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In the xy-coordinate plane how many circles can be constructed that have a center at (r, s), where r and s are integers, that have a radius with an integer length, and that in no portion lie outside the square region defined by \(0 \leq x \leq 5\) and \(0 \leq y \leq 5\)?
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26 Aug 2019, 00:38
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Asked: In the xy-coordinate plane how many circles can be constructed that have a center at (r, s), where r and s are integers, that have a radius with an integer length, and that in no portion lie outside the square region defined by 0≤x≤5 and 0≤y≤5?
Let r = 1; Centers of circle may be at (x,y) where x = {1,2,3,4} and y={1,2,3,4} Total cases = 4 * 4 = 16 Let r =2; Centers of circle may be at (x,y) where x = {2,3} and y={2,3} Total cases = 2 * 2 = 4 Let r=3; Any circle is NOT POSSIBLE since diameter = 6 >5 and portion of the circles lie outside the given region defined by 0≤x≤5 and 0≤y≤5 Total cases = 16 + 4 = 20
IMO B
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26 Aug 2019, 04:03
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First of all, the square is of dimension 5x5 as 0 ≤ x ≤ 5 and 0 ≤ y ≤ 5 and centers of all circles have coordinates as integer values and so are lengths of radii of all the circles.
Now, circles can be of either radius 1 unit or 2 units only since radius of 3 would result in diameter of 6 units whereas as per question no portion of a circle should be outside the square of dimension 5 x 5.
Consider the snapshot as attached.
Here, points except purple color and those on x & y axis can only be centers of circles. 1. Blue points can have length of 1 unit only to suffice the condition. Total blue points = 12 i.e. total circles are 12x1 = 12. 2. Pink points can have length of 1 and 2 units to suffice the condition. Total pink points = 4 i.e. total circles are 4x2 = 8.
Hence total circles are 20.
Answer (B).
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26 Aug 2019, 04:04
Given range in the coordinate plane where the circles have to be drawn is 0<=x<=5 and 0<=y<=5. Radius of each circle must be an integer, the origin of the circle must have cordinates (r,s) where r and s are integers. Possible radius that can be drawn within a square of sides 5x5 is 1 and 2.
Since the circles must be within the region defined in the coordinate plane, r and r have the following possible integer values r= {1,2,3, and 4} and s={1,2,3, and 4}. There are 4*4 =16 circles that can be drawn with a radius of 1 unit.
With a radius of 2, there possible values that r and s can take so that the circle can be drawn within the region defined above, are as follows: r={2 and 3} and s={2 and 3} 2*2 circles of radius 2 units can be drawn within the region defined above.
Total number of circles = 16+4=20. The answer is B imo.
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26 Aug 2019, 06:46
In the xy-coordinate plane how many circles can be constructed that have a center at (r, s), where r and s are integers, that have a radius with an integer length, and that in no portion lie outside the square region defined by 0≤x≤50≤x≤5 and 0≤y≤50≤y≤5?
A. 16 B. 20 C. 21 D. 24 E. 25
I plotted the values and solved with each center x=1 ; ( 1,1) ( 1,2) ( 1,3) ( 1,4) x=2 ; ( 2,1) ( 2,2) ( 2,2) ( 2,3) ( 2,3) ( 2,4) x=3; (3,1) ( 3,2) ( 3,2) ( 3,3) ( 3,3) ( 3,4) x=4 ; ( 4,1) ( 4,2) ( 4,3) ( 4,4) radius of lengths will be either 1 or 2 total center points ;20 IMO B
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26 Aug 2019, 10:28
In the xy-coordinate plane how many circles can be constructed that have a center at (r, s), where r and s are integers, that have a radius with an integer length, and that in no portion lie outside the square region defined by 0≤x≤5 and 0≤y≤5?
The circle should lie within the the square defined by the region 0<=x<=5 & 0<=y<=5 The radius of the circle should be an integer, so the radius value could be 1 or 2 And the center (r, s) are integers, so we can construct circles with radius 1 at center (r, s) as = (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) and with radius 2 and at center (r, s) as = (2,2), (2,3), (3,2), and (3,3) So, the total number of circles is 20 Answer is (B)
gmatclubot
Re: In the xy-coordinate plane how many circles can be constructed that ha
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49% (02:07) correct 
