Exam Mathematics Olympiad [MO] Department IIT Bombay [IITB] Download Sample Question Paper Year 2023 Document Type PDF Official Website https://www.math.iitb.ac.in/

## IIT Bombay Mathematics Olympiad Sample Questions

1. Assume that the quadratic equations ax2 + bx + c = 0 and dx2 + ex + f = 0 have exactly one solution α in common, then α = .
(a) af − cd/ bd − ae
(b) bf − ce/ bd − ae
(c) af − cd/bf − ce
(d) None of the above

2. Consider the following statements.
I. Any continuous function f : (0, 1) → R can be extended to a continuous function ˜f : R → R such that ˜f(x) = f(x) for all x ∈ (0, 1).
II. Any continuous function f : [0, 1] → R can be extended to a continuous function ˜f : R → R such that ˜f(x) = f(x) for all x ∈ [0, 1].
III. For any continuous function f : [0, 1] → [0, 1], there exists an x ∈ [0, 1] such that f(x) = x.
IV. For any continuous function f : (0, 1) → (0, 1), there exists an x ∈ (0, 1) such that f(x) = x.

Which of the following is true?
(a) Only statements II and III are correct
(b) Only statements I and IV are correct
(c) Only statements I, II and III are correct
(d) All four statements are correct

3. Let C1 and C2 be circles whose centers are 12 cm apart, and whose radii are 2 cm and 4 cm respectively. Let x cm2 be the area of the region formed by all points M for which there exist points X on C1 and Y on C2 such that M is the midpoint of the line segment XY . Then the largest integer which is less than or equal to x is .
(a) 25
(b) 26
(c) 27
(d) 28

4. Mahavir plays a game on the number line. He starts at zero. At each second, he either stays where he is, or jumps some distance left or right. The distance he can jump at second s is 3s units (he does not have to jump every second, but when he does, he jumps exactly 3s units left or right). His time starts at s = 0. His goal is to reach the point 2023. He has devised a path to do the same in the minimum amount of time. How many times will he jump left on this path?
(a) 1
(b) 2
(c) 3
(d) 4

General Instructions:

• Do not open this booklet until instructed by the invigilators.
• Questions must be answered on Candidate Response Sheet (CRS) by darkening the appropriate bubbles (marked a, b, c, d or 0 − 9) using blue/black ball pen.
• Write your registration number, your name, and the Question Paper code (A, B, C or D) in the space provided on the CRS.
• The question paper consists of two sections: Section 1 and Section 2.

For Section 1:
• There are 15 Objective Questions each of which has exactly one correct answer.
• Each correct answer will be awarded +2 marks and each incorrect answer will be awarded −1 mark.
• Darken exactly one bubble for each question using black/blue ball pen.

For Section 2:
• There are 15 Numerical Answer Type Questions of which the answer is an integer between 000 and 999 (both inclusive).
• Each correct answer will be awarded +2 marks and there will be no negative marking.
• Darken three bubbles for each digit for each question using black/blue ball pen, e.g., 9 should be bubbled as 009, 10 should be bubbled as 010, 831 should be bubbled as 831.

In case of ties, the following order will be followed to rank the candidates:
1. Higher number of correct asterisk(∗) questions.
2. Lower number of wrong answers in Section 1.
3. Higher score in Section 2.

Note:
• This question paper contains 10 printed pages. Please check all pages and report if there is any discrepancy.
• You won’t be allowed to leave the examination hall during the exam.

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