Mathematicos

BY INFINITIO
Round 1
ELIMINATION
Rules For Round 1:
1. Each Question carries 4 marks i.e. each correct attempt will give 4 marks
2. Each Wrong attempt will be evaluated as -1 mark
3. Use of calculator/mobile or any other electronic device would cause disqualification
4. A simplified answer would be taken as correct.
5. If a question has more than one right answers, any of them would bring full marks.
Qs 1

The largest real value of a for which the equation |x+a|+|x−1|=2 has an infinite number of
solutions for x is

a) 1
b) 0
c) -1
d) 2

Ans: a
Qs 2
1 1
If x is a positive real number such that 𝑥 8 + =47, then the value of 𝑥 9 + 9 Is
𝑥8 𝑥

a) 34 5
b) 40 5
c) 30 5
d) 35 5

Ans: a
Qs 3

Let 𝑎𝑛 = 46+8n and 𝑏𝑛 = 98+4n be two sequences for natural numbers n≤100. Then, the sum of
all terms common to both the sequences is

a) 14900
b) 25200
c) 12400
d) 28500

Ans: a
Qs 4

For n positive integers, if their product is 𝑛𝑛 , then what will be their sum?

a) Equal to n+(1/n)
b) Less than n
c) A negative integer
d) Never less than 𝑛2

Ans : d
Qs 5

If 𝑥 + 1 + 𝑥 2 𝑦 + 1 + 𝑦 2 = 1 then x+y=

a) 1
b) 2
c) -1
d) 0

Ans: d
Qs 6

4
If 𝑥 𝑥 =64 then x=?

a) 2
b) 2 2
4
c) 8
4
d) 2 2

Ans: 𝑐
 Qs 7

1 3 3
If 3 3 = 𝑎 25 + 𝑏 5 + 𝑐, where a,b,c are rational number then value of a+b+c=?
25+ 5+1

a) 0
b) 1/4
c) - 2/5
d) 1

Ans : a
Qs 8

3
If a,b,c are roots of the equation 𝑥 3 +3𝑥 2 − 24𝑥 + 1 = 0, then 3 𝑎+ 𝑏+ 3 𝑐 =?

a) 0
b) 1
c) -1
d) -3

Ans : a
Qs 9

If xy+x+y=34 and x,y are positive integers then Find x+y

a) 14
b) 25
c) 12
d) 7

Ans: c
Qs 10

Cards are drawn one by one without replacement from a pack of 52 cards till all the aces are
drawn out. What is the probability that only two cards are left unturned when all aces are out ?
a) (47x48)/(51x52)
b) (4x47x48)/(50x51x52)
c) 1/52
d) 4/13

Ans: b) (4x47x48)/(50x51x52)
Qs 11

If 𝑎𝑖 represents co-efficient of 𝑥 𝑖 in the expansion of (1 + 𝑥 + 𝑥 2 + 𝑥 3 )20 , then what is the

value of 𝑎1 −𝑎3 +𝑎5 − 𝑎7 + ⋯ + 𝑎17 − 𝑎19 ?

a) 0
b) 252
c) 190
d) 220

Ans : a
Qs 12
p,q,r,s,t are choosen randomly from decimal digits(repeation allowed). What is the probability
that 2𝑝 +3𝑞 +5𝑟 +7𝑠 +9𝑡 is divisible by 4 ?

a) 9/20
b) 7/10
c) 3/25
d) 36789/100000

Ans : a
Qs 13

ABCD is a square with AB=2 unit. O is center of the square. A circle passes through O and touches AB
an BC. What is the radius of the circle ?

a) 2 unit
b) 2- 2 unit
c) 1+ 2 unit
d) 1 unit

Ans : b
Qs 14
Let a, b, m and n be natural numbers such that a>1and b>1 If 𝑎𝑚 𝑏 𝑛 =144145 , then the largest
possible value of n−m is

a) 580
b) 290
c) 579
d) 289

Ans : c
Qs 15
Let p be a prime number, which is greater than 10. Which one is false ?

a) If p+2 is also a prime number, then p+1 is divisible by 6.

b) Last digit of 𝑝4 is always 1.
c) There exists atleast one value of p, for which p+2 and p+4 both are prime numbers.
d) 𝑝2 − 1 is always divisible by 12.

Ans : c
Round 2
RAPID FIRE
Rules for Round 2

1. Each Question carries 10 marks i.e. each correct attempt will give 10 marks
2. Each Wrong attempt will be evaluated as -5 mark
3. Use of calculator/mobile or any other electronic device would cause disqualification
4. If a question has more than one right answers, any of them would bring full marks.
5. Only the first correct respondent will get the points.
Qs 1

Which one is greater 99! Or 5099 ?

