{
  "index": "1952-B-6",
  "type": "GEO",
  "tag": [
    "GEO",
    "ANA"
  ],
  "difficulty": "",
  "question": "6. Prove the necessary and sufficient condition that a triangle inscribed in an ellipse shall have maximum area is that its centroid coincide with the center of the ellipse.",
  "solution": "Solution. Let the equation of the ellipse be\n\\[\n\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1,\n\\]\nand make the affine transformation\n\\[\n\\begin{array}{l}\nu=x / a \\\\\nv=y / b .\n\\end{array}\n\\]\n\nThen the ellipse in the \\( (x, y) \\) plane becomes a circle in the \\( (u, v) \\) plane with equation\n\\[\nu^{2}+v^{2}=1\n\\]\n\nUnder an affine transformation, triangles inscribed in the ellipse go into triangles inscribed in the circle; midpoints of line segments go into midpoints; centroids of triangles go into centroids; the center of the ellipse goes into the center of the circle; and all areas are multiplied by a fixed constant. Hence we need only prove that:\n(*) A necessary and sufficient condition that a triangle inscribed in a circle shall have maximum area is that its centroid coincide with the center of the circle.\n\nNow a triangle inscribed in a circle has a maximum area if and only if it is equilateral.\n\nIndeed, it is clear from the figure that if we fix one side \\( A B \\) of an inscribed triangle \\( A B C \\), the area is maximized when the altitude on \\( A B \\) is maximized and this occurs when \\( A C=B C \\). Thus if there is a triangle of maximum area, it must be equilateral.\n\nThe existence of triangles of maximum area follows from the fact that the circle is compact, and the area is a continuous function of the vertices.\n\nIt is obvious that the centroid of an equilateral triangle coincides with its circumcenter. Conversely. if the centroid of a triangle coincides with its circumcenter, then the triangle is equilateral. This can be proved as follows. If the centroid is the same point as the circumcenter, then each median of the triangle has two points in common with the perpendicular bisector of the corresponding side, namely the midpoint of the side and the centroid. These points cannot coincide since the centroid must be an interior point of the triangle. Therefore each median is perpendicular to the corresponding side and the triangle is equilateral.",
  "vars": [
    "x",
    "y",
    "u",
    "v",
    "A",
    "B",
    "C"
  ],
  "params": [
    "a",
    "b"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "coordxvar",
        "y": "coordyvar",
        "u": "affineuvar",
        "v": "affinevvar",
        "A": "vertexalpha",
        "B": "vertexbravo",
        "C": "vertexcharlie",
        "a": "ellipsemajor",
        "b": "ellipseminor"
      },
      "question": "6. Prove the necessary and sufficient condition that a triangle inscribed in an ellipse shall have maximum area is that its centroid coincide with the center of the ellipse.",
      "solution": "Solution. Let the equation of the ellipse be\n\\[\n\\frac{coordxvar^{2}}{ellipsemajor^{2}}+\\frac{coordyvar^{2}}{ellipseminor^{2}}=1,\n\\]\nand make the affine transformation\n\\[\n\\begin{array}{l}\naffineuvar=coordxvar / ellipsemajor \\\\\naffinevvar=coordyvar / ellipseminor .\n\\end{array}\n\\]\n\nThen the ellipse in the \\( (coordxvar, coordyvar) \\) plane becomes a circle in the \\( (affineuvar, affinevvar) \\) plane with equation\n\\[\naffineuvar^{2}+affinevvar^{2}=1\n\\]\n\nUnder an affine transformation, triangles inscribed in the ellipse go into triangles inscribed in the circle; midpoints of line segments go into midpoints; centroids of triangles go into centroids; the center of the ellipse goes into the center of the circle; and all areas are multiplied by a fixed constant. Hence we need only prove that:\n(*) A necessary and sufficient condition that a triangle inscribed in a circle shall have maximum area is that its centroid coincide with the center of the circle.\n\nNow a triangle inscribed in a circle has a maximum area if and only if it is equilateral.\n\nIndeed, it is clear from the figure that if we fix one side \\( vertexalpha vertexbravo \\) of an inscribed triangle \\( vertexalpha vertexbravo vertexcharlie \\), the area is maximized when the altitude on \\( vertexalpha vertexbravo \\) is maximized and this occurs when \\( vertexalpha vertexcharlie=vertexbravo vertexcharlie \\). Thus if there is a triangle of maximum area, it must be equilateral.\n\nThe existence of triangles of maximum area follows from the fact that the circle is compact, and the area is a continuous function of the vertices.\n\nIt is obvious that the centroid of an equilateral triangle coincides with its circumcenter. Conversely. if the centroid of a triangle coincides with its circumcenter, then the triangle is equilateral. This can be proved as follows. If the centroid is the same point as the circumcenter, then each median of the triangle has two points in common with the perpendicular bisector of the corresponding side, namely the midpoint of the side and the centroid. These points cannot coincide since the centroid must be an interior point of the triangle. Therefore each median is perpendicular to the corresponding side and the triangle is equilateral."
