QuantoLogic: cat

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Saturday, August 5, 2017

Problem of the week 6th August 2017

ABC is a right angled triangle with hypotenuse AC such that a circle of radius 6 passes through B and C. Th circle cuts the side AB and AC at points D and E respectively such that AD=5 and AE=4. Find BC

Wednesday, October 12, 2011

Problem of the Day 12 Oct 2011

How many factors of $2010^{2010}$ have last digit 2?

Monday, October 10, 2011

Problem of the Day 10 Oct 2011

Ten identical boxes of dimensions $2 \times 3 \times 5$ are stacked flat on top of each other, with the orientation of each box being independent and random. If $\frac{m}{n}$ is the probability that the height of the stack is $31$ and $\gcd(m, n)=1$ , find $m$ . All units are in feet.

Sunday, October 9, 2011

Problem of the Day 9th Oct 2011

In an increasing Arithmetic Progression, the product of the 5th term and the 6th term is 300. When
the 9th term of this A.P. is divided by the 5th term, the quotient is 5 and the remainder is 4. What is
the first term of the A.P.?
(a) 12 (b) –40 (c) –16 (d) –5

Saturday, October 8, 2011

Problem of the day 8th oct 2011

Rohan is asked to figure out the marks scored by Sunil in three different subjects with the help of
certain clues. He is told that the product of the marks obtained by Sunil is 72 and the sum of the
marks obtained by Sunil is equal to the Rohan’s current age (in completed years). Rohan could not
answer the question with this information. When he was also told that Sunil got the highest marks
in Physics among the three subjects, he immediately answered the question correctly. What is the
sum of the marks scored by Sunil in the two subjects other than Physics?

(a) 6 (b) 8 (c) 10 (d) Cannot be determined

Friday, October 7, 2011

Problem of the day 7th Oct 2011

Ten books are arranged in a row on a bookshelf. A student has to select three out of these ten books
in such a way that no two books selected by him must have been lying adjacently. In how many
ways can he make the selection?
(a) 56 (b) 64 (c) 72 (d) None of these

Problem of the Day 6th Oct 2011

If xΔ(y +1) = yΔ(x +1), xΔ x = 1 and (x − y)Δ(x + y) = xΔ y, then what is the value of 1001Δ1?
(a) 1000 (b) 100 (c) 10 (d) 1

Wednesday, October 5, 2011

Problem of the day 5th Oct 2011

The set S contains nine numbers. The mean of the numbers in S is 202. The mean of the five smallest of the numbers in S is 100. The mean of the five largest numbers in S is 300. What is the median of the numbers in S?

Tuesday, October 4, 2011

Problem of the Day 4th Oct 2011

Find $x$ and $y$ , where the variables are natural numbers:

$\frac{xy + y}{x + y} = \frac{15}{7}$
$\frac{x^2 + y}{2x + y} = \frac{19}{11}$

Monday, October 3, 2011

Problem of the Day 3rd Oct 2011

How many ordered pairs of positive integers $(m, n)$ satisfy the system

$\begin{align*}\gcd (m^3, n^2) & = 2^2 \cdot 3^2,\\ \text{LCM} [m^2, n^3] & = 2^4 \cdot 3^4 \cdot 5^6,\end{align*}$

where $\gcd(a, b)$ and $\text{LCM}[a, b]$ denote the greatest common divisor and least common multiple of $a$ and $b$ , respectively?

(A) 0 (B) 1 (C) 2 (3) More than 2

Sunday, October 2, 2011

Problem of The day 2nd Oct 2011

Let [x] and {x} respectively denote the integer and fractional part of of a real number x. If {n} + {3n}=1.4, find the sum of all possible values of 100{n}.

(A) 180 (B) 145 (C) 85 (d) 102

Saturday, October 1, 2011

Problem of the day 1 oct 2011

In 3-dimensional space, there are 3 rays leaving point $P$ . Any pair of 2 rays make a 60 degree angle with each other in their respective planes. Points $A$ , $B$ , and $C$ are situated on the rays (one per ray) such that $PA$ , $PB$ , and $PC$ are all integers, and $PA<PB<PC$ . if $PC=2010$ and $PB$ is odd, then determine the value of $PA$ if $\angle ABC = 90^{\circ}$ .

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Friday, September 30, 2011

Problem of the day 30 Sept 2011

What is the largest number in the sequence $\sqrt{50},2\sqrt{49},3\sqrt{48},\cdots,49\sqrt{2},50$ ?

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Thursday, September 29, 2011

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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Wednesday, September 28, 2011

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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Tuesday, September 27, 2011

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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Monday, September 26, 2011

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