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data.jsonl

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+{"messages": [{"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 4, from vertex 4 to vertex 8, from vertex 10 to vertex 4, from vertex 6 to vertex 3, from vertex 5 to vertex 8, from vertex 7 to vertex 9, from vertex 9 to vertex 2, from vertex 2 to vertex 6, from vertex 10 to vertex 5, from vertex 4 to vertex 6, from vertex 1 to vertex 3, from vertex 4 to vertex 9, from vertex 2 to vertex 4, from vertex 8 to vertex 1, from vertex 2 to vertex 10, from vertex 8 to vertex 3, from vertex 9 to vertex 5, from vertex 2 to vertex 5, and from vertex 8 to vertex 10. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 12, from vertex 1 to vertex 4, from vertex 6 to vertex 10, from vertex 11 to vertex 5, from vertex 12 to vertex 5, from vertex 7 to vertex 5, from vertex 9 to vertex 5, from vertex 4 to vertex 11, from vertex 8 to vertex 6, from vertex 6 to vertex 1, from vertex 2 to vertex 6, from vertex 10 to vertex 4, from vertex 3 to vertex 5, from vertex 3 to vertex 1, from vertex 11 to vertex 1, from vertex 10 to vertex 1, and from vertex 11 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 7, from vertex 4 to vertex 7, from vertex 2 to vertex 5, from vertex 4 to vertex 1, from vertex 1 to vertex 7, from vertex 1 to vertex 5, from vertex 1 to vertex 6, from vertex 6 to vertex 3, from vertex 4 to vertex 8, from vertex 1 to vertex 3, from vertex 6 to vertex 4, from vertex 5 to vertex 7, from vertex 6 to vertex 2, from vertex 2 to vertex 7, from vertex 6 to vertex 8, from vertex 8 to vertex 2, and from vertex 3 to vertex 5. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 12, from vertex 9 to vertex 5, from vertex 11 to vertex 7, from vertex 12 to vertex 6, from vertex 9 to vertex 3, from vertex 12 to vertex 4, from vertex 10 to vertex 8, from vertex 6 to vertex 4, from vertex 8 to vertex 4, from vertex 11 to vertex 3, from vertex 4 to vertex 10, from vertex 6 to vertex 2, from vertex 9 to vertex 2, from vertex 6 to vertex 9, from vertex 5 to vertex 1, from vertex 12 to vertex 3, and from vertex 1 to vertex 9. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 2, from vertex 3 to vertex 8, from vertex 10 to vertex 8, from vertex 7 to vertex 1, from vertex 9 to vertex 1, from vertex 6 to vertex 10, from vertex 2 to vertex 10, from vertex 5 to vertex 3, from vertex 1 to vertex 3, from vertex 9 to vertex 4, from vertex 9 to vertex 2, from vertex 1 to vertex 6, from vertex 9 to vertex 3, from vertex 4 to vertex 6, and from vertex 7 to vertex 5. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 7, from vertex 3 to vertex 7, from vertex 6 to vertex 12, from vertex 6 to vertex 10, from vertex 5 to vertex 7, from vertex 2 to vertex 4, from vertex 4 to vertex 11, from vertex 7 to vertex 1, from vertex 10 to vertex 8, from vertex 6 to vertex 2, from vertex 12 to vertex 7, from vertex 11 to vertex 3, from vertex 10 to vertex 12, from vertex 10 to vertex 4, from vertex 3 to vertex 6, from vertex 11 to vertex 5, from vertex 9 to vertex 4, and from vertex 6 to vertex 9. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 8, from vertex 9 to vertex 2, from vertex 6 to vertex 1, from vertex 5 to vertex 2, from vertex 2 to vertex 7, from vertex 7 to vertex 1, from vertex 8 to vertex 10, from vertex 3 to vertex 5, from vertex 7 to vertex 10, from vertex 8 to vertex 4, from vertex 5 to vertex 1, from vertex 2 to vertex 4, from vertex 8 to vertex 5, from vertex 1 to vertex 3, from vertex 5 to vertex 10, and from vertex 6 to vertex 8. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 1, from vertex 5 to vertex 7, from vertex 6 to vertex 3, from vertex 5 to vertex 8, from vertex 6 to vertex 8, from vertex 8 to vertex 4, from vertex 2 to vertex 5, from vertex 4 to vertex 1, from vertex 3 to vertex 1, from vertex 2 to vertex 7, from vertex 1 to vertex 5, from vertex 4 to vertex 6, from vertex 7 to vertex 3, from vertex 2 to vertex 8, and from vertex 8 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 3, from vertex 6 to vertex 1, from vertex 3 to vertex 8, from vertex 8 to vertex 4, from vertex 4 to vertex 1, from vertex 5 to vertex 2, from vertex 4 to vertex 6, from vertex 4 to vertex 7, from vertex 2 to vertex 6, from vertex 1 to vertex 3, from vertex 3 to vertex 5, from vertex 7 to vertex 1, from vertex 2 to vertex 8, and from vertex 6 to vertex 8. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 9, from vertex 9 to vertex 3, from vertex 11 to vertex 2, from vertex 1 to vertex 9, from vertex 12 to vertex 8, from vertex 8 to vertex 1, from vertex 5 to vertex 12, from vertex 12 to vertex 9, from vertex 1 to vertex 5, from vertex 6 to vertex 8, from vertex 3 to vertex 5, from vertex 10 to vertex 8, from vertex 6 to vertex 1, from vertex 5 to vertex 10, from vertex 3 to vertex 6, from vertex 6 to vertex 4, and from vertex 2 to vertex 10. