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What is the greatest possible number of points at which 11 circles wit

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What is the greatest possible number of points at which 11 circles wit  [#permalink]

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New post 19 Dec 2015, 07:43
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A
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Math Revolution and GMAT Club Contest Starts!



QUESTION #14:

What is the greatest possible number of points at which 11 circles with different radii intersected one another?

A. 45
B. 60
C. 85
D. 90
E. 110


Check conditions below:



Math Revolution and GMAT Club Contest

The Contest Starts November 28th in Quant Forum


We are happy to announce a Math Revolution and GMAT Club Contest

For the following four (!) weekends we'll be publishing 4 FRESH math questions per weekend (2 on Saturday and 2 on Sunday).

To participate, you will have to reply with your best answer/solution to the new questions that will be posted on Saturday and Sunday at 9 AM Pacific.
Then a week later, the forum moderator will be selecting 2 winners who provided most correct answers to the questions, along with best solutions. Those winners will get 6-months access to GMAT Club Tests.

PLUS! Based on the answers and solutions for all the questions published during the project ONE user will be awarded with ONE Grand prize:

PS + DS course with 502 videos that is worth $299!



All announcements and winnings are final and no whining :-) GMAT Club reserves the rights to modify the terms of this offer at any time.


NOTE: Test Prep Experts and Tutors are asked not to participate. We would like to have the members maximize their learning and problem solving process.

Thank you!


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Re: What is the greatest possible number of points at which 11 circles wit  [#permalink]

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New post 20 Dec 2015, 03:30
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Any Circle intersects another circle at maximum two intersection points.

With two circles : greatest number of intersection points = 2
With third circle introduced to the first two (already intersecting at two points), the third circle can intersect both the circles at two different intersection points, introducing 4 more intersection points ( third circle cuts each of the first two at two different poitns) . in addition to the two intersection points of first two.
i.e., with three circles , max # of intersection points = 2 + 2*2 = 6
with 4th circle introduced, it cuts the first three circles at two intersection points each i.e, 2*3 additional intersection points introduced now.
with 4 circles , max # of intersection points = (2 + 2*2) (intersection points for 3 circles) + 2*3 (additional intersection points introduced by 4th circle) = 2 + 2*2 + 2*3
Similarily , with 5th circle . total = 2 + 2*2 + 2*3 + 2*4
we see a series emerging here
greatest number of of intersection points for m circles = 2 + 2*2 + 2*3 + .....+ 2*(m-1)

Applying the series extrapolation for 11 circles, greatest number of intersection points = 2 + 2*2 + 2*3 + .....+ 2*10

= 2 * (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)
= 2 * (110)/2 = 110 ( calculation for the sum shown below)
-- Calculation for the sum.
since the sum in the paranthesis is a sum of consecutive integers , we can calculate the average by (last + first) / 2
Sum in paranthesis = average * # of terms = 10*((1 + 10)/2) = 110/2
--

Correct Answer : E
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Re: What is the greatest possible number of points at which 11 circles wit  [#permalink]

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New post 19 Dec 2015, 09:43
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Answer is E i.e 110

Two circles can intersect at maximum 2 points.

Three circles can intersect at maximum 6 points..

Because we select 2 circles from 3 and each two circles can intersect at maximum 2 points

2\(c^3_2\) ... Three circles can intersect at 6 points

So 11 circles can intersect at maximum

\(2c^{11}_2\) ... 2 * 11 * 5 = 110
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Re: What is the greatest possible number of points at which 11 circles wit  [#permalink]

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New post 19 Dec 2015, 11:05
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The first 2 circles intersect at 2 points.
3rd circle can intersect the first two circles at 4 points. refer to the figure.

i.e Circle 1 = no intersection
Circle 2 = 2 points
Circle 3 = 2+4 points
Circle 4 = 2+4+8 points
.
.
.
Circle 11= 2+4+8+10+12+14+16+18+20 = n(n+1)= 10*11=110 points.
Implies N circles can be drawn to intersect N(N-1) points . In this case, 11 circles can be drawn that can intersect at 11*10=110 points

Ans E
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Re: What is the greatest possible number of points at which 11 circles wit  [#permalink]

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New post 20 Dec 2015, 19:46
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1st Circle doesn't intersect itself - 2*(1-1)=0 points
2nd Circle intersects 1st at 2*(2-1)=2 Points
3rd Circle intersect 2nd and 3rd at 2*(3-1)=4 Points
4th Circle intersects other Circles at 2*(4-1)=6 points
...
11th circle intersects other circles at 2*(11-1)=20 points

Therefore total no of points = 2 + 4 + 6 + ... + 20
\(Sum = 10/2 * (20+2) = 5*22 =110\)

Answer is E
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Re: What is the greatest possible number of points at which 11 circles wit  [#permalink]

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New post 21 Dec 2015, 16:54
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Answer is E -> 110.

