# Exact Squared Distances, Signs, and Tie Groups

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261008-advance-20261010-r03-distance-formula-foundations · 2026-10-08 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[0, 0], [3, 4]] in order. Give their two squared L2 distances.

View answer and explanation(0, 25)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (0, 25).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0001/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 2. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[1, -1], [2, 3]] in order. Give their two squared L2 distances.

View answer and explanation(2, 13)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (2, 13).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0002/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 3. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[2, -1], [3, 4]] in order. Give their two squared L2 distances.

View answer and explanation(5, 25)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (5, 25).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0003/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 4. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[3, -1], [4, 2]] in order. Give their two squared L2 distances.

View answer and explanation(10, 20)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (10, 20).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0004/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 5. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[4, -1], [5, 3]] in order. Give their two squared L2 distances.

View answer and explanation(17, 34)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (17, 34).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0005/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 6. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[5, -1], [6, 4]] in order. Give their two squared L2 distances.

View answer and explanation(26, 52)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (26, 52).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0006/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 7. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[6, -1], [7, 2]] in order. Give their two squared L2 distances.

View answer and explanation(37, 53)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (37, 53).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0007/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 8. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[7, -1], [8, 3]] in order. Give their two squared L2 distances.

View answer and explanation(50, 73)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (50, 73).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0008/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 9. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[8, -1], [9, 4]] in order. Give their two squared L2 distances.

View answer and explanation(65, 97)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (65, 97).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0009/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 10. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[9, -1], [10, 2]] in order. Give their two squared L2 distances.

View answer and explanation(82, 104)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (82, 104).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0010/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 11. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[10, -1], [11, 3]] in order. Give their two squared L2 distances.

View answer and explanation(101, 130)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (101, 130).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0011/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 12. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[11, -1], [12, 4]] in order. Give their two squared L2 distances.

View answer and explanation(122, 160)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (122, 160).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0012/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 13. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[12, -1], [13, 2]] in order. Give their two squared L2 distances.

View answer and explanation(145, 173)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (145, 173).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0013/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 14. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[13, -1], [14, 3]] in order. Give their two squared L2 distances.

View answer and explanation(170, 205)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (170, 205).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0014/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 15. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[14, -1], [15, 4]] in order. Give their two squared L2 distances.

View answer and explanation(197, 241)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (197, 241).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0015/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 16. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[15, -1], [16, 2]] in order. Give their two squared L2 distances.

View answer and explanation(226, 260)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (226, 260).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0016/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 17. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[16, -1], [17, 3]] in order. Give their two squared L2 distances.

View answer and explanation(257, 298)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (257, 298).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0017/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 18. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[17, -1], [18, 4]] in order. Give their two squared L2 distances.

View answer and explanation(290, 340)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (290, 340).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0018/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 19. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[18, -1], [19, 2]] in order. Give their two squared L2 distances.

View answer and explanation(325, 365)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (325, 365).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0019/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 20. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[19, -1], [20, 3]] in order. Give their two squared L2 distances.

View answer and explanation(362, 409)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (362, 409).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0020/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 21. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[20, -1], [21, 4]] in order. Give their two squared L2 distances.

View answer and explanation(401, 457)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (401, 457).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0021/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 22. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[21, -1], [22, 2]] in order. Give their two squared L2 distances.

View answer and explanation(442, 488)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (442, 488).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0022/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 23. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[22, -1], [23, 3]] in order. Give their two squared L2 distances.

View answer and explanation(485, 538)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (485, 538).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0023/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 24. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[23, -1], [24, 4]] in order. Give their two squared L2 distances.

View answer and explanation(530, 592)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (530, 592).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0024/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 25. Ordered Pair of Squared Euclidean Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[24, -1], [25, 2]] in order. Give their two squared L2 distances.

View answer and explanation(577, 629)

Sum the two coordinate squares for each vector separately; zero is allowed, and the result is a squared distance, not the distance itself. The exact result, retaining the requested order, is (577, 629).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0025/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-squares-pair · candidate





## 26. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[3, 4], integer candidate c=-5. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(No, 0)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0026/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 27. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[3, 4], integer candidate c=5. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(Yes, 0)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0027/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 28. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[3, 4], integer candidate c=4. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(Yes, -9)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0028/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 29. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[0, 0], integer candidate c=0. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(Yes, 0)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0029/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 30. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[4, 5], integer candidate c=7. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(Yes, 8)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0030/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 31. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[5, 6], integer candidate c=-8. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(No, 3)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0031/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 32. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[6, 7], integer candidate c=9. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(Yes, -4)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0032/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 33. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[7, 8], integer candidate c=-10. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(No, -13)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -13).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0033/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 34. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[8, 9], integer candidate c=11. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(Yes, -24)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -24).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0034/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 35. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[9, 10], integer candidate c=-12. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(No, -37)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -37).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0035/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 36. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[10, 11], integer candidate c=13. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(Yes, -52)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -52).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0036/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 37. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[11, 12], integer candidate c=-14. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(No, -69)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -69).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0037/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 38. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[12, 13], integer candidate c=15. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(Yes, -88)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -88).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0038/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 39. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[13, 14], integer candidate c=-16. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(No, -109)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -109).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0039/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 40. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[14, 15], integer candidate c=17. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(Yes, -132)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -132).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0040/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 41. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[15, 16], integer candidate c=-18. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(No, -157)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -157).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0041/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 42. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[16, 17], integer candidate c=19. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(Yes, -184)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -184).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0042/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 43. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[17, 18], integer candidate c=-20. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(No, -213)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -213).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0043/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 44. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[18, 19], integer candidate c=21. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(Yes, -244)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -244).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0044/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 45. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[19, 20], integer candidate c=-22. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(No, -277)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -277).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0045/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 46. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[20, 21], integer candidate c=23. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(Yes, -312)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -312).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0046/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 47. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[21, 22], integer candidate c=-24. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(No, -349)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -349).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0047/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 48. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[22, 23], integer candidate c=25. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(Yes, -388)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -388).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0048/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 49. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[23, 24], integer candidate c=-26. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(No, -429)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (No, -429).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0049/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 50. Candidate Sign and Squared-Distance Defect