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 2, from vertex 2 to vertex 7, from vertex 8 to vertex 3, from vertex 3 to vertex 1, from vertex 8 to vertex 6, from vertex 4 to vertex 6, from vertex 1 to vertex 4, from vertex 2 to vertex 6, from vertex 8 to vertex 4, from vertex 1 to vertex 6, from vertex 2 to vertex 5, from vertex 5 to vertex 1, from vertex 3 to vertex 5, and from vertex 8 to vertex 5. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 9 to vertex 4, from vertex 4 to vertex 2, from vertex 8 to vertex 6, from vertex 8 to vertex 4, from vertex 3 to vertex 10, from vertex 6 to vertex 9, from vertex 9 to vertex 5, from vertex 1 to vertex 3, from vertex 2 to vertex 8, from vertex 10 to vertex 4, from vertex 5 to vertex 10, from vertex 4 to vertex 1, from vertex 1 to vertex 7, from vertex 1 to vertex 8, from vertex 9 to vertex 7, from vertex 3 to vertex 9, from vertex 7 to vertex 5, from vertex 2 to vertex 9, from vertex 4 to vertex 6, and from vertex 7 to vertex 10. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 2, from vertex 6 to vertex 8, from vertex 1 to vertex 4, from vertex 3 to vertex 1, from vertex 6 to vertex 4, from vertex 7 to vertex 4, from vertex 3 to vertex 8, from vertex 5 to vertex 3, from vertex 8 to vertex 5, from vertex 3 to vertex 7, from vertex 5 to vertex 2, from vertex 6 to vertex 3, from vertex 2 to vertex 7, from vertex 1 to vertex 5, from vertex 2 to vertex 4, from vertex 6 to vertex 1, and from vertex 7 to vertex 1. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 1, from vertex 4 to vertex 11, from vertex 7 to vertex 10, from vertex 6 to vertex 12, from vertex 2 to vertex 8, from vertex 12 to vertex 4, from vertex 5 to vertex 12, from vertex 8 to vertex 1, from vertex 4 to vertex 6, from vertex 7 to vertex 12, from vertex 8 to vertex 5, from vertex 12 to vertex 8, from vertex 3 to vertex 1, from vertex 2 to vertex 7, from vertex 6 to vertex 10, from vertex 5 to vertex 9, from vertex 5 to vertex 11, from vertex 9 to vertex 6, and from vertex 11 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 4, from vertex 3 to vertex 7, from vertex 3 to vertex 6, from vertex 8 to vertex 2, from vertex 5 to vertex 7, from vertex 7 to vertex 4, from vertex 2 to vertex 6, from vertex 5 to vertex 2, from vertex 4 to vertex 2, from vertex 5 to vertex 3, from vertex 3 to vertex 1, from vertex 6 to vertex 1, from vertex 5 to vertex 1, from vertex 7 to vertex 2, from vertex 3 to vertex 8, from vertex 4 to vertex 8, from vertex 6 to vertex 8, from vertex 1 to vertex 4, from vertex 5 to vertex 8, and from vertex 7 to vertex 1. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 9-gon, with vertices numbered 1 through 9 in counterclockwise order. Draw the diagonal from vertex 9 to vertex 6, from vertex 2 to vertex 7, from vertex 2 to vertex 6, from vertex 7 to vertex 3, from vertex 9 to vertex 7, from vertex 6 to vertex 3, from vertex 4 to vertex 1, from vertex 7 to vertex 5, from vertex 8 to vertex 6, from vertex 6 to vertex 4, from vertex 2 to vertex 8, from vertex 1 to vertex 6, from vertex 7 to vertex 1, from vertex 3 to vertex 9, from vertex 2 to vertex 5, and from vertex 4 to vertex 9. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 8, from vertex 1 to vertex 4, from vertex 4 to vertex 2, from vertex 5 to vertex 8, from vertex 8 to vertex 6, from vertex 1 to vertex 6, from vertex 6 to vertex 2, from vertex 3 to vertex 7, from vertex 8 to vertex 3, from vertex 7 to vertex 1, from vertex 4 to vertex 7, from vertex 5 to vertex 3, from vertex 3 to vertex 1, from vertex 4 to vertex 6, from vertex 6 to vertex 3, and from vertex 8 to vertex 4. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 1, from vertex 8 to vertex 4, from vertex 7 to vertex 5, from vertex 2 to vertex 7, from vertex 3 to vertex 8, from vertex 7 to vertex 3, from vertex 3 to vertex 5, from vertex 4 to vertex 2, from vertex 8 to vertex 5, from vertex 7 to vertex 1, from vertex 7 to vertex 4, from vertex 8 to vertex 6, from vertex 2 to vertex 6, from vertex 1 to vertex 3, from vertex 6 to vertex 1, from vertex 8 to vertex 2, and from vertex 5 to vertex 2. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 1, from vertex 10 to vertex 3, from vertex 7 to vertex 9, from vertex 10 to vertex 2, from vertex 8 to vertex 1, from vertex 9 to vertex 2, from vertex 2 to vertex 4, from vertex 2 to vertex 7, from vertex 7 to vertex 5, from vertex 4 to vertex 7, from vertex 7 to vertex 3, from vertex 2 to vertex 8, from vertex 5 to vertex 3, from vertex 1 to vertex 4, from vertex 1 to vertex 6, from vertex 5 to vertex 9, from vertex 6 to vertex 2, from vertex 6 to vertex 3, and from vertex 1 to vertex 9. