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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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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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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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

B 20.

16 circles of radius 1
and 4 circles of radius 2.
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An easy one.

For radius 1,
we have 16 circles with center (x,y) 1<=X<=4, 1<=Y<=4(subtract 1 from both ends)

For radius 2,
WE have 4 circles with center (x,y) 2<=x<=3, 2<=x<=3(Subtract 2 from both ends)

for radius 3, such circle is not possible. Try subtracting 3 in above limit.

Hence total=20.

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)

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