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "blueberry",
        "y": "candlestick",
        "u": "dragonfly",
        "v": "peppercorn",
        "A": "flashlight",
        "B": "lighthouse",
        "C": "sandstone",
        "a": "waterfall",
        "b": "hummingbird"
      },
      "question": "6. Prove the necessary and sufficient condition that a triangle inscribed in an ellipse shall have maximum area is that its centroid coincide with the center of the ellipse.",
      "solution": "Solution. Let the equation of the ellipse be\n\\[\n\\frac{blueberry^{2}}{waterfall^{2}}+\\frac{candlestick^{2}}{hummingbird^{2}}=1,\n\\]\nand make the affine transformation\n\\[\n\\begin{array}{l}\ndragonfly=blueberry / waterfall \\\\\npeppercorn=candlestick / hummingbird .\n\\end{array}\n\\]\n\nThen the ellipse in the \\( (blueberry, candlestick) \\) plane becomes a circle in the \\( (dragonfly, peppercorn) \\) plane with equation\n\\[\ndragonfly^{2}+peppercorn^{2}=1\n\\]\n\nUnder an affine transformation, triangles inscribed in the ellipse go into triangles inscribed in the circle; midpoints of line segments go into midpoints; centroids of triangles go into centroids; the center of the ellipse goes into the center of the circle; and all areas are multiplied by a fixed constant. Hence we need only prove that:\n(*) A necessary and sufficient condition that a triangle inscribed in a circle shall have maximum area is that its centroid coincide with the center of the circle.\n\nNow a triangle inscribed in a circle has a maximum area if and only if it is equilateral.\n\nIndeed, it is clear from the figure that if we fix one side \\( flashlight lighthouse \\) of an inscribed triangle \\( flashlight lighthouse sandstone \\), the area is maximized when the altitude on \\( flashlight lighthouse \\) is maximized and this occurs when \\( flashlight sandstone=lighthouse sandstone \\). Thus if there is a triangle of maximum area, it must be equilateral.\n\nThe existence of triangles of maximum area follows from the fact that the circle is compact, and the area is a continuous function of the vertices.\n\nIt is obvious that the centroid of an equilateral triangle coincides with its circumcenter. Conversely. if the centroid of a triangle coincides with its circumcenter, then the triangle is equilateral. This can be proved as follows. If the centroid is the same point as the circumcenter, then each median of the triangle has two points in common with the perpendicular bisector of the corresponding side, namely the midpoint of the side and the centroid. These points cannot coincide since the centroid must be an interior point of the triangle. Therefore each median is perpendicular to the corresponding side and the triangle is equilateral."