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 4, from vertex 9 to vertex 11, from vertex 10 to vertex 1, from vertex 11 to vertex 8, from vertex 3 to vertex 11, from vertex 7 to vertex 1, from vertex 6 to vertex 3, from vertex 11 to vertex 7, from vertex 9 to vertex 5, from vertex 1 to vertex 4, from vertex 11 to vertex 5, from vertex 2 to vertex 9, from vertex 3 to vertex 9, from vertex 8 to vertex 10, from vertex 4 to vertex 7, from vertex 4 to vertex 8, and from vertex 8 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 6, from vertex 8 to vertex 4, from vertex 1 to vertex 7, from vertex 2 to vertex 7, from vertex 1 to vertex 3, from vertex 8 to vertex 6, from vertex 5 to vertex 3, from vertex 8 to vertex 5, from vertex 6 to vertex 1, from vertex 2 to vertex 5, from vertex 8 to vertex 3, from vertex 6 to vertex 2, from vertex 7 to vertex 3, from vertex 7 to vertex 4, from vertex 7 to vertex 5, from vertex 8 to vertex 2, from vertex 1 to vertex 4, from vertex 5 to vertex 1, from vertex 2 to vertex 4, and from vertex 4 to vertex 6. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 7, from vertex 6 to vertex 8, from vertex 6 to vertex 4, from vertex 4 to vertex 1, from vertex 4 to vertex 8, from vertex 6 to vertex 3, from vertex 8 to vertex 3, from vertex 2 to vertex 7, from vertex 6 to vertex 1, from vertex 3 to vertex 1, from vertex 3 to vertex 5, from vertex 7 to vertex 1, from vertex 1 to vertex 5, from vertex 8 to vertex 5, from vertex 2 to vertex 5, from vertex 7 to vertex 3, from vertex 4 to vertex 2, and from vertex 6 to vertex 2. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 8, from vertex 1 to vertex 6, from vertex 8 to vertex 1, from vertex 6 to vertex 3, from vertex 8 to vertex 6, from vertex 1 to vertex 4, from vertex 6 to vertex 4, from vertex 1 to vertex 9, from vertex 4 to vertex 10, from vertex 7 to vertex 5, from vertex 5 to vertex 2, from vertex 8 to vertex 3, from vertex 2 to vertex 9, from vertex 6 to vertex 2, and from vertex 5 to vertex 8. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 6, from vertex 9 to vertex 5, from vertex 7 to vertex 1, from vertex 3 to vertex 6, from vertex 7 to vertex 3, from vertex 10 to vertex 2, from vertex 9 to vertex 4, from vertex 8 to vertex 11, from vertex 3 to vertex 10, from vertex 2 to vertex 11, from vertex 12 to vertex 2, from vertex 2 to vertex 7, from vertex 4 to vertex 7, from vertex 3 to vertex 8, from vertex 1 to vertex 8, from vertex 8 to vertex 5, and from vertex 11 to vertex 7. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 8, from vertex 9 to vertex 5, from vertex 1 to vertex 5, from vertex 9 to vertex 1, from vertex 5 to vertex 3, from vertex 7 to vertex 3, from vertex 8 to vertex 12, from vertex 11 to vertex 7, from vertex 10 to vertex 7, from vertex 1 to vertex 3, from vertex 2 to vertex 11, from vertex 5 to vertex 7, from vertex 4 to vertex 9, and from vertex 8 to vertex 6. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 5, from vertex 10 to vertex 6, from vertex 10 to vertex 2, from vertex 5 to vertex 1, from vertex 2 to vertex 4, from vertex 10 to vertex 3, from vertex 1 to vertex 8, from vertex 9 to vertex 2, from vertex 2 to vertex 6, from vertex 10 to vertex 7, from vertex 7 to vertex 9, from vertex 1 to vertex 4, from vertex 8 to vertex 3, from vertex 10 to vertex 8, from vertex 7 to vertex 1, from vertex 3 to vertex 7, from vertex 7 to vertex 4, and from vertex 2 to vertex 5. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 2, from vertex 4 to vertex 1, from vertex 8 to vertex 2, from vertex 10 to vertex 8, from vertex 9 to vertex 7, from vertex 9 to vertex 6, from vertex 1 to vertex 7, from vertex 1 to vertex 8, from vertex 4 to vertex 10, from vertex 7 to vertex 5, from vertex 9 to vertex 2, from vertex 5 to vertex 3, from vertex 7 to vertex 4, from vertex 10 to vertex 3, from vertex 1 to vertex 9, from vertex 5 to vertex 10, from vertex 9 to vertex 3, from vertex 6 to vertex 4, from vertex 5 to vertex 9, and from vertex 2 to vertex 4. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 7, from vertex 5 to vertex 7, from vertex 5 to vertex 2, from vertex 4 to vertex 10, from vertex 7 to vertex 3, from vertex 3 to vertex 10, from vertex 2 to vertex 8, from vertex 9 to vertex 6, from vertex 7 to vertex 10, from vertex 6 to vertex 3, from vertex 9 to vertex 1, from vertex 1 to vertex 5, from vertex 5 to vertex 10, from vertex 7 to vertex 2, and from vertex 2 to vertex 6. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 11 to vertex 3, from vertex 2 to vertex 4, from vertex 3 to vertex 12, from vertex 2 to vertex 10, from vertex 10 to vertex 12, from vertex 10 to vertex 1, from vertex 1 to vertex 6, from vertex 5 to vertex 12, from vertex 4 to vertex 9, from vertex 10 to vertex 3, from vertex 1 to vertex 5, from vertex 9 to vertex 11, from vertex 4 to vertex 12, from vertex 10 to vertex 8, from vertex 6 to vertex 2, from vertex 4 to vertex 11, from vertex 5 to vertex 9, from vertex 10 to vertex 6, and from vertex 11 to vertex 1. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 9 to vertex 3, from vertex 1 to vertex 8, from vertex 10 to vertex 8, from vertex 1 to vertex 4, from vertex 9 to vertex 1, from vertex 11 to vertex 7, from vertex 6 to vertex 2, from vertex 10 to vertex 2, from vertex 4 to vertex 9, from vertex 3 to vertex 11, from vertex 7 to vertex 1, from vertex 4 to vertex 7, from vertex 8 to vertex 2, and from vertex 6 to vertex 1. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 4, from vertex 1 to vertex 6, from vertex 6 to vertex 8, from vertex 6 to vertex 3, from vertex 1 to vertex 7, from vertex 3 to vertex 5, from vertex 2 to vertex 7, from vertex 7 to vertex 5, from vertex 1 to vertex 3, from vertex 1 to vertex 5, from vertex 3 to vertex 8, from vertex 8 to vertex 5, from vertex 8 to vertex 2, from vertex 4 to vertex 7, from vertex 2 to vertex 6, from vertex 4 to vertex 6, and from vertex 7 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 1, from vertex 6 to vertex 3, from vertex 3 to vertex 8, from vertex 2 to vertex 4, from vertex 6 to vertex 4, from vertex 2 to vertex 8, from vertex 7 to vertex 1, from vertex 4 to vertex 1, from vertex 8 to vertex 5, from vertex 1 to vertex 5, from vertex 5 to vertex 7, from vertex 4 to vertex 8, from vertex 5 to vertex 3, from vertex 2 to vertex 6, and from vertex 7 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 8, from vertex 8 to vertex 3, from vertex 5 to vertex 1, from vertex 2 to vertex 7, from vertex 8 to vertex 2, from vertex 6 to vertex 2, from vertex 7 to vertex 4, from vertex 5 to vertex 3, from vertex 6 to vertex 1, from vertex 6 to vertex 4, from vertex 5 to vertex 7, from vertex 3 to vertex 6, from vertex 3 to vertex 7, from vertex 8 to vertex 4, from vertex 1 to vertex 4, from vertex 5 to vertex 2, and from vertex 1 to vertex 7. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 8, from vertex 7 to vertex 3, from vertex 10 to vertex 3, from vertex 3 to vertex 8, from vertex 2 to vertex 7, from vertex 9 to vertex 4, from vertex 1 to vertex 9, from vertex 10 to vertex 8, from vertex 4 to vertex 2, from vertex 10 to vertex 4, from vertex 7 to vertex 1, from vertex 7 to vertex 9, from vertex 5 to vertex 3, from vertex 10 to vertex 7, and from vertex 6 to vertex 1. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 2, from vertex 6 to vertex 1, from vertex 1 to vertex 7, from vertex 8 to vertex 6, from vertex 1 to vertex 5, from vertex 4 to vertex 1, from vertex 3 to vertex 1, from vertex 2 to vertex 5, from vertex 4 to vertex 8, from vertex 2 to vertex 8, from vertex 6 to vertex 2, from vertex 5 to vertex 3, from vertex 4 to vertex 6, from vertex 2 to vertex 7, and from vertex 7 to vertex 5. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 6, from vertex 5 to vertex 1, from vertex 5 to vertex 8, from vertex 6 to vertex 8, from vertex 2 to vertex 10, from vertex 2 to vertex 9, from vertex 1 to vertex 7, from vertex 7 to vertex 5, from vertex 10 to vertex 5, from vertex 4 to vertex 10, from vertex 9 to vertex 4, from vertex 8 to vertex 10, from vertex 7 to vertex 10, from vertex 1 to vertex 3, from vertex 8 to vertex 2, from vertex 9 to vertex 6, from vertex 6 to vertex 1, from vertex 5 to vertex 2, and from vertex 7 to vertex 2. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 5, from vertex 5 to vertex 2, from vertex 3 to vertex 6, from vertex 11 to vertex 1, from vertex 8 to vertex 4, from vertex 10 to vertex 5, from vertex 2 to vertex 7, from vertex 2 to vertex 9, from vertex 3 to vertex 11, from vertex 6 to vertex 10, from vertex 6 to vertex 9, from vertex 6 to vertex 1, from vertex 12 to vertex 4, from vertex 5 to vertex 3, from vertex 9 to vertex 11, from vertex 6 to vertex 4, from vertex 10 to vertex 8, from vertex 5 to vertex 12, and from vertex 4 to vertex 11. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 5, from vertex 5 to vertex 10, from vertex 2 to vertex 10, from vertex 5 to vertex 7, from vertex 4 to vertex 10, from vertex 8 to vertex 6, from vertex 3 to vertex 1, from vertex 1 to vertex 4, from vertex 8 to vertex 10, from vertex 4 to vertex 2, from vertex 4 to vertex 7, from vertex 8 to vertex 5, from vertex 5 to vertex 1, from vertex 3 to vertex 9, from vertex 4 to vertex 9, from vertex 1 to vertex 6, from vertex 9 to vertex 7, from vertex 7 to vertex 10, from vertex 3 to vertex 6, and from vertex 9 to vertex 1. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 6, from vertex 1 to vertex 4, from vertex 8 to vertex 5, from vertex 7 to vertex 9, from vertex 11 to vertex 3, from vertex 9 to vertex 12, from vertex 4 to vertex 11, from vertex 7 to vertex 2, from vertex 8 to vertex 10, from vertex 6 to vertex 10, from vertex 9 to vertex 4, from vertex 10 to vertex 7, from vertex 1 to vertex 10, from vertex 6 to vertex 2, from vertex 12 to vertex 2, from vertex 5 to vertex 3, and from vertex 8 to vertex 2. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 7, from vertex 2 to vertex 8, from vertex 1 to vertex 4, from vertex 3 to vertex 8, from vertex 3 to vertex 6, from vertex 3 to vertex 5, from vertex 4 to vertex 2, from vertex 5 to vertex 2, from vertex 5 to vertex 8, from vertex 1 to vertex 6, from vertex 1 to vertex 5, from vertex 5 to vertex 7, from vertex 6 to vertex 2, from vertex 8 to vertex 4, and from vertex 7 to vertex 2. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 2, from vertex 11 to vertex 9, from vertex 10 to vertex 6, from vertex 8 to vertex 5, from vertex 4 to vertex 8, from vertex 5 to vertex 12, from vertex 5 to vertex 1, from vertex 8 to vertex 11, from vertex 6 to vertex 4, from vertex 8 to vertex 1, from vertex 8 to vertex 2, from vertex 10 to vertex 8, from vertex 12 to vertex 8, and from vertex 11 to vertex 4. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 10, from vertex 1 to vertex 7, from vertex 9 to vertex 12, from vertex 11 to vertex 9, from vertex 2 to vertex 9, from vertex 10 to vertex 5, from vertex 10 to vertex 1, from vertex 8 to vertex 2, from vertex 1 to vertex 9, from vertex 2 to vertex 6, from vertex 12 to vertex 4, from vertex 4 to vertex 6, from vertex 11 to vertex 2, from vertex 5 to vertex 12, from vertex 11 to vertex 6, from vertex 10 to vertex 7, from vertex 12 to vertex 10, from vertex 2 to vertex 10, and from vertex 3 to vertex 11. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 5, from vertex 2 to vertex 6, from vertex 1 to vertex 4, from vertex 4 to vertex 8, from vertex 5 to vertex 8, from vertex 8 to vertex 3, from vertex 7 to vertex 2, from vertex 4 to vertex 6, from vertex 8 to vertex 2, from vertex 7 to vertex 4, from vertex 3 to vertex 7, from vertex 6 to vertex 8, from vertex 1 to vertex 3, from vertex 4 to vertex 2, from vertex 7 to vertex 5, from vertex 1 to vertex 7, from vertex 6 to vertex 1, and from vertex 5 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 7, from vertex 4 to vertex 6, from vertex 7 to vertex 3, from vertex 8 to vertex 2, from vertex 6 to vertex 3, from vertex 5 to vertex 1, from vertex 4 to vertex 7, from vertex 7 to vertex 5, from vertex 8 to vertex 6, from vertex 7 to vertex 1, from vertex 8 to vertex 4, from vertex 4 to vertex 2, from vertex 4 to vertex 1, from vertex 2 to vertex 6, from vertex 5 to vertex 8, from vertex 2 to vertex 5, and from vertex 3 to vertex 1. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 8, from vertex 4 to vertex 2, from vertex 7 to vertex 5, from vertex 1 to vertex 5, from vertex 8 to vertex 2, from vertex 7 to vertex 1, from vertex 3 to vertex 6, from vertex 6 to vertex 8, from vertex 1 to vertex 4, from vertex 6 to vertex 1, from vertex 2 to vertex 5, from vertex 4 to vertex 8, from vertex 8 to vertex 3, from vertex 4 to vertex 6, from vertex 2 to vertex 6, from vertex 3 to vertex 5, from vertex 1 to vertex 3, from vertex 7 to vertex 4, and from vertex 2 to vertex 7. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 10, from vertex 7 to vertex 4, from vertex 4 to vertex 6, from vertex 10 to vertex 2, from vertex 6 to vertex 8, from vertex 4 to vertex 1, from vertex 3 to vertex 6, from vertex 5 to vertex 9, from vertex 8 to vertex 1, from vertex 2 to vertex 8, from vertex 6 to vertex 1, from vertex 7 to vertex 3, from vertex 10 to vertex 3, and from vertex 5 to vertex 10. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 6, from vertex 7 to vertex 2, from vertex 9 to vertex 2, from vertex 6 to vertex 4, from vertex 3 to vertex 1, from vertex 10 to vertex 4, from vertex 2 to vertex 5, from vertex 7 to vertex 5, from vertex 3 to vertex 8, from vertex 4 to vertex 2, from vertex 5 to vertex 3, from vertex 1 to vertex 7, from vertex 1 to vertex 6, from vertex 3 to vertex 9, from vertex 5 to vertex 10, from vertex 9 to vertex 6, from vertex 8 to vertex 1, from vertex 8 to vertex 5, and from vertex 1 to vertex 4. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 2, from vertex 6 to vertex 2, from vertex 3 to vertex 7, from vertex 8 to vertex 6, from vertex 7 to vertex 5, from vertex 5 to vertex 2, from vertex 2 to vertex 4, from vertex 5 to vertex 3, from vertex 7 to vertex 1, from vertex 8 to vertex 3, from vertex 3 to vertex 6, from vertex 1 to vertex 5, from vertex 4 to vertex 7, from vertex 8 to vertex 5, from vertex 3 to vertex 1, from vertex 4 to vertex 1, and from vertex 2 to vertex 7. