating block in its decimal expansion. For example, the number r = 0.123123123… has period 3. If S denotes the set of all rational numbers r of the form r ? 0.abcdefgh having period 8, find the sum of all the elements of S.
Sol. Sum of numbers with period 1 is 1 2 3 ….. 8 4 9?
Sum of numbers with period 2 is 1 2 3 ….. 98 4 45 99?
Sum of numbers with period 4 is 1 2 3 ….. 9998 4 45 4950 9999
Sum of numbers with period 8 is 1 2 3 ….. 99999998 4 45 4950 49995000 99999999

4. Let E denote the set of 25 points (m, n) in the xy-plane, where m, n are natural numbers, 1 ? m ? 5, 1 ? n ? 5. Suppose the points of E are arbitrarily coloured using two colours, red and blue. Show that there always exist four points in the set E of the form (a, b), (a + k, b), (a + k, b + k), (a, b + k) for some positive integer k such that at least three of these four points have the same colour. (That is, there always exist four points in the set E which form the vertices of a square with sides parallel to axes and having at least three points of the same colour).
Sol. Every row has atleast 3 vertices of same colour. Without loss of generality we can assume this to be red for first row, so 4 possibilities arises In each figure any vertex denoted by ?? can not be of red colour otherwise there will be a square whose atleast three vertices will be of red colour. If all will be of blue colour then there will be a square whose atleast three vertices will be blue In fourth case the vertex which is indicated by the arrow will be either red or blue. Again we ill get a square whose three vertices are of same colour

Alternate solution :
If we try to create 5 ? 5 grid such that no ‘unit’ square exists such that if has atleast three points of same colour, then we will have to colour them alternatively along atleast one of the side of square grid. In doing so, it is obvious that a square of side 2 units will be formed which has atleast 3 points of same colour as (a, b) will have same colour as (a + 2, b) and (a, b + 2).

5. Find all natural numbers n such that 1? ?? 2n?? divides 2n. (For any real number x, [x] denotes the largest integer not exceeding x).

6. Let ABC be an acute-angled triangle with AB < AC. Let I be the incentre of triangle ABC, and let D, E, F be the points at which its incircle touches the sides BC, CA, AB, respectively. Let BI, CI meet the line EF at Y, X, respectively. Further assume that both X and Y are outside the triangle ABC. Prove that
(i) B, C, Y, X are concyclic; and
(ii) I is also the incentre of triangle DY X.

PRE-RMO-2018

Time : 3 hours
Date : August 19, 2018

1. A book is published in three volumes, the pages being numbered from 1 onwards. The page numbers are continued from the first volume to the second volume to the third. The number of pages in the second volume is 50 more than that in the first volume, and the number of pages in the third volume is one and a half times that in the second. The sum of the page numbers on the first pages of the three volumes is 1709. If n is the last page number, what is the largest prime factor of n?
Sol. 17
Let number of pages in volume 1 be ‘x’
1 + (x + 1) + (2x + 51) = 1709
x = 552
n = 2057 = 11 ? 11 ? 17
Largest prime factor = 17

2. In a quadrilateral ABCD, it is given that AB = AD = 13, BC = CD = 20, BD = 24. If r is the radius of the circle inscribable in the quadrilateral, then what is the integer closest to r?

3. Consider all 6-digit numbers of the form abccba where b is odd. Determine the number of all such 6-digit numbers that are divisible by 7.
Sol. 70
If abccba is divisible by 7, then a + 3b + 2c + 6c + 4b + 5a is divisible by ‘7’
7b + 6a + 8c is divisible by ‘7’
c – a is divisible by ‘7’ and ‘b’ is odd
(5 + 9) ? 5 = 70 such numbers are possible