2 circle intersects at 2 points, 3 circles intersect at 4 points, 4 circles intersect at 6 points, 5 circles at 8 points, etc., Hence max # of points will be the result of addition of these points - as you can draw a circle anyway you want to ensure there are distinct points with each of these circle's intersection.

S = 2+4+6+... +20 (n=10) as there are 11 circles
S = (a1+an)/2*n = (2+20)/2*10 = 110
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Re: What is the greatest possible number of points at which 11 circles wit  [#permalink]

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New post 21 Dec 2015, 19:43
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Given: Number of circles with different radii = 11

We can draw a few circles to obtain the below results

For 2 circles, max number of points of intersection = 2 = 2 * 1
For 3 circles, max number of points of intersection = 6 = 3 * 2
For 4 circles, max number of points of intersection = 12 = 4 * 3
We can observe a pattern here.

For n circles with different radii, max number of points of intersection can be deduced from the above results as n * (n-1).

Therefore, for 11 circles with different radii, max number of points of intersection = 11 * 10 = 110

Answer: E
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Re: What is the greatest possible number of points at which 11 circles wit  [#permalink]

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New post 24 Dec 2015, 09:21
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The answer is E
By drawing circles and deducing
2 circles intersect at 2 points(1x2)
3 circles intersect at 6 points(2x3)
4 circles intersect at 12 points(3x4)
11 circles intersect at 10x11=110 points
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Re: What is the greatest possible number of points at which 11 circles wit  [#permalink]

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New post 25 Dec 2015, 04:52
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2 circles can intersect at the most 2 points. ---> which is 2P2
3 circles can intersect at the most 6 points. ----> which is 3P2

Therefore it is similar to number of arrangements possible for 11 objects taken two at a time.
Therefore 11 circles with different radii will intersect at 11P2 = 11!/9! = 11*10= 110 points. Answer choice E.
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Re: What is the greatest possible number of points at which 11 circles wit  [#permalink]

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New post 27 Dec 2015, 02:14
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1 circles have 0 intersection point ( 1 x 0)
2 circles have 2 intersection points (2 x 1)
3 circles have 6 intersection points (3 x 2)
4 circles have 12 intersection points (4 x 3)
....
....
a pattern develops!
So, 11 circles have (11 x 10) = 110 intersection points

However, an expression exists as well ie; C x (C-1) = I, where C is the number of circles and I the number of intersections.
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Re: What is the greatest possible number of points at which 11 circles wit  [#permalink]

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New post 27 Dec 2015, 11:17
Bunuel wrote:

Math Revolution and GMAT Club Contest Starts!



QUESTION #14:

What is the greatest possible number of points at which 11 circles with different radii intersected one another?

A. 45
B. 60
C. 85
D. 90
E. 110


Check conditions below:



Math Revolution and GMAT Club Contest

The Contest Starts November 28th in Quant Forum


We are happy to announce a Math Revolution and GMAT Club Contest

For the following four (!) weekends we'll be publishing 4 FRESH math questions per weekend (2 on Saturday and 2 on Sunday).

To participate, you will have to reply with your best answer/solution to the new questions that will be posted on Saturday and Sunday at 9 AM Pacific.
Then a week later, the forum moderator will be selecting 2 winners who provided most correct answers to the questions, along with best solutions. Those winners will get 6-months access to GMAT Club Tests.

PLUS! Based on the answers and solutions for all the questions published during the project ONE user will be awarded with ONE Grand prize:

PS + DS course with 502 videos that is worth $299!



All announcements and winnings are final and no whining :-) GMAT Club reserves the rights to modify the terms of this offer at any time.


NOTE: Test Prep Experts and Tutors are asked not to participate. We would like to have the members maximize their learning and problem solving process.

Thank you!



MATH REVOLUTION OFFICIAL SOLUTION:

The greatest possible number of points of intersection between two circles is 2. The greatest possible number of points of intersection of three circles is 2+2*2 (The greatest possible number of points of intersection between two circles + The greatest possible number of points of intersection that one circle can have with other two circles). The greatest possible number of points of intersection of four circles is 2+2*2+2*3 (The greatest possible number of points of intersection of three circles + The greatest possible number of points of intersection that one circle can have with other two circles).

So, the greatest possible number of points of intersection of 11 circles is, 2+2*2+2*3+…..2*10=2(1+2+…..+10)=2[10(10+1)/2]=110. The correct answer is E.

Note: 1+2+…..+n=n(n+1)/2
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Re: What is the greatest possible number of points at which 11 circles wit   [#permalink] 27 Dec 2015, 11:17

What is the greatest possible number of points at which 11 circles wit

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