Circle Boundary Squares and Zero-Radius Cases
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batch-20261010-advance-20261014-r07-circle-foundations · 2026-10-10 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[0, 0], r=0, ordered points=[[0, 0], [1, 0], [0, -1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 1, 1], [0, 1, 1], [on-boundary, outside, outside], [Yes, No, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 1, 1], [0, 1, 1], [on-boundary, outside, outside], [Yes, No, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
2. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[0, 0], r=1, ordered points=[[0, 0], [1, 0], [2, 1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 1, 5], [-1, 0, 4], [inside, on-boundary, outside], [No, Yes, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 1, 5], [-1, 0, 4], [inside, on-boundary, outside], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
3. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[1, -1], r=2, ordered points=[[1, -1], [3, -1], [4, 0]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 4, 10], [-4, 0, 6], [inside, on-boundary, outside], [No, Yes, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 4, 10], [-4, 0, 6], [inside, on-boundary, outside], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
4. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[-1, -2], r=3, ordered points=[[-1, -2], [2, -2], [3, -1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 9, 17], [-9, 0, 8], [inside, on-boundary, outside], [No, Yes, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 9, 17], [-9, 0, 8], [inside, on-boundary, outside], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
5. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[0, 1], r=4, ordered points=[[0, 1], [4, 1], [5, 2]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 16, 26], [-16, 0, 10], [inside, on-boundary, outside], [No, Yes, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 16, 26], [-16, 0, 10], [inside, on-boundary, outside], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
6. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[1, 0], r=5, ordered points=[[1, 0], [6, 0], [7, 1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 25, 37], [-25, 0, 12], [inside, on-boundary, outside], [No, Yes, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 25, 37], [-25, 0, 12], [inside, on-boundary, outside], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
7. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[-1, -1], r=6, ordered points=[[-1, -1], [5, -1], [6, 0]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 36, 50], [-36, 0, 14], [inside, on-boundary, outside], [No, Yes, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 36, 50], [-36, 0, 14], [inside, on-boundary, outside], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
8. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[0, -2], r=7, ordered points=[[0, -2], [7, -2], [8, -1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 49, 65], [-49, 0, 16], [inside, on-boundary, outside], [No, Yes, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 49, 65], [-49, 0, 16], [inside, on-boundary, outside], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
9. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[1, 1], r=8, ordered points=[[1, 1], [9, 1], [10, 2]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 64, 82], [-64, 0, 18], [inside, on-boundary, outside], [No, Yes, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 64, 82], [-64, 0, 18], [inside, on-boundary, outside], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
10. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[-1, 0], r=9, ordered points=[[-1, 0], [8, 0], [9, 1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 81, 101], [-81, 0, 20], [inside, on-boundary, outside], [No, Yes, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 81, 101], [-81, 0, 20], [inside, on-boundary, outside], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
11. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[0, -1], r=10, ordered points=[[0, -1], [10, -1], [11, 0]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 100, 122], [-100, 0, 22], [inside, on-boundary, outside], [No, Yes, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 100, 122], [-100, 0, 22], [inside, on-boundary, outside], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
12. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[1, -2], r=11, ordered points=[[1, -2], [12, -2], [13, -1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 121, 145], [-121, 0, 24], [inside, on-boundary, outside], [No, Yes, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 121, 145], [-121, 0, 24], [inside, on-boundary, outside], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
13. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[-1, 1], r=12, ordered points=[[-1, 1], [11, 1], [12, 2]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 144, 170], [-144, 0, 26], [inside, on-boundary, outside], [No, Yes, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 144, 170], [-144, 0, 26], [inside, on-boundary, outside], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
14. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[0, 0], r=13, ordered points=[[0, 0], [13, 0], [14, 1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 169, 197], [-169, 0, 28], [inside, on-boundary, outside], [No, Yes, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 169, 197], [-169, 0, 28], [inside, on-boundary, outside], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
15. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[1, -1], r=14, ordered points=[[1, -1], [15, -1], [16, 0]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 196, 226], [-196, 0, 30], [inside, on-boundary, outside], [No, Yes, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 196, 226], [-196, 0, 30], [inside, on-boundary, outside], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
16. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[-1, -2], r=15, ordered points=[[-1, -2], [14, -2], [15, -1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 225, 257], [-225, 0, 32], [inside, on-boundary, outside], [No, Yes, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 225, 257], [-225, 0, 32], [inside, on-boundary, outside], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
17. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[0, 1], r=16, ordered points=[[0, 1], [16, 1], [17, 2]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 256, 290], [-256, 0, 34], [inside, on-boundary, outside], [No, Yes, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 256, 290], [-256, 0, 34], [inside, on-boundary, outside], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
18. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[1, 0], r=17, ordered points=[[1, 0], [18, 0], [19, 1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 289, 325], [-289, 0, 36], [inside, on-boundary, outside], [No, Yes, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 289, 325], [-289, 0, 36], [inside, on-boundary, outside], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
19. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[-1, -1], r=18, ordered points=[[-1, -1], [17, -1], [18, 0]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 324, 362], [-324, 0, 38], [inside, on-boundary, outside], [No, Yes, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 324, 362], [-324, 0, 38], [inside, on-boundary, outside], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
20. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[0, -2], r=19, ordered points=[[0, -2], [19, -2], [20, -1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 361, 401], [-361, 0, 40], [inside, on-boundary, outside], [No, Yes, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 361, 401], [-361, 0, 40], [inside, on-boundary, outside], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
21. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[1, 1], r=20, ordered points=[[1, 1], [21, 1], [22, 2]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 400, 442], [-400, 0, 42], [inside, on-boundary, outside], [No, Yes, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 400, 442], [-400, 0, 42], [inside, on-boundary, outside], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
22. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[-1, 0], r=21, ordered points=[[-1, 0], [20, 0], [21, 1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 441, 485], [-441, 0, 44], [inside, on-boundary, outside], [No, Yes, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 441, 485], [-441, 0, 44], [inside, on-boundary, outside], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
23. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[0, -1], r=22, ordered points=[[0, -1], [22, -1], [23, 0]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 484, 530], [-484, 0, 46], [inside, on-boundary, outside], [No, Yes, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 484, 530], [-484, 0, 46], [inside, on-boundary, outside], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
24. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[1, -2], r=23, ordered points=[[1, -2], [24, -2], [25, -1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 529, 577], [-529, 0, 48], [inside, on-boundary, outside], [No, Yes, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 529, 577], [-529, 0, 48], [inside, on-boundary, outside], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
25. Squared Circle Distances, Residuals, and Boundary Membership
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[-1, 1], r=24, ordered points=[[-1, 1], [23, 1], [24, 2]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).
View answer and explanation
([0, 576, 626], [-576, 0, 50], [inside, on-boundary, outside], [No, Yes, No])
Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 576, 626], [-576, 0, 50], [inside, on-boundary, outside], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-offset-residual-triple · candidate
26. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[0, 0], r=0, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([0, 0], [0, 0], 0, 0)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([0, 0], [0, 0], 0, 0).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
27. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[0, 0], r=1, axis=y. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([0, 1], [0, -1], 4, 2)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([0, 1], [0, -1], 4, 2).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
28. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[0, -10], r=2, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([2, -10], [-2, -10], 16, 4)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([2, -10], [-2, -10], 16, 4).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
29. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[1, -9], r=3, axis=y. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([1, -6], [1, -12], 36, 6)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([1, -6], [1, -12], 36, 6).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
30. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[-2, -8], r=4, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([2, -8], [-6, -8], 64, 8)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([2, -8], [-6, -8], 64, 8).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
31. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[-1, -7], r=5, axis=y. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([-1, -2], [-1, -12], 100, 10)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([-1, -2], [-1, -12], 100, 10).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
32. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[0, -6], r=6, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([6, -6], [-6, -6], 144, 12)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([6, -6], [-6, -6], 144, 12).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
33. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[1, -5], r=7, axis=y. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([1, 2], [1, -12], 196, 14)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([1, 2], [1, -12], 196, 14).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
34. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[-2, -4], r=8, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([6, -4], [-10, -4], 256, 16)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([6, -4], [-10, -4], 256, 16).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
35. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[-1, -3], r=9, axis=y. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([-1, 6], [-1, -12], 324, 18)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([-1, 6], [-1, -12], 324, 18).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
36. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[0, -2], r=10, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([10, -2], [-10, -2], 400, 20)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([10, -2], [-10, -2], 400, 20).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
37. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[1, -1], r=11, axis=y. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([1, 10], [1, -12], 484, 22)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([1, 10], [1, -12], 484, 22).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
38. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[-2, 0], r=12, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([10, 0], [-14, 0], 576, 24)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([10, 0], [-14, 0], 576, 24).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
39. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[-1, 1], r=13, axis=y. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([-1, 14], [-1, -12], 676, 26)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([-1, 14], [-1, -12], 676, 26).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
40. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[0, 2], r=14, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([14, 2], [-14, 2], 784, 28)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([14, 2], [-14, 2], 784, 28).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
41. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[1, 3], r=15, axis=y. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([1, 18], [1, -12], 900, 30)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([1, 18], [1, -12], 900, 30).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
42. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[-2, 4], r=16, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([14, 4], [-18, 4], 1024, 32)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([14, 4], [-18, 4], 1024, 32).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
43. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[-1, 5], r=17, axis=y. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([-1, 22], [-1, -12], 1156, 34)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([-1, 22], [-1, -12], 1156, 34).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
44. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[0, 6], r=18, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([18, 6], [-18, 6], 1296, 36)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([18, 6], [-18, 6], 1296, 36).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
45. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[1, 7], r=19, axis=y. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([1, 26], [1, -12], 1444, 38)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([1, 26], [1, -12], 1444, 38).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
46. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[-2, 8], r=20, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([18, 8], [-22, 8], 1600, 40)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([18, 8], [-22, 8], 1600, 40).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
47. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[-1, 9], r=21, axis=y. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([-1, 30], [-1, -12], 1764, 42)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([-1, 30], [-1, -12], 1764, 42).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
48. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[0, 10], r=22, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([22, 10], [-22, 10], 1936, 44)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([22, 10], [-22, 10], 1936, 44).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
49. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[1, 11], r=23, axis=y. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([1, 34], [1, -12], 2116, 46)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([1, 34], [1, -12], 2116, 46).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
50. Axis Antipodes, Squared Separation, and Diameter
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[-2, 12], r=24, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.