Qs 2

What is the year 1989 in Roman Numerals?

Qs 3

1 1 1
What is the value of + + ?
1+log2 15 1+log3 10 1+log5 6
Qs 4

Find the sum of the series :

1 1 1 1
+ + …. + + ?
10−9 +1 10−8 +1 108 +1 109 +1
Qs 5

What will be the reminder if the octal number 1256 is divided by 7 ?

Qs 6

What is value of Kaprekar’s Constant ?

Qs 7

If p, q, r, s, t are fifth root of 1 then what is the value of the given determinant ?
𝑝 𝑞 𝑟 𝑡 𝑠
𝑞 𝑟 𝑝 𝑡 𝑠
𝑡 𝑝 𝑞 𝑟 𝑠
𝑟 𝑡 𝑠 𝑞 𝑝
𝑠 𝑟 𝑞 𝑝 𝑡
Qs 8

For some natural number n, assume that (15,000)! is divisible by (n!)!What is the largest
possible value of n ?
Qs 9

What is the theme of international mathematics day, 2024 ?

Qs 10

How many sides does an “ icosagon” have ?

Qs 11

To write numbers from 100 to 1000 (both inclusively) how many times zero(0) is used ?
Qs 12

What is the value of 3 upto 3 decimal places?

Qs 13

What is Euler's Number in mathematics?

Qs 14

There are 4 ants sitting on four corners of a square. All ants randomly pick an edge of square and
start moving along it. What is the probability that any two ants collide?
Assume they have same speed.
Qs 15

What is 56 in binary number system ?

Qs 16

There are 1 white ball, 10 identical red balls, 10 identical green balls. How many different ways
you can arrange them in a row such that the row is symmetric about the middle ball ?
Qs 17

On a new scale of temperature (which is linear) and called the W scale, the freezing and boiling
points of water are -35°W and 35°W respectively. What will be the temperature on the new
scale, corresponding to a temperature of 50°C on the Celsius scale ?
Qs 18

Find the radius of a circle inscribed in a triangle with side lengths of 5 cm, 5 cm, and 6 cm ?
Qs 19

A man fell in a 50m deep well. He climbs 4 meters up and slips 3 meters down in one day. How
many days would it take for him to come out of the well?
Qs 20

The first term of a sequence is 1 and the n-th term of the sequence is the sum of previous
(n-1) terms. What is the 100th term of the sequence ?
Qs 21

0 1 0
If A = 0 0 1 , then what is the value of 𝐴𝐴𝑇 ?
1 0 0
Qs 22

Find the no. of zeroes at the end of P(71, 37)?

Qs 23

What is the 10 digit smallest palindrome divisible by 11 ?

Qs 24

The angles of a triangle are in the ratio 1:1:6. What is the measure of the largest angle?
Qs 25

You begin in the top left corner of a 6×6 grid, and your objective is to move to the bottom right
corner. There are just two directions you can move: right or down. Both diagonal and backward
movements are prohibited. How many different ways are there to get from the start to the
finish?
ANSWER
10. 20 19. 47 days
1. 5099
2. MCMLXXXIX
11. 183 20. 298
12. 1.732 21. 𝐼3
3. 1
13. e (2.71828) 22. 9
4. 9.5
14. 7/8 23. 109 +1
5. 0
15. 111000 24. 135°
6. 6174
16. 10C5 25. 12C6
7. 0
17. 0°W
8. 7
18. 1.5 cm
9. ‘Playing with Math’
Round 3
PUZZLE
What is the opposite of ‘G’

I P I

O P O R G B
What is the opposite of ‘O’

Face ‘O’ is on the opposite side of ‘G’

Poison & Rat

There are 1000 wine bottles. One of the bottles contains poisoned wine. A rat dies after one
hour of drinking the poisoned wine. How many minimum rats are needed to figure out which
bottle contains poison in hour.
Poison & Rat

7 rats
We need to figure out in hour. We need 7 rats to figure out the poisoned bottle. The result is
based on binary number system. We get 7 using ? log2128 ?.
The idea is to number bottles from 1 to 128 and write their corresponding binary numbers on
the bottle. Each rat is assigned a position in the binary numbers written on bottles. Let us take
an example. Rat 1 represents first bit in every bottle, rat 2 represents second bit and so on. If rat
numbers 2, 4 and 6 die, then bottle number 42 (Binary 0101010) is poisoned.
 Solve the FUTOSHIKI
Fill up the white boxes using 1 to 4 so that each row/column contains 1 to 4 without repetition
and the given relations(greater than or smaller than) holds.

