Category A

Objective
11 January, 2025

Instructions

• Duration of the Examination: 2 hours

• Total Points: 73

• This Paper Consists of 2 sections:

Section 1: Single Correct Questions

∗ There are 16 questions in this section.

∗ Each question will have 4 choices of which only one is correct.

∗ Each question is worth 3 points.

Section 2: Numerical Type Questions

∗ There are 5 questions in this section.

∗ Each question will have an integer type answer between 000–999.

∗ Answers should be entered as an integer in this range.

∗ Each question is worth 5 points. 0 deduction for wrong answers.

• All questions are compulsory.

• In case the time runs out, the answers saved shall be automatically submitted.
Objective Questions LIMIT Paper A

Single Choice Correct Questions

1. In ancient Europe, there was a place called Rafland. The people of Rafland had
a unique number system. Although the base of their number system was 10, their
number definitions were strange. They had the following digits:

4<1<7<3<6<9<5<8<0<2

A traveller came from England and was surprised to see such a number system. One
night he was trying to evaluate the expression

∇ · 6 × 3 + 22 − (3 + 5)

in this strange number system by putting different values in the box.

What should go inside the box in the above expression, so that the final answer is
the same in the original number system and this strange number system?

(a) 68

(b) 86

(c) 14

(d) none of these

2. Consider the set S = {64, 65, . . . , 128}. A move consists of choosing 2 distinct
elements a, b ∈ S and replacing them with their product ab.

Find the minimum number of times we must make the move to ensure that gcd(S) >
1. Here, gcd(S) is defined as the greatest common divisor of the elements in the set
S.

(a) 24

(b) 28

(c) 32

(d) 48

3. You are given a quadrilateral ABCD whose sides AB and CD when extended meet
at P , and AD and BC when extended meet at Q. The diagonal AC intersects P Q
at R and BD intersects P Q at S.

Which of the following is true?

(a) |P R| · |RQ| = |RQ| · |QS|

(b) |P Q| · |QS| = |RS| · |RQ|

(c) |P R| · |QS| = |RQ| · |P S|

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Objective Questions LIMIT Paper A

(d) |P Q| · |QB| = |CR| · |RQ|

4. The LIMIT 2022 Superclock is marked 1 to 20222022 , and we can shift the numbers
in the following way:

(i) 1 cannot be moved.

(ii) Any number other than 1 can JUMP OVER exactly 2 numbers in the anti-
clockwise direction provided 1 is NOT among the 2.

(iii) If it has to jump over 1, it jumps, then 1 changes to 20222022 AND all the other
numbers get decremented by 1.

How many JUMPS are necessary AT THE LEAST to have an arrangement where
1 and 20222022 change places?

(a) 20222022
2022
(b) 20222022

(c) 20222022 !

(d) none of these

5. You are given m columns and in each column you can put a non-negative number
of balls. The total number of balls is at least 1 and at most n. Assume all the balls
are identical. Let the total number of arrangements be T .

For n = 20 and m = 10, find the maximum value of k such that 3k | (T + 1).

(a) 0

(b) 1

(c) 2

(d) 3

6. There are 2 classes in grade 1. The first class has x students and the second class
has x + 1 students. All the students from grade 1 advance to grade 2 and some new
students join in grade 2 as well. The total strength of grade 2 is 24 students.

Assuming that 2 students that were from the same class won’t fight, find the maxi-
mum number of fights that take place.

(a) 276

(b) 185

(c) 200

(d) 191

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Objective Questions LIMIT Paper A

7. Evaluate ∞  
Y i
z= 1+ 3
r=1
r
√
where i = −1.

In which quadrant does z lie?

(a) I Quadrant

(b) II Quadrant

(c) III Quadrant

(d) IV Quadrant
P∞ 1
Hint: Use the fact that n=1 n3 ≈ 1.202.

8. Two circles of radii 1 and 2 intersect at an angle of 60◦ . Find the length of the
common chord.
√
2√ 3
(a) 7

2√
(b) 5+2 3
√
2 √2
(c) 5+2 2

(d) √4
5

9. Let ABC be a triangle with circumcenter O, circumradius 4 units and inradius 2

units. Let X and Y be the intersections of BO and CO with the circumcircle of
△ABC.

Find the value of AX 2 + AY 2 + XY 2 .