4. The equation 166 ? 56 = 8590 is valid in some base b ? 10 (that is, 1, 6, 5, 8, 9, 0 are digits in base b in the above equation). Find the sum of all possible values of b ? 10 satisfying the equation.
Sol. 12
(b2 + 6b + 6)(5b + 6) = (8b3 + 5b2 + 9b)
3b3 – 31b2 – 57b – 36 = 0
(b – 12)(3b2 + 5b + 3) = 0
b = 12 is the only possibility

5. Let ABCD be a trapezium in which AB || CD and AD ? AB. Suppose ABCD has an incircle which touches AB at Q and CD at P. Given that PC = 36 and QB = 49, find PQ. Sol. 84
Let radius of circle b ‘r’
CE2 + BE2 = CB2
(49 – 36)2 + (2r)2 = (49 + 36)2
r = 42
PQ = 2r = 84 units

6. Integers a, b, c satisfy a + b ? c = 1 and a2 + b2 ? c2 = ?1. What is the sum of all possible values of a2 + b2 + c2?
Sol. 18
a + b – c = 1
a2 + b2 – c2 = 1 + 2c – 2ab = –1
c = ab – 1
a + b = ab
(a – 1)(b – 1) = 1
As, a, b, c are integers
a = b = 2, c = 3
a2 + b2 + c2 = 17
and a = b = 0, c = –1
Sum of possible value of a2 + b2 + c2 is 18

7. A point P in the interior of a regular hexagon is at distances 8, 8, 16 units from three consecutive vertices of the hexagon, respectively. If r is radius of the circumscribed circle of the hexagon, what is the integer closest to r?

8. Let AB be a chord of a circle with centre O. Let C be a point on the circle such that ABC = 30º and O lies inside the triangle ABC. Let D be a point on AB such that DCO = ?OCB = 20°. Find the measure of ?CDO in degrees.
Sol. 80
Shown in the figure
CDO = 80º

9. Suppose a, b are integers and a + b is a root of x2 + ax + b = 0. What is the maximum possible value of b2?
Sol. 81
If a + b is a root it satisfies the equation
2a2 + 3ba + (b2 + b) = 0
Now, since ‘a’ is an integer, discriminant is a perfect square
So, (b – 4 + p)(b – 4 – p) = 16
b – 4 + p = ?8, b – 4 – p = ?2, b – 4 + p = b – 4 – p = ? 4
b – 4 = 5, –5, 4, –4
b = 9, –1, 8, 0 (b2) maximum = 81

10. In a triangle ABC, the median from B to CA is perpendicular to the median from C to AB. If the median from A to BC is 30, determine (BC2 + CA2 + AB2)/100.
Sol. 24
CD = BD = GD (? right triangle)
AB2 = (2BF)2 = 4(x2 + 4y2)
AC2 = (2CE)2 = 4(y2 + 4×2)
BC2 = 202 = 4(x2 + y2)
AB2 + BC2 + AC2 = 24

11. There are several tea cups in the kitchen, some with handles and the others without handles. The number of ways of selecting two cups without a handle and three with a handle is exactly 1200. What is the maximum possible number of cups in the kitchen?
Sol. 29
n1 = cups with handles
n2 = cups without handles
n1 = n1 ?
C2 C3 1200 ? Only prime factors of n1
C3 and n2
C2 are 2, 3, 5 and n1
C3 | 1200 n1 ?
C3 1200 n1 < 21

12. Determine the number of 8-tuples (?1, ?2, …. , ?8) such that 1, 2, …. 8 ? {1, 1} and 1 + 22 + 33 + …. + 88 is a multiple of 3.
Sol. 88
Expression ? (?1 + ?4 + ?7) – (?2 + ?5 + ?8) (mod 3)
Number of ways = 2 . 2 . [1 + 1 + 3 . 3 . 2 + 1 + 1] = 88

15. Let a and b be natural numbers such that 2a ? b, a ? 2b and a + b are all distinct squares. What is the smallest possible value of b?
Sol. 21
2a ? b = 2
k1 …(1)
a ? 2b = 22
k …(2)
a + b = 23
k …(3)

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