SOLUTION: Find the number of pairs of integers, x and y, such that (x^2 + y^2) < 100. An integer may be positive, negative, or zero.

Let's count the integer points in the first quadrant area  0 <  < 100,  0 < argument <= 90 degrees.


    y = 1;  x from 0 to 9  (10 points);

    y = 2;  x from 0 to 9  (10 points);

    y = 3;  x from 0 to 9  (10 points);

    y = 4;  x from 0 to 9  (10 points);

    y = 5;  x from 0 to 8  (9 points);

    y = 6;  x from 0 to 7  (8 points);

    y = 7;  x from 0 to 7  (8 points);

    y = 8;  x from 0 to 5  (6 points);

    y = 9;  x from 0 to 4  (5 points);

    y = 10;  no points     (0 points).


So, in the first quadrant we have 


    10 + 10 + 10 + 10 + 9 + 8 + 8 + 6 + 5 = 76 points.


We take this amount 4 times and add the point (0,0).


In all, we have 4*76 + 1 = 305 points.      ANSWER



    Notice that in my count for QI, I include the points on the vertical axis x= 0 (0 < y < 10),
    but do not include the points on the horizontal axis y= 0 (0 < x < 10).


    Thus, I can multiply 76 by 4 without doubling counted points.