Problem of the day 1 oct 2011

In 3-dimensional space, there are 3 rays leaving point . Any pair of 2 rays make a 60 degree angle with each other in their respective planes. Points , , and  are situated on the rays (one per ray) such that , , and  are all integers, and . if  and  is odd, then determine the value of  if .






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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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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 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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Bonus Question 26.07.09

The perimeter of a right triangle is 60. The height to the hypotenuse is 12 what is the area?
(A) 75 (B) 144 (C) 150 (D) 300 (E) none of these

Problem of the day 17.07.09

Find the area of right angle triangle whose inradius is 4 and circumradius
is 10?
a) 28                   b) 56                    c) 96                     d) 192                   e) none of these

Problem of The day 13.07.09

On a circle 26 equidistant points are marked. these points are joined to form a triangles. Of the triangles formed, how many of them will have their circumcenter on one of their sides.?

A) 318 B) 312 C) 288 d) 624 e) None of these

Problem of the Day 3.07.09

In a triangle ABC, perpendiculars BD and CE are drawn to the sides AC and AB. Points  D and E are joined, then the ratio of the area of ADE to the area of ABC is:

1) Cos²A 2) Sin²A 3) Cot²A 4) Tan²A  5) None of these

Problem of The Day 2.07.09

Two triangles are considered distinct if they cannot be superimposed on each other by rotation. How many distinct triangles, with integer sides,  exist, such that there perimeter is 30?

A) 19    B) 57    C) 114    d) 38    E) 36

Problem Of The Week 49

Find the maximum area that can be bound by four line segments of length 1, 2, 3 and 4. (you are not allowed to break a segment, you may join two :) )

1) 6√2

2) 2√6

3) 4√6

4) 3√2

5) None of these

Problem Of The Week 45

Find the largest value for for pairs of real numbers which satisfy

Problem Of The Week 32

Four Equilateral triangles are formed taking one of their sides as the sides of the square, the third vertices of equilateral triangles being inside the square. The ratio of the area of fig formed by the third vertices of the triangles to that of the square is nearly

Problem Of The Week 27

A golden rectangle is a rectangle in which the ratio of the width to length is the same as that of the length to the sum of the length and width. Which of the following is also true about a golden rectangle?
I. The ratio of the length to width is the same as the ratio of the width to the difference of the length and width.
II. The product of the length and width is equal to the product of the sum of the two sides and the difference of the two sides.
III. The length has to be greater than two times the width.

(1) Only I and II (2) Only II and III
(3) Only I and III (4) All the three statements
(5) None of these

Problem Of The Week 19

New Problem: A fresh one, I created it this morning, trying to find the most adequate data!

In a triangle ABC, altitude AD=6 is drawn to cut BC at D. From D, altitude DE=3 is drawn to cut AC at E. If it is know that AB =12. Find the ratio of the area of ABC to area of DEC?

A) 4:1 B) 16:1 C) 25:1 D) 5:1 E) Cannot be determined

Problem Of The Week 17

Two points are selected randomly on the surface of a sphere of radius R. What is the expected distance between them, along the surface of the sphere?

Problem is simple if you think logically, else you will have issues. This question is taken from teh CAT quant Blog by Suresh, the link of which you can see on the right!

Tipster:  The sum of the two legs of a right angled triangle is equal to sum of diameters of incircle and circumcircle :)

Concept 3 Circle and Triangles ( Part 1)

Geometry as a section wa spretty popular in CAT till 2004, consiting of 1/3rd of the problems and people used to think if they could handle it well they are clear with quant cutoff. And really that used to be the case. Cat has changed the trend reducing geometry every year since and last year in CAT 2007, there was not a single problem from geometry.  But, we can for sure be ready for nice geometry problems, so that if they come, we are up and ready for it.

Geometry has some major theorems. One should be clear about them, the ones on similarity of triangles, congruency of triangles, pythagoras, area and volume formula. Kindly refer to a text book for revising such concepts, I would recommend Quantum Cat By Arihant Publishers. Lets roll then !

