15) A square garden with a side of 20 meters is watered with point sprinklers. Each sprinkler waters all points that are at most 10 meters away from it. What is the minimum number of sprinklers needed to water the entire garden?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5.
15) The answer is $(D)$

It is immediate to verify that 4 sprinklers are sufficient, just place them at the midpoints of the sides. To see that 3 sprinklers are not sufficient, we first observe that each sprinkler can only cover the points of a circle with a radius of $10 \mathrm{~m}$, in particular, two points more than $20 \mathrm{~m}$ apart cannot be covered by the same sprinkler. Since the vertices of the square are 4, if three sprinklers were sufficient, there would be two vertices of the square, $A$ and $B$, covered by the same sprinkler, and $A$ and $B$ should be adjacent 
(two opposite vertices have a distance of $20 \sqrt{2}>20 \mathrm{~m}$) and the irrigation point should coincide with the midpoint $M$ of the segment $A B$. Referring to the figure below, the sprinkler placed at point $M$ will cover the points of the semicircle with center $M$ and radius 10. Since the points $P$ on $A D$ 1 meter away from $A$, $Q$ on $B C$ 2 meters away from $B$, and $R$ the midpoint of $C D$ do not belong to this semicircle and have, pairwise, a distance greater than 20, at least three other distinct sprinklers are needed to cover these 3 points.

![](https://cdn.mathpix.com/cropped/2024_04_17_e738d1c220e6fc403f5eg-3.jpg?height=397&width=402&top_left_y=544&top_left_x=884)