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# Pre RMO | IOQM 2020-21 Question Paper with Answer Keys & Solutions

Hi Students, welcome to Amans Maths Blogs (AMB). On this post, you will get Pre RMO IOQM 2020-21 Question Paper with Answer Keys & SolutionsIn 2020, PRMO (Pre RMO) exam now changed to IOQM (Indian Olympiad Qualifier in Mathematics) exam.

On 17 January 2021, IOQM 2020-21 exam is conducted and it is organized by the Mathematics Teachers’ Association (India) – MTA(I) with support from the Indian Association of Physics Teachers (IAPT) and Homi Bhabha Centre for Science Education (HBCSE).

IOQM is a three hour paper and pen exam with responses to be written on OMR sheet. The IOQM will have 30 questions with each question having an integer answer in the range 00-99. The syllabus and standard of this examination will be the same as that of PRMO of Previous Years.

## IOQM 2020-21 Question Paper

Pre RMO / IOQM 2020-21 is conducted on 17 January 2021.

## Pre RMO 2020 / IOQM 2020-21 Answer Keys

Pre RMO / IOQM 2020-21
Ques No Ans Key Ques No Ans Key Ques No Ans Key Ques No Ans Key
1 08 9 02 17 30 25 56
2 64 10 40 18 80 26 .
3 99 11 24 19 40 27 .
4 19 12 24 20 75 28 .
5 18 13 . 21 . 29 .
6 15 14 20 22 84 30 .
7 07 15 96 23 74
8 15 16 87 24 21

## IOQM 2020-21 Solutions

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 1

Let ABCD be a trapezium in which AB || CD and AB = 3CD. Let E be the midpoint of the diagonal BD. If [ABCD] = n[CDE]. What is the value of n? Here [????] denotes the area of the geometrical figure ????.

Solution:

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 2

A number N in base 10, is 503 in base b and 305 in base b+2. What is the product of the digits of N?

Solution:

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 3

If then determine the value of N.

Solution:

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 4

Let ABCD be a rectangle in which AB + BC + CD = 20 and AE = 9 where E is the mid point of the side BC. Find the area of the rectangle.

Solution:

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 5

Find the number of integer solutions to  ||x| – 2020| < 5.

Solution:

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 6

What is the least positive integer by which 25.36.43.53.67 should be multiplied so that the product is a perfect square?

Solution:

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 7

Let ABC be a triangle with AB = AC. Let D be a point on the segment BC such that BD = 48 1/61 and DC = 61. Let E be a point on AD such that CE is perpendicular to AD and DE = 11. Find AE.

Solution:

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 8

A 5-digit number (in base 10) has digits k, k+1, k+2, 3k, k+3 in that order, from left to right. If this number is m2 for some natural number m, find the sum of the digits of m.

Solution:

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 9

Let ABC be a triangle with AB = 5, AC = 4, BC = 6. The internal angle bisector of C intersects the side AB at D. Points M and N are taken on the sides BC and AC respectively, such that DM || AC and DN || BC. If (MN)2 = p/q, where p and q are relatively prime positive integers then what is the sum of the digits of |p – q|?

Solution:

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 10

Five students take a test on which any integer score from 0 to 100 inclusive is possible. What is the largest possible difference between the median and the mean of the scores? (The median of a set of scores is the middlemost score when the data is arranged in increasing order. Is is exactly the middle score when there are an odd number of scores and it is the average of the two middle scores when there are an even number of scores.)

Solution:

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 11

Let X = {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5} and S = {(a, b) ∈ X × X : x2 + ax + b and x3 + bx + a have at least a common real zero}. How many elements are there in S?

Solution:

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 12

Given a pair of concentric circles, chords AB, BC, CD, … of the outer circles are drawn such that they all touch the inner circle. If angle ABC = 75 degree, how many chords can be drawn before returning to the starting point?

Solution:

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 13

Find the sum of all positive integers n for which |2n + 5n – 65| is a perfect square.

Solution: Solving…

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 14

The product 55 × 60 × 65 is written as the product of five distinct positive integers. What is the least possible value of the largest of these integers?