< 3
<

<

<
Solve the FUTOSHIKI

4 3 2 1

2 < 4 1 3
<

1 2 < 3 4

<
3 1 4 2
Solve the KAKURO
Fill up the boxes using digits from 1 to 9, so that no number repeats in an row or column. Sum of
the digits in a row/column is provided by the arrow.

5 5 19
13 4
12
3
Solve the KAKURO

5 5 19
13 4 9 4
12 1 8 3
3 2 1
Pay the gold bar

You have got someone working for you for five days and a gold bar to pay him.
You must give them a piece of gold at the end of every day. What are the fewest
number of cuts to the bar of gold that will allow you to pay him 1/5th each day?
Pay the gold bar
2 cuts required.
Solution- 1 :
After Two cut there are three pieces of [1 unit and two 2 units]
◦ First day : Pay the worker Gold Bar with 1 unit.
◦ Second day : Pay the worker Gold with 2 units and take back the gold bar with 1 unit.
◦ Third day : Pay the worker Gold Bar with 1 unit.
◦ Fourth day : Pay the worker Gold with 2 units and take back the gold bar with 1 unit.
◦ Fifth day : Pay the worker with only left Gold Bar with 1 unit.
Solution- 2 :
After Two cut there are three pieces of [1 unit and two 2 units]
◦ First day : Pay the worker Gold Bar with 1 unit.
◦ Second day : Pay the worker Gold with other 1 units.
◦ Third day : Pay the worker Gold Bar with 3 unit and take back all the gold bar with 1 unit.
◦ Fourth day : Pay the worker Gold with 1 unit.
◦ Fifth day : Pay the worker with only left Gold Bar with 1 unit.
Mislabeled Jars
There are 3 jars, namely, A, B, C. All of them are mislabeled. Following are the labels of each of
the jars:
•A: Candies
•B: Sweets
•C: Candies and Sweets (mixed in a random proportion)
You can put your hand in a jar and pick only one eatable at a time. Tell the minimum number of
eatable(s) that has/have to be picked in order to label the jars correctly.
Mislabeled Jars
Jar A B C
Label Candies Sweets Mixed
Possible Eatable in it - 1 Sweets Mixed Candies
Possible Eatable in it-2 Mixed Candies Sweets

1 eatable is picked up from jar-C and tasted.

If it is candy, then A has sweets, B has mixed and C has candies I it.
If it is sweet, then A has mixed, B has candies, C has sweets in it.
Find the day

If 20th March 2024 is on Wednesday, then what day was 19th March 1995 ?
Find the day

19th March 1995 was Sunday.

Find out the student
Consider there are three rooms in a building and there are three student Ram,
Shyam, Jadu and we have three departments CSE, IT, ME. We don’t know which
of the student resides in which room neither do we know which department
they belong to. But you are given these clues.
 Shyam is from IT.
 CSE student stays in the room just to the left of Ram's room.
 The middle room does not belong to ME.
Now you have to find out the student who stays in the middle room.
Find out the student
Given CSE student stays in the left room where Ram stays & ECE is not middle
Room Left Middle Right Room Left Middle Right Room Left Middle Right
Dept. CSE IT ME Dept. ME CSE IT Dept. IT CSE ME
Name Ram Name Ram Name Ram

As Shyam is IT student. Then table 1 & 2 doesn’t provide solution. So third one is
solution. And the IT student is Shyam.
So,the middle one is occupied by Jadu.
Sum of Digits

Let f(x) = Sum of all the digits of x

Then what is the value of f(f(f(825 ))) ?
Sum of Digits

Actually f(f(f(825 ))) repeatedly calculates sum of digits till the sum<=9
It equals to the reminder when the number is divided by 9.
So, the answer is 8.
Contaminated Pills

You have 5 jars of pills, each contains 10 pills. Each pill weighs 10 grams, except
for contaminated pills contained in one jar, where each pill weighs 9
grams. Given a scale, how could you tell which jar had the contaminated pills in
just one measurement?
Contaminated Pills
To find the contaminated Jar, follow this step-wise approach.

Step 1: Take out 1 pill from jar 1, 2 pills from jar 2, 3 pills from jar 3, 4 pills from jar 4 and 5 pills
from jar 5.
Step 2: Put all these 15 pills on the scale. The correct weight is 150 (15*10). But one of the jars
has contaminated pills. So the weight will definitely be less than 150.
Step 3: If the weight is 149 then jar 1 has contaminated pills because there is only one
contaminated pill. If the weight is 148 then jar 2, if the weight is 147 then jar 3, if 146 then jar 4,
if 145 then jar 5.