View answer and explanation
([22, 12], [-26, 12], 2304, 48)
Add/subtract the radius on the specified axis, then compute squared separation and diameter; do not fabricate distinct zero-radius points. The exact result, retaining the requested order, is ([22, 12], [-26, 12], 2304, 48).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-axis-antipodal-pair · candidate
51. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=0, scale k=5. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(0, 0, 0, 0, No)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (0, 0, 0, 0, No).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
52. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=1, scale k=0. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(0, 1, 0, 0, No)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (0, 1, 0, 0, No).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
53. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=2, scale k=3. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(6, 4, 36, 12, Yes)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (6, 4, 36, 12, Yes).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
54. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=3, scale k=1. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(3, 9, 9, 6, Yes)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (3, 9, 9, 6, Yes).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
55. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=4, scale k=2. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(8, 16, 64, 16, Yes)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (8, 16, 64, 16, Yes).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
56. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=5, scale k=3. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(15, 25, 225, 30, Yes)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (15, 25, 225, 30, Yes).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
57. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=6, scale k=1. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(6, 36, 36, 12, Yes)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (6, 36, 36, 12, Yes).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
58. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=7, scale k=2. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(14, 49, 196, 28, Yes)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (14, 49, 196, 28, Yes).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
59. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=8, scale k=3. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(24, 64, 576, 48, Yes)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (24, 64, 576, 48, Yes).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
60. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=9, scale k=1. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(9, 81, 81, 18, Yes)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (9, 81, 81, 18, Yes).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
61. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=10, scale k=2. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(20, 100, 400, 40, Yes)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (20, 100, 400, 40, Yes).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
62. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=11, scale k=3. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(33, 121, 1089, 66, Yes)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (33, 121, 1089, 66, Yes).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
63. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=12, scale k=1. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(12, 144, 144, 24, Yes)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (12, 144, 144, 24, Yes).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
64. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=13, scale k=2. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(26, 169, 676, 52, Yes)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (26, 169, 676, 52, Yes).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
65. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=14, scale k=3. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(42, 196, 1764, 84, Yes)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (42, 196, 1764, 84, Yes).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
66. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=15, scale k=1. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(15, 225, 225, 30, Yes)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (15, 225, 225, 30, Yes).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
67. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=16, scale k=2. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(32, 256, 1024, 64, Yes)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (32, 256, 1024, 64, Yes).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
68. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=17, scale k=3. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(51, 289, 2601, 102, Yes)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (51, 289, 2601, 102, Yes).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
69. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=18, scale k=1. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(18, 324, 324, 36, Yes)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (18, 324, 324, 36, Yes).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
70. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=19, scale k=2. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(38, 361, 1444, 76, Yes)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (38, 361, 1444, 76, Yes).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
71. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=20, scale k=3. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(60, 400, 3600, 120, Yes)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (60, 400, 3600, 120, Yes).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
72. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=21, scale k=1. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(21, 441, 441, 42, Yes)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (21, 441, 441, 42, Yes).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
73. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=22, scale k=2. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(44, 484, 1936, 88, Yes)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (44, 484, 1936, 88, Yes).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
74. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=23, scale k=3. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(69, 529, 4761, 138, Yes)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (69, 529, 4761, 138, Yes).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
75. Radius-Parameter Scaling and Positivity
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=24, scale k=1. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked.