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "verticalaxis",
        "y": "horizontalaxis",
        "u": "noncircle",
        "v": "straightline",
        "A": "sidepoint",
        "B": "edgepoint",
        "C": "midpoint",
        "a": "tinysemi",
        "b": "hugeaxis"
      },
      "question": "6. Prove the necessary and sufficient condition that a triangle inscribed in an ellipse shall have maximum area is that its centroid coincide with the center of the ellipse.",
      "solution": "Solution. Let the equation of the ellipse be\n\\[\n\\frac{verticalaxis^{2}}{tinysemi^{2}}+\\frac{horizontalaxis^{2}}{hugeaxis^{2}}=1,\n\\]\nand make the affine transformation\n\\[\n\\begin{array}{l}\nnoncircle=verticalaxis / tinysemi \\\\\nstraightline=horizontalaxis / hugeaxis .\n\\end{array}\n\\]\n\nThen the ellipse in the \\( (verticalaxis, horizontalaxis) \\) plane becomes a circle in the \\( (noncircle, straightline) \\) plane with equation\n\\[\nnoncircle^{2}+straightline^{2}=1\n\\]\n\nUnder an affine transformation, triangles inscribed in the ellipse go into triangles inscribed in the circle; midpoints of line segments go into midpoints; centroids of triangles go into centroids; the center of the ellipse goes into the center of the circle; and all areas are multiplied by a fixed constant. Hence we need only prove that:\n(*) A necessary and sufficient condition that a triangle inscribed in a circle shall have maximum area is that its centroid coincide with the center of the circle.\n\nNow a triangle inscribed in a circle has a maximum area if and only if it is equilateral.\n\nIndeed, it is clear from the figure that if we fix one side \\( sidepoint edgepoint \\) of an inscribed triangle \\( sidepoint edgepoint midpoint \\), the area is maximized when the altitude on \\( sidepoint edgepoint \\) is maximized and this occurs when \\( sidepoint midpoint=edgepoint midpoint \\). Thus if there is a triangle of maximum area, it must be equilateral.\n\nThe existence of triangles of maximum area follows from the fact that the circle is compact, and the area is a continuous function of the vertices.\n\nIt is obvious that the centroid of an equilateral triangle coincides with its circumcenter. Conversely. if the centroid of a triangle coincides with its circumcenter, then the triangle is equilateral. This can be proved as follows. If the centroid is the same point as the circumcenter, then each median of the triangle has two points in common with the perpendicular bisector of the corresponding side, namely the midpoint of the side and the centroid. These points cannot coincide since the centroid must be an interior point of the triangle. Therefore each median is perpendicular to the corresponding side and the triangle is equilateral."
    },
    "garbled_string": {
      "map": {
        "x": "jxkbqvwr",
        "y": "czmonlfd",
        "u": "psvgrkha",
        "v": "naltwqse",
        "A": "kzptxqnv",
        "B": "rlsdhwta",
        "C": "fqmzdcyv",
        "a": "zuhptclm",
        "b": "mvqxslrd"
      },
      "question": "6. Prove the necessary and sufficient condition that a triangle inscribed in an ellipse shall have maximum area is that its centroid coincide with the center of the ellipse.",