Solution:

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 15

Three couples sit for a photgraph in 2 rows of three people each such that no couple is sitting in the same row next to each other or in the same column one behind the other. How many arrangements are possible?

Solution:

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 16

The sides x and y of a scalene triangle satisfy x + 2∆ / x = y + 2∆ / y. where ∆ is the area of the triangle. If x = 60, y = 63, what is the length of the largest side of the triangle?

Solution:

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 17

How many two digit numbers have exactly 4 positive factors? (here 1 and the number n are also considered as factor of n).

Solution:

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 18

If where a, b, c ∈ N, b <c, gcd(n, c) = 1, then what is the value of (a + b)?

Solution:

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 19

Let ABCD be a parallelogram. Let E and F be midpoints of AB and BC respectively. The lines EC and FD intersect in P and form four triangles APB, BPC, CPD and DPA. If the area of the parallelogram is 100 sq units. What is the maximum area in sq units of a triangle among these four triangles?

Solution:

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 20

A group of women working together at the same rate can build a wall in 45 hours. When the work started, all the women did not start working together. They joined the work over a period of time, one by one at equal intervals. Once at work, each one stayed till the work was completed. If the first women worked 5 times as many hours as the last woman, for how many hours did the first women work?

Solution:

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 21

A total fixed amount of N thousand rupees is given to three persons A, B, C, every year, each being given an amount proportional to her age. In the first year, A got half the total amount. When the sixth payment was made. A got six-seventh of the amount that she had in the first year; B got Rs. 1000 less than that she had in the first year; and C got twice of that she had in the first year. Find N.

Solution: Solving…

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 22

In triangle ABC, let P and R be the feet of the perpendiculars from A onto the external and internal bisector of angle ABC respectively and let Q and S be the feet of the perpendiculars from A onto the internal and external bisectors of angle ACB respectively. If PQ = 7 and QR = 6 and RS = 8, what is the area of triangle ABC?

Solution:

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 23

The incircle ???? of a scalene triangle ABC touches BC at D, CA at E and AB at F. Let rA be the radius of the circle inside ABC which is tangent to ???? and the sides AB and AC. Define rB and rC similarly. If rA = 16, rB = 25 and rC = 36, determine the radius of ????.

Solution:

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 24

A light source at the point (0, 16) in the coordinate plane casts light in all directions. A disc (a circle along with its interior) of radius 2 with center at (6, 10) casts a shadow on the X axis. The length of the shadow can be written in the form of m√n where m and n are positive integers and n is a square free. Find (m+n).

Solution:

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 25

For a positive integer n, let〈n〉denote the perfect square integer closest to n. For example: 〈74〉= 81, 〈18〉= 16. If N is the smallest positive integer such that 〈91〉⋅〈120〉⋅〈143〉⋅〈180〉⋅〈N〉= 91 ⋅ 120 ⋅ 143 ⋅ 180 ⋅ N. Find the sum of the squares of the digits of N?

Solution:

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 26

In the figure below, 4 of the 6 disks are to be coloured black and 2 are to be colored white. Two coloring that can be obtained from one another by a rotation of a reflection of the entire figure are considered the same.

There are only four such colorings for the given two colors, as shown in figure 1. In how many ways can we color the 6 disks such that 2 are colored black, 2 are colored white, 2 are colored blue with the given identification condition?

Solution: Solving…

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 27

A bug travels in the coordinate plane moving only along the lines that are parallel to the x-axis or y-axis. Let A = (-3, 2) and B = (3, -2). Consider all possible paths of the bug from A to B of length at most 14. How many points with integer coordinates lie on at least one of these paths?

Solution: Solving…

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 28

A natural number n is said to be good if n is the sum of r consecutive positive integers, for some r ≧ 2. Find the number of good numbers in the set {1,2,3,4,…,100}.

Solution: Solving…

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 29

Positive integers a, b, c satisfy ab/(a – b) = c. What is the largest possible value of (a + b + c) not exceeding 99?

Solution: Solving…

IOQM 2020-21 Question Paper With Answer Keys Solutions: Question No 30

Find the number of the pairs (a, b) of a natural numbers such that b is a 3-digit number, (a+1) divides (b-1) and b divides (a2 + a + 2).

Solution: Solving…

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