View answer and explanation
(24, 576, 576, 48, Yes)
Multiply the nonnegative radius parameter, square it, and double it; parameter arithmetic is not a similarity-theorem proof. The exact result, retaining the requested order, is (24, 576, 576, 48, Yes).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-radius-scale-witness · candidate
76. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=0, horizontal displacement dx=0, ordered candidate y values=[0, 1, -1] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(0, [0, 1, 1], [Yes, No, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (0, [0, 1, 1], [Yes, No, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
77. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=5, horizontal displacement dx=3, ordered candidate y values=[4, -4, 0] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(16, [0, 0, -16], [Yes, Yes, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (16, [0, 0, -16], [Yes, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
78. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=1, horizontal displacement dx=2, ordered candidate y values=[0, 1, 2] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(-3, [3, 4, 7], [No, No, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (-3, [3, 4, 7], [No, No, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
79. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=4, horizontal displacement dx=-1, ordered candidate y values=[0, 4, 5] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(15, [-15, 1, 10], [No, No, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (15, [-15, 1, 10], [No, No, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
80. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=5, horizontal displacement dx=0, ordered candidate y values=[0, 5, 6] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(25, [-25, 0, 11], [No, Yes, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (25, [-25, 0, 11], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
81. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=6, horizontal displacement dx=1, ordered candidate y values=[0, 6, 7] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(35, [-35, 1, 14], [No, No, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (35, [-35, 1, 14], [No, No, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
82. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=7, horizontal displacement dx=-1, ordered candidate y values=[0, 7, 8] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(48, [-48, 1, 16], [No, No, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (48, [-48, 1, 16], [No, No, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
83. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=8, horizontal displacement dx=0, ordered candidate y values=[0, 8, 9] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(64, [-64, 0, 17], [No, Yes, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (64, [-64, 0, 17], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
84. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=9, horizontal displacement dx=1, ordered candidate y values=[0, 9, 10] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(80, [-80, 1, 20], [No, No, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (80, [-80, 1, 20], [No, No, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
85. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=10, horizontal displacement dx=-1, ordered candidate y values=[0, 10, 11] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(99, [-99, 1, 22], [No, No, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (99, [-99, 1, 22], [No, No, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
86. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=11, horizontal displacement dx=0, ordered candidate y values=[0, 11, 12] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(121, [-121, 0, 23], [No, Yes, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (121, [-121, 0, 23], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
87. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=12, horizontal displacement dx=1, ordered candidate y values=[0, 12, 13] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(143, [-143, 1, 26], [No, No, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (143, [-143, 1, 26], [No, No, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
88. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=13, horizontal displacement dx=-1, ordered candidate y values=[0, 13, 14] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(168, [-168, 1, 28], [No, No, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (168, [-168, 1, 28], [No, No, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
89. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=14, horizontal displacement dx=0, ordered candidate y values=[0, 14, 15] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(196, [-196, 0, 29], [No, Yes, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (196, [-196, 0, 29], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
90. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=15, horizontal displacement dx=1, ordered candidate y values=[0, 15, 16] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(224, [-224, 1, 32], [No, No, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (224, [-224, 1, 32], [No, No, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
91. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=16, horizontal displacement dx=-1, ordered candidate y values=[0, 16, 17] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(255, [-255, 1, 34], [No, No, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (255, [-255, 1, 34], [No, No, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
92. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=17, horizontal displacement dx=0, ordered candidate y values=[0, 17, 18] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(289, [-289, 0, 35], [No, Yes, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (289, [-289, 0, 35], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
93. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=18, horizontal displacement dx=1, ordered candidate y values=[0, 18, 19] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(323, [-323, 1, 38], [No, No, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (323, [-323, 1, 38], [No, No, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
94. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=19, horizontal displacement dx=-1, ordered candidate y values=[0, 19, 20] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(360, [-360, 1, 40], [No, No, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (360, [-360, 1, 40], [No, No, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
95. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=20, horizontal displacement dx=0, ordered candidate y values=[0, 20, 21] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(400, [-400, 0, 41], [No, Yes, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (400, [-400, 0, 41], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
96. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=21, horizontal displacement dx=1, ordered candidate y values=[0, 21, 22] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(440, [-440, 1, 44], [No, No, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (440, [-440, 1, 44], [No, No, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
97. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=22, horizontal displacement dx=-1, ordered candidate y values=[0, 22, 23] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(483, [-483, 1, 46], [No, No, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (483, [-483, 1, 46], [No, No, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
98. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=23, horizontal displacement dx=0, ordered candidate y values=[0, 23, 24] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(529, [-529, 0, 47], [No, Yes, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (529, [-529, 0, 47], [No, Yes, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
99. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=24, horizontal displacement dx=1, ordered candidate y values=[0, 24, 25] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(575, [-575, 1, 50], [No, No, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (575, [-575, 1, 50], [No, No, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
100. Circle Squared Remainder at a Fixed Horizontal Offset
Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. r=25, horizontal displacement dx=-1, ordered candidate y values=[0, 25, 26] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).
View answer and explanation
(624, [-624, 1, 52], [No, No, No])
Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (624, [-624, 1, 52], [No, No, No]).
Reviewed step-by-step derivation
Circle boundaries and zero-radius degeneracy · qf-density14-circle-horizontal-section-candidates · candidate
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