Problem of the day 30 Sept 2011

What is the largest number in the sequence ?




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Problem of the Day 29 Sept 2011

Let a, b, c be three distinct odd natural numbers. Which of the following can be the sum of the squares of a, b and c?

(a) 3333                            (b) 5555                                (c) 9999                        (d) 7777

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Problem of the Day 28 Sept 2011

A good approximation of π is 3.14. Find the least positive integer d such that if the area of a circle with

diameter d is calculated using the approximation 3.14, the error will exceed 1.

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Permutations and Combinations- Free Lecture Notes

Many of you have emailed about some easy reference for Permutations and Combinations, Number theory etc. I cannot say if it is an easy reference, but it is one concise one.. It is from one of our Professors at IIT Kanpur, Dr A K Lal, great chap, I must say. This is one gem of a compilation. Hope it helps you.. If the material is too tough, just ignore!

http://home.iitk.ac.in/~arlal/book/mth202.pdf

Enjoy!

Problem of the Day 27 Sept 2011

The diagram below shows some small squares each with area 3 enclosed inside a larger square. Squares that touch each other do so with the corner of one square coinciding with the midpoint of a side of the other square. Find integer n such that the area of the shaded region inside the larger square but outside the smaller squares is √n

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Problem of the day 26 Sept 2011

Let a₁= 2 and a_n+1 = 2a_n + 1. Find the least value of a_n which is not prime.

(a) 47                  (b) 4                    (c) 5                 (d) 95

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Problem of the Day 25 Sept 2011

Let P be the set of all the vertices of a regular polygon of 25 sides with its center at C. How many triangles have vertices in P and contain the point C in the interior of the triangles?


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Problem of the Day 24 Sept 2011

In triangles ABC and DEF, DE=4AB, EF=4BC, FD=4CA The area of triangle DEF is 360 units more than the area of triangle ABC. Compute the area of triangle ABC



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Problem of the Day 23 Sept 2011

Darryl has a six-sided die with faces 1; 2; 3; 4; 5; 6. He knows the die is weighted so that one face
comes up with probability 1/2 and the other five faces have equal probability of coming up. He
unfortunately does not know which side is weighted, but he knows each face is equally likely
to be the weighted one.

He rolls the die 5 times and gets a 1; 2; 3; 4 and 5 in some unspecified order. Compute the probability that his next roll is a 6.



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How to use the options to maximize scores in QA/DI/LR in IIM CAT

The good thing about CAT is, that it is not a mathematics test. So, it helps that you know the process and the steps but you should be smart and try to minimize the work. This is why there are options to help you. A smart use of options will save a lot of time and help you increase your score.

Some examples

Two different two-digit natural numbers are written beside each other such that the larger number is written on the left. When the absolute difference of the two numbers is subtracted from the four-digit number so formed, the number obtained is 5481. What is the sum of the two two-digit numbers?


(a) 70         (b) 71                (c) 72                         (d) 73


Now it is pretty much obvious the number is of the form 55xy
To find xy we can just subtract 55 from the options which gives xy= 15, 16, 17, 18

If you are lucky you will start with 73 and you will know it is right, else even if you start with 15, you will soon reach 18 and get your answer.


The direct method will be a bit long.

 Let n be the total number of different 5-digit numbers with all distinct digits, formed using 2, 3, 4, 5 and 6 and divisible by 4. What is the value of n?

1] 44 2] 32 3] 36 4] 38 5] 40

Permutations & Combinations is probably the most ‘hated’ topic. However, if you understand the basics and use logic, then it is the most fun-filled topic of all. Let us get to this question. As mentioned, we need to find out the 5 digit numbers divisible by 4 formed by the digits given in the question. To be divisible by 4, the last 2 digits should be divisible by 4. So to arrive at the answer, the first step is to find out combinations of the 2 digits from 2,3,4,5 & 6 that are divisible by 4 – eg: 24. Then, for each such combination, the last 2 digits are fixed. The remaining 3 digits can be arranged in 3! Ways = 6 ways. So the answer would be - 6 multiplied by the total no. of combinations of 2 digits divisible by 4. The answer necessarily should be a multiple of 6 and therefore the answer is 36 – option [3]. We just got lucky here, by the way, since there is just one option that is divisible by 6!

Problem of the Day 22 Sept 2011

Susan plays a game in which she rolls two fair standard six-sided dice with sides labeled one through six. She wins if the number on one of the dice is three times the number on the other die. If Susan plays this game three times, compute the probability that she wins at least once.

Problem of the Week 21 Sept 2011

The leftmost digit of an integer of length 2000 digits is 3. In this integer, any two consecutive digits must be divisible by 17 or 23. The 2000th digit may be either a or b. What is the value of a+b?

Problem of the Day 20th Sept 2011

Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters?


