{"status": "success", "data": {"description_md": "Every point in $\\mathbb R^3$ is coloured either red or blue such that if two points $A, B$ have the same colour, then their midpoints share this colour. A set of points is *monochromatic* if they are all the same colour.\n\n___\n\nProve that there exists a monochromatic regular octahedron (the set of $8$ vertices only). A regular octahedron has eight flat triangular faces which are all equal in dimensions, twelve equal edges, and six vertices. An image of a regular octahedron is shown below:\n\n<img src=\"https://upload.wikimedia.org/wikipedia/commons/2/27/Octahedron.jpg?20050106205758\" width=\"200px\" class=\"problem-image\">", "description_html": "<p>Every point in <span class=\"katex--inline\">\\mathbb R^3</span> is coloured either red or blue such that if two points <span class=\"katex--inline\">A, B</span> have the same colour, then their midpoints share this colour. A set of points is <em>monochromatic</em> if they are all the same colour.</p>&#10;<hr/>&#10;<p>Prove that there exists a monochromatic regular octahedron (the set of <span class=\"katex--inline\">8</span> vertices only). A regular octahedron has eight flat triangular faces which are all equal in dimensions, twelve equal edges, and six vertices. An image of a regular octahedron is shown below:</p>&#10;<img src=\"https://upload.wikimedia.org/wikipedia/commons/2/27/Octahedron.jpg?20050106205758\" width=\"200px\" class=\"problem-image\"/>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 6, "problem_name": "Mock Euclid 2025 - Problem 10 Part C", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}