(a) 100

(b) 72

(c) 80

(d) 60

10. Statement 1: x4 + 4x3 + 16x2 + 24x + 16 is reducible over Z, i.e., can be factorized
into smaller degree polynomials with integer coefficients.

Statement 2: Number of polynomials P with real coefficients satisfying the fol-

lowing polynomial equation
P (x3 ) = P (x + 3)P (x − 1)P (x − 2) ∀x ∈ R
(Note: We do not mean the number of roots of the polynomials.)

The answers are respectively:

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Objective Questions LIMIT Paper A

(a) True, 3

(b) True, 5

(c) False, 3

(d) False, 5

11. Take a positive number N (where N > 10). Square its digits and add them to make
a new number. Which of the following is true about the process when repeated
indefinitely?

(a) The process always becomes constant in the long run.

(b) The process ends either in 1 or hits a fixed particular cycle.

(c) The process may hit 1 or end in different cycles for different inputs.

(d) It is not necessary for the process to end in 1 or some cycle; it is always
bounded, but can oscillate without any pattern whatsoever.

12. The number of solutions of the equation x2 ≡ 2 (mod m) when m = 162, 98, and
196 are respectively:

(a) 1, 2, 2

(b) 0, 2, 4

(c) 1, 2, 4

(d) 0, 2, 2

13. Find the number of binary strings of length 10 such that no three consecutive 1’s
are adjacent and no two consecutive 0’s are adjacent.

(a) 30

(b) 28

(c) 20

(d) 24

14. Let S be a nonempty finite set. Define uniform probability of a set A ⊆ S as

P (A) = |A|
|S|
.

Two events A and B are said to be independent if P (A ∩ B) = P (A)P (B).

Which of the following conditions suffice to prove events A and B are independent?

(a) A, B ̸= ∅ but A ∩ B = ∅

(b) |A ∩ B| = |A| + |B|

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Objective Questions LIMIT Paper A

(c) |A ∪ B| = |A| × |B|

(d) none of these

15. If a, b, c are sides of ANY triangle, then we have

a−b b−c c−a 1

+ + ≤
a+b b+c c+a K

where K ∈ N.

Find the largest possible value of K that satisfies the above statement for all trian-
gles.

Tip: Feel free to use graphing.

(a) 18

(b) 22

(c) 30

(d) 40
N  
X N
16. Find which option is equivalent to for all N ∈ N.
i=1
i
√
Here, g = ⌊ N ⌋ and h = N 2 .
h    
X N N
(a) −
i=1
i2 (i + 1)2
g    
X N N
(b) − ×g
i=1
i2 (i + 1)2

g     ⌊N/(g+1)⌋  
X N N X N
(c) − ×i+
i=1
i i+1 i=1
i

h     ⌊h/(N +1)⌋  
X N N h X N
(d) − × +
i=1
i i+1 i i=1
i

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Objective Questions LIMIT Paper A

Numerical Type Questions

1. Let dk (n) = dk (for brevity) denote the k-th divisor of n (arranged in ascending
order). Find n ∈ N which satisfies the following properties:
n
(i) d6 + d7 = d9
.

(ii) d9 + d10 = 27.

2. Define X n
g(n) = σ(d)ϕ
d
d|n

Define a 4 × 4 matrix A such that:


4
g(j )
 if i = j
aij = g(j 4 i5 ) if i ≤ j and i | j

 4
g(j ) otherwise

det(A)
Determine (4!)4
.

3. Let △ABC be an acute-angled triangle. For n = 1, 2, 3, we define:

xn = 2n−3 (cosn A + cosn B + cosn C) + cos A cos B cos C

If x1 + x2 + x3 = k, find the minimum possible value of 16k.

4. Shankara and Labonya are playing a game where they are given a number and each
of them can perform a certain operation once in their turn. If they are given a
number N , in each move they subtract a number d from N , where d is a divisor of
N , d ̸= 1 and d ̸= N . Shankara goes first and the number N is updated each time.
Whoever cannot do the operation loses.

Given the list of 10 numbers below, in how many of them as initial values of N does
Labonya win, provided both play optimally?

1006, 1003, 132, 512, 3072, 9000, 1331, 343, 625, 1729

5. Let x be the smallest prime factor of 2130! + 1.

Let y be the number of solutions of

log(x) = sin(πx) + sin(sin(sin(πx)))

in [0, 3].

Find the sum of digits of x + y.