(a) 200   (b) 300 (c) 350  (4) 550

Problem of the Day 19 September 2011

Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly  mintues. The probability that either one arrives while the other is in the cafeteria is  and where  and  are positive integers, and  is not divisible by the square of any prime. Find 

Problem of the day 15 sept 2011

Rachel and Brian are playing a game in a grid with 1 row of 2011 squares. Initially, there is one
white checker in each of the first two squares from the left, and one black checker in the third square
from the left. At each stage, Rachel can choose to either run or ght. If Rachel runs, she moves the
black checker 1 unit to the right, and Brian moves each of the white checkers one unit to the right. If
Rachel chooses to fight, she pushes the checker immediately to the left of the black checker 1 unit to
the left, the black checker is moved 1 unit to the right, and Brian places a new white checker in the
cell immediately to the left of the black one. The game ends when the black checker reaches the last
cell. How many different final configurations are possible?

a) 2011   b) 2010  c) 2009 d) None

Problem of the day 14 Sep 2011

3 men -A,B and C played a Dart game.

1.Each of the dart lodged in the game board scored 1,5,10,25,50, or 100 points.
2.Each man threw 9 darts that lodged in the board.
3.Each man's total score was the same as any other man's total score.
4.No number of points scored by a dart was scored by more than 1 man.
5.A scored all the 5s and B scored all the 10s.

Who scored all the 100's?


1) A           2) B                   3) C      4) Cannot be determined

Problem of the Day 13 Sept 2011

Let X be the set of three digit prime numbers with the following properties: 1) Each digit of the elements of X are distinct. 2) Each digit of the elements of X are prime. Let K be the sum of all the elements of X. Find the sum of the digits of K?

Tips: Tournament Problems

Tournament Problems
There are 16 teams and they are divided into 2 pools of 8 each. Each team in a group plays against one another on a round-robin basis. Draws in the competition are not allowed. The top four teams from each group will qualify for the next round i.e round 2. In case of teams having the same number of wins, the team with better run-rate would be ranked ahead.

1. Minimum number of wins required to qualify for the next round _____?
2. Minimum number of wins required to guarantee qualification in the next round _____?

Now, i don't know how many of you are aware of the following method. But 1 thing I mention in advance that this should take only 30 seconds to solve
1.
1 group is consisting of 8 teams. So each team will play 7 match each. Suppose each of the 8 teams were seeded and we consider the case where a higher seeded team will always win.

So the number of wins for the 8 teams would be 7,6,5,4,3,2,1,0 with highest seeded team winning all and lowest seeded team losing all.

For minimum number of wins we allow 3 teams to win maximum number of matches. Of the remaining 5 teams just find out the mean of their number of wins.

In this case it would be (4+3+2+1+0)/5=2.

So 5 teams can end up with 2 wins each and a team with better run rate will qualify with 2 wins.

2.

In this case consider the mean of first 5 higher seeded teams (7+6+5+4+3)/5=5

So it may be the case that 5 teams can end up having 5 wins each. And hence 1 team will miss the second round birth. So minimum number of wins to guarantee a place would be 6.

The trick is to consider wording, qualify means best case scenario while guarantee qualify means worst case scenario.

Problem of the Day 12 Sept 2011

'We__Aaron,Brian and Clyde_each have some children.
1.Aaron has at least 1 girl and twice as many boys as girls.
2.Brian has at least 1 girl and three times as many boys as girls.
3.Clyde has at least 1 girl and three more boys than girls.
4.When i tell you the number of children wee have altogether,a number less than 25,you will now how many children i have,but not how many children the others has.Altogether we have..."

Who is the speaker?

A) Aaron b) Brian C) Clyde D) Cannot be determined

My house has a number

1.If my house number is a multiple of 3,then its a number through 50-59.
2.If my house number is not a multiple of 4,then it's number through 60-69.
3.If my house number is not a multiple of 6,then it is a number from 70 through 79.

What is my house number?

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Problem Set 9 Sept 2011

Q1 The integers from 1 to n are written in increasing order from left to right on a blackboard. David and Goliath play the following game: starting with David, the two players alternate erasing any two consecutive numbers and replacing them with their sum or product. Play continues until only one number on the board remains. If it is odd, David wins, but if it is even, Goliath wins. Find the probability that Goliath wins if n=2011?

a) 0 b) 1 c) 1/2 d) None of these


Q2 A classroom has 30 students and 30 desks arranged in 5 rows of 6.The class has 15 boys and 15 girls.If the students be placed in the chairs such that no boy is sitting in front of, behind, or next to another boy, and no girl is sitting in front of, behind, or next to another girl in x(y!)(z!), where x, y and z are positive integers. Find x+y+z?

a) 32 b) 30 c) 27 d) 29


Q3. Find the sum of all integers x such that 2x^2 + x- 6 is a positive integral power of a prime positive integer?

a) 7 b) 5 c) 4 d) 12

We are BACK!

Well, for starters Quantologic has moved from Wordpress and will now solely be on Blogger and hosted on www.quantologic.in. This marks the beginning of a new era as the site was dormant for a long while. Hope to keep the fire burning!

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