The major theorems which we always need are :

Theorem 3.1 Pythagoras Theorem : a^2+b^2=c^2 where a,b,c are sides of a right angled triangle.

Clearly, C is the largest side, we call it hypotenuse.

The triplets of real numbers (a,b,c) which satisfy the above theorem is called pythagorean triplets. They are of real interest in all kinds of work.

Example 3.1 The length of one of the legs of a right triangle exceeds the length of the other leg by 10 cm but is smaller than that of the hypotenuse by 10 cm. Find the hypotenuse.

The obvious solution is  (a-10)^2+a^2=(a+10)^2 ( I have jumped a step)

solving we have a^2-20a=20a =>a=40 ( a can't be zero, its side of a triangle)

hypo is a+10=50

P.s : we have avoided the cumbersome assumption of sides as a,a+10 and a+20

Tipster clue: See this , the smallest integer pythagorean triplet is (3,4,5) so all numbers of the form (3k,4k,5k) will be pythagorean!

Practice Problem 3.1 FInd the sum of the lengths of the sides of a right angled triangle if the Circumradius=15 and inradius=6

Theorem 3.2  Sin law

a/Sin A=b/SinB =c/SIn C=2R   where a,b,c are sides opposite <A , <B and <C respectively and R is circumradius of Triangle ABC.

Very useful theorem, though we have entered the domain of trigonometry, but trigonometry, plane geometry and coordinate geometry are very important for each other to co exist.

Theorem 3.3 Cosine law

a^2=b^2+c^2-2bcCos A ( the notations remain the same as Theorem 3.2). The theorem can be similarly used for other angles too.

Practice Problem 3.2 Find the angle between the diagonal of a rectangle with perimeter 2p and area (3/16)p^2

Example 3.2 Find the length of the base of an isosceles triangle with area S and vertical angle A.

How do we start with this, we can offcourse going to need some basic geometry knowledge. let me tell you all of it. First the vertical angle of an isosceles triangle is the angle between the two equal sides( unless otherwise mentioned). The Perpendicular dropped on the unequal side from the opposite vertex, bisects the vertical angle as well as bisects the side. It means if we have a triangle ABC with AC=AB and AD perpendicular to BC then BD=BC and <BAD=<CAD.

The last thing we need is that area of a triangle is (1/2)bcsinA or (1/2)b^2Sin A for an isosceles triangle as b=c

now given (1/2) b^2sin A=S.......(1)

Now as AD bisects the vertical angle and then use BD=bsin(A/2)

hence BC=2BD=2bSin(A/2)

we can put the value of b from (1) and we are done !

Practice Problem 3.3  Find the largest angle of a triangle in which the altitude and the median drawn from the same vertex divide the angle at the vertex into three equal parts

Lets do a more involved example. This came in IMS SimCat 9. Nice and easy problem, but it might scare you for a moment if you look at the figure they drew. So I am not giving it :)

Example 3.3 In Triangle ABC, AD,BE and CF are the medians which intersect at G. ABCH is trapezium with AH=5units , and BC=10units  and Area( Tr BHC)=35 Sq units. Find the ratio of Area( BDFG): Area( ABCH). ( note we have  H and C on same side of B :) )

Here we again need to know this. The three medians divide the triangle into three triangle of equal area . Also they divide it into three quadrilaterals of equal area. So Area( BDFG)= (1/3)Area(ABC)

Next comes, the traingles drawn on the same base and between same parallel lines have equal area. Hence Area(ABC)=Area(BHC)=35 as we know the base BC, we know the altitude D= 2Area/base=70/10=7

Area of trapezium =(1/2)altitude( sum of paralle sides)= (1/2)7(10+5)

so our ratio is (35/3)/(15.7/2)=2/9

We are done :)

Problem Of The Week 16

Points M and N are taken on the hypotenuse of a right triangle ABC so that BC=BM and AC=AN.. Find <MCN

A relatively easy problem, still try to do it. You will benefit from it

Problem Of The Week 11

Triangle ABC is right angled at C. m<ABC=60 and AB=10. Let P be a point randomly chosen inside triangle ABC, such that BP extended meets CA at D. What is the probability that BD>5root(2)?