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 6, from vertex 10 to vertex 6, from vertex 12 to vertex 10, from vertex 8 to vertex 4, from vertex 7 to vertex 12, from vertex 11 to vertex 8, from vertex 8 to vertex 12, from vertex 12 to vertex 9, from vertex 10 to vertex 7, from vertex 1 to vertex 9, from vertex 9 to vertex 6, from vertex 5 to vertex 2, from vertex 2 to vertex 8, from vertex 10 to vertex 5, from vertex 11 to vertex 2, from vertex 4 to vertex 7, and from vertex 12 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 4, from vertex 1 to vertex 4, from vertex 4 to vertex 7, from vertex 7 to vertex 5, from vertex 6 to vertex 3, from vertex 8 to vertex 3, from vertex 3 to vertex 5, from vertex 2 to vertex 8, from vertex 1 to vertex 5, from vertex 4 to vertex 8, from vertex 7 to vertex 3, from vertex 6 to vertex 4, from vertex 1 to vertex 3, from vertex 6 to vertex 1, from vertex 6 to vertex 2, from vertex 8 to vertex 5, from vertex 7 to vertex 2, and from vertex 1 to vertex 7. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 6, from vertex 1 to vertex 9, from vertex 2 to vertex 9, from vertex 1 to vertex 3, from vertex 6 to vertex 4, from vertex 7 to vertex 5, from vertex 5 to vertex 8, from vertex 4 to vertex 10, from vertex 10 to vertex 5, from vertex 3 to vertex 8, from vertex 4 to vertex 1, from vertex 8 to vertex 2, from vertex 3 to vertex 9, from vertex 5 to vertex 1, from vertex 5 to vertex 9, from vertex 6 to vertex 10, from vertex 5 to vertex 2, and from vertex 3 to vertex 6. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 4, from vertex 1 to vertex 4, from vertex 2 to vertex 7, from vertex 3 to vertex 8, from vertex 1 to vertex 3, from vertex 7 to vertex 4, from vertex 6 to vertex 3, from vertex 1 to vertex 5, from vertex 8 to vertex 6, from vertex 2 to vertex 6, from vertex 5 to vertex 7, from vertex 3 to vertex 7, from vertex 2 to vertex 4, from vertex 8 to vertex 4, from vertex 5 to vertex 8, and from vertex 6 to vertex 1. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 8, from vertex 2 to vertex 8, from vertex 1 to vertex 5, from vertex 5 to vertex 10, from vertex 10 to vertex 4, from vertex 6 to vertex 4, from vertex 6 to vertex 8, from vertex 4 to vertex 9, from vertex 10 to vertex 6, from vertex 3 to vertex 6, from vertex 10 to vertex 3, from vertex 6 to vertex 2, from vertex 7 to vertex 3, from vertex 6 to vertex 9, from vertex 1 to vertex 8, from vertex 1 to vertex 9, from vertex 5 to vertex 2, and from vertex 2 to vertex 7. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 2, from vertex 8 to vertex 2, from vertex 6 to vertex 1, from vertex 3 to vertex 1, from vertex 7 to vertex 5, from vertex 3 to vertex 6, from vertex 1 to vertex 7, from vertex 4 to vertex 6, from vertex 8 to vertex 5, from vertex 7 to vertex 2, from vertex 8 to vertex 3, from vertex 2 to vertex 5, from vertex 4 to vertex 8, from vertex 5 to vertex 3, from vertex 3 to vertex 7, from vertex 6 to vertex 2, from vertex 7 to vertex 4, from vertex 5 to vertex 1, and from vertex 1 to vertex 4. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 10 to vertex 5, from vertex 1 to vertex 4, from vertex 4 to vertex 6, from vertex 7 to vertex 1, from vertex 10 to vertex 1, from vertex 3 to vertex 11, from vertex 4 to vertex 9, from vertex 1 to vertex 6, from vertex 9 to vertex 7, from vertex 4 to vertex 7, from vertex 11 to vertex 7, from vertex 1 to vertex 9, from vertex 10 to vertex 12, from vertex 8 to vertex 1, from vertex 4 to vertex 12, from vertex 3 to vertex 12, and from vertex 11 to vertex 5. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 3, from vertex 5 to vertex 9, from vertex 2 to vertex 12, from vertex 11 to vertex 5, from vertex 2 to vertex 11, from vertex 4 to vertex 9, from vertex 9 to vertex 12, from vertex 4 to vertex 6, from vertex 9 to vertex 3, from vertex 11 to vertex 1, from vertex 6 to vertex 9, from vertex 10 to vertex 8, from vertex 1 to vertex 8, from vertex 3 to vertex 5, from vertex 3 to vertex 7, from vertex 4 to vertex 11, from vertex 4 to vertex 10, from vertex 7 to vertex 12, and from vertex 5 to vertex 2. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 8, from vertex 5 to vertex 8, from vertex 7 to vertex 2, from vertex 6 to vertex 3, from vertex 8 to vertex 4, from vertex 5 to vertex 1, from vertex 4 to vertex 6, from vertex 4 to vertex 7, from vertex 2 to vertex 5, from vertex 2 to vertex 8, from vertex 6 to vertex 8, from vertex 4 to vertex 1, from vertex 1 to vertex 7, from vertex 4 to vertex 2, from vertex 5 to vertex 3, from vertex 7 to vertex 5, from vertex 2 to vertex 6, and from vertex 1 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 9-gon, with vertices numbered 1 through 9 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 7, from vertex 8 to vertex 4, from vertex 3 to vertex 6, from vertex 1 to vertex 6, from vertex 2 to vertex 8, from vertex 8 to vertex 3, from vertex 9 to vertex 6, from vertex 6 to vertex 4, from vertex 5 to vertex 1, from vertex 8 to vertex 1, from vertex 9 to vertex 4, from vertex 4 to vertex 2, from vertex 2 to vertex 9, and from vertex 4 to vertex 7. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 5, from vertex 9 to vertex 2, from vertex 7 to vertex 4, from vertex 7 to vertex 10, from vertex 7 to vertex 1, from vertex 6 to vertex 2, from vertex 10 to vertex 2, from vertex 9 to vertex 6, from vertex 2 to vertex 8, from vertex 8 to vertex 6, from vertex 10 to vertex 3, from vertex 7 to vertex 2, from vertex 1 to vertex 9, from vertex 10 to vertex 6, and from vertex 1 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 4, from vertex 12 to vertex 6, from vertex 5 to vertex 2, from vertex 10 to vertex 5, from vertex 11 to vertex 8, from vertex 11 to vertex 2, from vertex 11 to vertex 6, from vertex 3 to vertex 7, from vertex 8 to vertex 2, from vertex 9 to vertex 5, from vertex 12 to vertex 10, from vertex 1 to vertex 4, from vertex 9 to vertex 6, from vertex 10 to vertex 1, from vertex 9 to vertex 4, and from vertex 6 to vertex 8. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 4, from vertex 1 to vertex 7, from vertex 6 to vertex 8, from vertex 2 to vertex 4, from vertex 3 to vertex 8, from vertex 7 to vertex 5, from vertex 1 to vertex 6, from vertex 8 to vertex 5, from vertex 7 to vertex 3, from vertex 5 to vertex 3, from vertex 1 to vertex 5, from vertex 3 to vertex 1, from vertex 2 to vertex 6, from vertex 4 to vertex 6, from vertex 8 to vertex 2, from vertex 4 to vertex 7, from vertex 8 to vertex 4, from vertex 7 to vertex 2, from vertex 5 to vertex 2, and from vertex 3 to vertex 6. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 10 to vertex 12, from vertex 1 to vertex 10, from vertex 9 to vertex 5, from vertex 11 to vertex 2, from vertex 7 to vertex 1, from vertex 7 to vertex 10, from vertex 10 to vertex 4, from vertex 10 to vertex 3, from vertex 9 to vertex 6, from vertex 8 to vertex 5, from vertex 2 to vertex 9, from vertex 10 to vertex 5, from vertex 2 to vertex 6, from vertex 8 to vertex 3, from vertex 12 to vertex 3, and from vertex 7 to vertex 9. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 2, from vertex 3 to vertex 1, from vertex 4 to vertex 7, from vertex 2 to vertex 8, from vertex 8 to vertex 3, from vertex 7 to vertex 3, from vertex 7 to vertex 1, from vertex 6 to vertex 1, from vertex 8 to vertex 6, from vertex 2 to vertex 7, from vertex 8 to vertex 5, from vertex 5 to vertex 2, from vertex 4 to vertex 2, from vertex 6 to vertex 4, from vertex 1 to vertex 5, from vertex 8 to vertex 4, from vertex 3 to vertex 6, and from vertex 4 to vertex 1. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 2, from vertex 7 to vertex 9, from vertex 1 to vertex 7, from vertex 3 to vertex 6, from vertex 2 to vertex 4, from vertex 8 to vertex 10, from vertex 7 to vertex 5, from vertex 3 to vertex 8, from vertex 1 to vertex 8, from vertex 4 to vertex 10, from vertex 6 to vertex 9, from vertex 5 to vertex 1, from vertex 2 to vertex 10, and from vertex 2 to vertex 9. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 10 to vertex 8, from vertex 8 to vertex 6, from vertex 11 to vertex 9, from vertex 7 to vertex 2, from vertex 11 to vertex 3, from vertex 5 to vertex 7, from vertex 4 to vertex 6, from vertex 1 to vertex 3, from vertex 11 to vertex 2, from vertex 4 to vertex 2, from vertex 3 to vertex 7, from vertex 1 to vertex 7, from vertex 2 to vertex 9, from vertex 6 to vertex 9, from vertex 6 to vertex 1, from vertex 4 to vertex 1, from vertex 8 to vertex 5, and from vertex 3 to vertex 5. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 9-gon, with vertices numbered 1 through 9 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 4, from vertex 2 to vertex 9, from vertex 6 to vertex 4, from vertex 5 to vertex 9, from vertex 1 to vertex 6, from vertex 1 to vertex 7, from vertex 3 to vertex 5, from vertex 7 to vertex 3, from vertex 8 to vertex 6, from vertex 9 to vertex 3, from vertex 9 to vertex 7, from vertex 1 to vertex 3, from vertex 4 to vertex 7, from vertex 6 to vertex 2, from vertex 1 to vertex 5, from vertex 8 to vertex 3, from vertex 8 to vertex 2, from vertex 8 to vertex 1, and from vertex 5 to vertex 7. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 5, from vertex 6 to vertex 8, from vertex 7 to vertex 3, from vertex 8 to vertex 4, from vertex 5 to vertex 8, from vertex 1 to vertex 4, from vertex 1 to vertex 3, from vertex 7 to vertex 1, from vertex 1 to vertex 5, from vertex 4 to vertex 6, from vertex 7 to vertex 5, from vertex 3 to vertex 5, from vertex 6 to vertex 1, from vertex 3 to vertex 6, from vertex 8 to vertex 2, from vertex 8 to vertex 3, and from vertex 2 to vertex 7. