Counts the number of squares with area 1, integer vertices, fully inside a circle of radius r.

maths/Drone_Domain.py

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+"""Drone's Domain
+We call a square valid on a 2d coordinate plane if it satisfies the following conditions:
+
+Area of the square should be equal to 1.
+All vertices of the square have integer coordinates.
+All vertices of square lie inside or on the circle of radius r, centered at origin.
+Given the radius of circle r, count the number of different valid squares. Two squares are different if atleast one of the vertices of first square is not a vertex of second square.
+
+Input Format
+The first line contains the integer r, the radius of the circle.
+
+Constraints
+1 <= r <= 2 * 10^5
+Use a 64-bit integer type (like long long in C++ or long in Java) for sum calculations to prevent overflow.
+
+Output Format
+Print the number of different squares in this circle of radius r.
+
+Sample Input 1
+Sample Output 0"""
+
+# Title: Count Axis-Aligned Unit Squares in Circle
+# Description: Counts the number of squares with area 1, integer vertices, fully inside a circle of radius r.
+
+import sys
+import math
+
+r=int(sys.stdin.readline())
+count=0
+for x in range(-r,r):
+    for y in range(-r,r):
+        # Check all four vertices are inside the circle
+        if (x*x+y*y<=r*r and
+            (x+1)*(x+1)+y*y<=r*r and
+            x*x+(y+1)*(y+1)<=r*r and
+            (x+1)*(x+1)+(y+1)*(y+1)<=r*r):
+            count += 1
+
+print(count)