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[24, 25], integer candidate c=27. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

View answer and explanation(Yes, -472)

Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -472).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0050/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-candidate-defect · candidate





## 51. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[0, 0], [0, 0], [0, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[1, 2, 3]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[1, 2, 3]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0051/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 52. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[1, 0], [0, -1], [2, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[1, 2], [3]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[1, 2], [3]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0052/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 53. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[2, 0], [1, 0], [1, 1]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[2], [3], [1]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2], [3], [1]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0053/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 54. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[1, 1], [1, 0], [-1, 1]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[2], [1, 3]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2], [1, 3]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0054/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 55. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[2, 1], [1, 2], [0, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[3], [1, 2]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[3], [1, 2]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0055/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 56. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[6, 0], [5, 1], [5, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[3], [2], [1]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[3], [2], [1]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0056/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 57. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[7, 1], [6, 0], [0, -6]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[2, 3], [1]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2, 3], [1]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0057/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 58. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[7, 0], [8, 0], [9, 1]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[1], [2], [3]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[1], [2], [3]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0058/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 59. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[9, 0], [8, 1], [8, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[3], [2], [1]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[3], [2], [1]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0059/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 60. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[10, 1], [9, 0], [0, -9]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[2, 3], [1]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2, 3], [1]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0060/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 61. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[10, 0], [11, 0], [12, 1]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[1], [2], [3]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[1], [2], [3]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0061/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 62. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[12, 0], [11, 1], [11, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[3], [2], [1]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[3], [2], [1]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0062/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 63. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[13, 1], [12, 0], [0, -12]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[2, 3], [1]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2, 3], [1]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0063/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 64. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[13, 0], [14, 0], [15, 1]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[1], [2], [3]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[1], [2], [3]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0064/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 65. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[15, 0], [14, 1], [14, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[3], [2], [1]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[3], [2], [1]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0065/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 66. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[16, 1], [15, 0], [0, -15]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[2, 3], [1]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2, 3], [1]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0066/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 67. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[16, 0], [17, 0], [18, 1]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[1], [2], [3]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[1], [2], [3]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0067/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 68. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[18, 0], [17, 1], [17, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[3], [2], [1]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[3], [2], [1]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0068/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 69. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[19, 1], [18, 0], [0, -18]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[2, 3], [1]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2, 3], [1]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0069/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 70. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[19, 0], [20, 0], [21, 1]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[1], [2], [3]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[1], [2], [3]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0070/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 71. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[21, 0], [20, 1], [20, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[3], [2], [1]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[3], [2], [1]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0071/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 72. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[22, 1], [21, 0], [0, -21]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[2, 3], [1]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2, 3], [1]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0072/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 73. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[22, 0], [23, 0], [24, 1]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[1], [2], [3]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[1], [2], [3]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0073/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 74. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[24, 0], [23, 1], [23, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[3], [2], [1]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[3], [2], [1]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0074/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 75. Distance Order with Explicit Tie Groups

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[25, 1], [24, 0], [0, -24]]. List index groups in increasing distance order, with indices increasing inside each tie group.

View answer and explanation[[2, 3], [1]]

Compare nonnegative squared distances; preserve original indices and do not replace distance order by coordinate order. The exact result, retaining the requested order, is [[2, 3], [1]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0075/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-three-order · candidate





## 76. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[0, 0], [0, 0]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation0

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0076/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate





## 77. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[3, 4], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation18

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 18.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0077/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate





## 78. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[0, 0], [3, 4]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation5

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0078/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate





## 79. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[9, 12], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation28

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 28.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0079/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate





## 80. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[12, 16], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation33

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 33.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0080/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate





## 81. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[15, 20], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation38

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 38.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0081/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate





## 82. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[18, 24], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation43

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 43.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0082/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate





## 83. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[21, 28], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation48

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 48.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0083/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate





## 84. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[24, 32], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation53

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 53.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0084/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate





## 85. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[27, 36], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation58

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 58.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0085/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate





## 86. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[30, 40], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation63

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 63.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0086/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate





## 87. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[33, 44], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation68

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 68.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0087/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate





## 88. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[36, 48], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation73

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 73.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0088/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate





## 89. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[39, 52], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation78

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 78.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0089/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate





## 90. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[42, 56], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation83

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 83.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0090/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate





## 91. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[45, 60], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation88

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 88.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0091/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate





## 92. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[48, 64], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation93

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 93.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0092/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate





## 93. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[51, 68], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation98

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 98.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0093/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate





## 94. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[54, 72], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation103

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 103.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0094/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate





## 95. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[57, 76], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation108

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 108.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0095/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate





## 96. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[60, 80], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation113

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 113.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0096/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate





## 97. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[63, 84], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation118

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 118.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0097/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate





## 98. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[66, 88], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation123

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 123.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0098/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate





## 99. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[69, 92], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation128

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 128.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0099/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate





## 100. Sum of Two Integer Principal Distances

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[72, 96], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.

View answer and explanation133

Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 133.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r03-distance-formula-0100/)

Euclidean distance in an equal-unit orthonormal frame · qf-density10-distance-integer-sum · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261010-r03-distance-formula-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261008-advance-20261010-r03-distance-formula-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261010-r03-distance-formula-foundations/validation-report.json)