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 1, from vertex 6 to vertex 2, from vertex 1 to vertex 5, from vertex 7 to vertex 5, from vertex 3 to vertex 8, from vertex 8 to vertex 5, from vertex 8 to vertex 2, from vertex 3 to vertex 6, from vertex 4 to vertex 8, from vertex 7 to vertex 2, from vertex 6 to vertex 4, from vertex 7 to vertex 4, from vertex 2 to vertex 5, from vertex 2 to vertex 4, and from vertex 7 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 1, from vertex 7 to vertex 1, from vertex 1 to vertex 3, from vertex 7 to vertex 4, from vertex 6 to vertex 2, from vertex 5 to vertex 7, from vertex 4 to vertex 6, from vertex 1 to vertex 4, from vertex 2 to vertex 8, from vertex 7 to vertex 3, from vertex 2 to vertex 4, from vertex 2 to vertex 5, from vertex 8 to vertex 3, from vertex 7 to vertex 2, from vertex 4 to vertex 8, from vertex 5 to vertex 8, from vertex 5 to vertex 1, from vertex 8 to vertex 6, and from vertex 6 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 6, from vertex 10 to vertex 7, from vertex 9 to vertex 5, from vertex 10 to vertex 6, from vertex 2 to vertex 9, from vertex 9 to vertex 6, from vertex 2 to vertex 5, from vertex 2 to vertex 4, from vertex 3 to vertex 8, from vertex 2 to vertex 10, from vertex 3 to vertex 1, from vertex 3 to vertex 6, from vertex 1 to vertex 5, from vertex 9 to vertex 4, from vertex 8 to vertex 10, and from vertex 7 to vertex 9. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 5, from vertex 1 to vertex 5, from vertex 2 to vertex 7, from vertex 6 to vertex 4, from vertex 8 to vertex 6, from vertex 5 to vertex 7, from vertex 3 to vertex 7, from vertex 6 to vertex 3, from vertex 6 to vertex 2, from vertex 4 to vertex 1, from vertex 1 to vertex 3, from vertex 7 to vertex 1, from vertex 2 to vertex 4, from vertex 7 to vertex 4, from vertex 3 to vertex 8, from vertex 5 to vertex 2, from vertex 2 to vertex 8, and from vertex 8 to vertex 4. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 3, from vertex 2 to vertex 7, from vertex 3 to vertex 5, from vertex 5 to vertex 1, from vertex 7 to vertex 5, from vertex 4 to vertex 7, from vertex 8 to vertex 6, from vertex 4 to vertex 6, from vertex 2 to vertex 6, from vertex 3 to vertex 8, from vertex 1 to vertex 6, from vertex 5 to vertex 8, from vertex 5 to vertex 2, from vertex 8 to vertex 2, from vertex 4 to vertex 1, and from vertex 1 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 9 to vertex 1, from vertex 5 to vertex 3, from vertex 1 to vertex 4, from vertex 8 to vertex 5, from vertex 5 to vertex 10, from vertex 4 to vertex 2, from vertex 1 to vertex 7, from vertex 9 to vertex 3, from vertex 6 to vertex 8, from vertex 3 to vertex 6, from vertex 5 to vertex 9, from vertex 6 to vertex 4, from vertex 9 to vertex 7, from vertex 8 to vertex 1, from vertex 10 to vertex 4, from vertex 1 to vertex 6, from vertex 3 to vertex 7, from vertex 5 to vertex 7, from vertex 9 to vertex 6, and from vertex 3 to vertex 10. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 7, from vertex 5 to vertex 9, from vertex 8 to vertex 1, from vertex 2 to vertex 5, from vertex 9 to vertex 1, from vertex 6 to vertex 2, from vertex 10 to vertex 8, from vertex 5 to vertex 1, from vertex 10 to vertex 3, from vertex 7 to vertex 10, from vertex 4 to vertex 9, from vertex 4 to vertex 2, from vertex 9 to vertex 6, from vertex 8 to vertex 5, from vertex 8 to vertex 3, from vertex 1 to vertex 4, from vertex 6 to vertex 8, and from vertex 4 to vertex 6. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 5, from vertex 7 to vertex 10, from vertex 7 to vertex 1, from vertex 12 to vertex 2, from vertex 9 to vertex 4, from vertex 4 to vertex 6, from vertex 2 to vertex 6, from vertex 9 to vertex 5, from vertex 5 to vertex 8, from vertex 10 to vertex 2, from vertex 8 to vertex 3, from vertex 1 to vertex 5, from vertex 7 to vertex 4, from vertex 1 to vertex 6, from vertex 6 to vertex 10, and from vertex 5 to vertex 12. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 8, from vertex 7 to vertex 5, from vertex 7 to vertex 3, from vertex 5 to vertex 2, from vertex 1 to vertex 3, from vertex 1 to vertex 4, from vertex 3 to vertex 5, from vertex 8 to vertex 6, from vertex 2 to vertex 4, from vertex 3 to vertex 8, from vertex 4 to vertex 6, from vertex 2 to vertex 7, from vertex 2 to vertex 6, from vertex 8 to vertex 4, and from vertex 1 to vertex 5. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
+{"messages": [{"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 6, from vertex 7 to vertex 3, from vertex 4 to vertex 2, from vertex 2 to vertex 5, from vertex 4 to vertex 6, from vertex 4 to vertex 7, from vertex 4 to vertex 1, from vertex 8 to vertex 6, from vertex 6 to vertex 3, from vertex 5 to vertex 1, from vertex 1 to vertex 3, from vertex 7 to vertex 1, from vertex 7 to vertex 5, from vertex 8 to vertex 2, and from vertex 7 to vertex 2. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}