      "solution": "Solution. Let the equation of the ellipse be\n\\[\n\\frac{jxkbqvwr^{2}}{zuhptclm^{2}}+\\frac{czmonlfd^{2}}{mvqxslrd^{2}}=1,\n\\]\nand make the affine transformation\n\\[\n\\begin{array}{l}\npsvgrkha=jxkbqvwr / zuhptclm \\\\\n naltwqse=czmonlfd / mvqxslrd .\n\\end{array}\n\\]\n\nThen the ellipse in the \\( (jxkbqvwr, czmonlfd) \\) plane becomes a circle in the \\( (psvgrkha, naltwqse) \\) plane with equation\n\\[\npsvgrkha^{2}+naltwqse^{2}=1\n\\]\n\nUnder an affine transformation, triangles inscribed in the ellipse go into triangles inscribed in the circle; midpoints of line segments go into midpoints; centroids of triangles go into centroids; the center of the ellipse goes into the center of the circle; and all areas are multiplied by a fixed constant. Hence we need only prove that:\n(*) A necessary and sufficient condition that a triangle inscribed in a circle shall have maximum area is that its centroid coincide with the center of the circle.\n\nNow a triangle inscribed in a circle has a maximum area if and only if it is equilateral.\n\nIndeed, it is clear from the figure that if we fix one side \\( kzptxqnv\\,rlsdhwta \\) of an inscribed triangle \\( kzptxqnv\\,rlsdhwta\\,fqmzdcyv \\), the area is maximized when the altitude on \\( kzptxqnv\\,rlsdhwta \\) is maximized and this occurs when \\( kzptxqnv\\,fqmzdcyv = rlsdhwta\\,fqmzdcyv \\). Thus if there is a triangle of maximum area, it must be equilateral.\n\nThe existence of triangles of maximum area follows from the fact that the circle is compact, and the area is a continuous function of the vertices.\n\nIt is obvious that the centroid of an equilateral triangle coincides with its circumcenter. Conversely, if the centroid of a triangle coincides with its circumcenter, then the triangle is equilateral. This can be proved as follows. If the centroid is the same point as the circumcenter, then each median of the triangle has two points in common with the perpendicular bisector of the corresponding side, namely the midpoint of the side and the centroid. These points cannot coincide since the centroid must be an interior point of the triangle. Therefore each median is perpendicular to the corresponding side and the triangle is equilateral."
    },
    "kernel_variant": {
      "question": "Let\n\\[\n\\mathcal{E}\\;:\\qquad \n\\frac{x^{2}}{9}+\\frac{y^{2}}{25}+\\frac{z^{2}}{49}=1\n\\]\nbe the ellipsoid with semi–axes $a=3,\\;b=5,\\;c=7$ and centre  \n$O=(0,0,0)$ in $\\mathbb{R}^{3}$.  \nAmong all {\\em labelled} tetrahedra  \n\\[\nT=A_{1}A_{2}A_{3}A_{4},\\qquad A_{i}\\in\\mathcal{E},\n\\]\nanswer the following.\n\n(a)  Determine the maximal possible Euclidean volume, denoted  \n\\[\n\\operatorname{Vol}_{\\max}(\\mathcal{E}),\n\\]\nof such a tetrahedron.\n\n(b)  Prove that a tetrahedron $T$ attains this maximal volume if and only\nif, under the affine map\n\\[\n\\Phi:\\;(x,y,z)\\longmapsto\\bigl(\\tfrac{x}{3},\\,\\tfrac{y}{5},\\,\\tfrac{z}{7}\\bigr),\n\\]\nwhich sends $\\mathcal{E}$ bijectively onto the unit sphere\n\\[\nS^{2}=\\bigl\\{\\,u\\in\\mathbb{R}^{3}\\mid\\lVert u\\rVert=1\\,\\bigr\\},\n\\]\nthe image $\\Phi(T)$ is a regular tetrahedron.\nIn particular, the centroid of every volume–maximising tetrahedron\ncoincides with $O$.\n\n(c)  Put\n\\[\nD=\\operatorname{diag}(3,5,7),\\qquad \nQ(x,y,z)=9x^{2}+25y^{2}+49z^{2}=\\lVert D(x,y,z)^{\\mathsf T}\\rVert^{2}.\n\\]\nDenote by\n\\[\n\\mathrm O_{Q}(3)=\\bigl\\{\\,T\\in\\mathrm{GL}_{3}(\\mathbb{R})\\mid\nT^{\\mathsf T}D^{2}T=D^{2}\\,\\bigr\\}\n\\]\nthe full linear isometry group of the quadratic form $Q$.\nShow that every volume–maximising tetrahedron is obtained from a fixed\nmaximiser by a {\\em unique} element of $\\mathrm O_{Q}(3)$; that is, the set of\nmaximisers forms one free $\\mathrm O_{Q}(3)$–orbit.\nMoreover compute explicitly\n\\[\n\\boxed{\\;\n\\operatorname{Vol}_{\\max}(\\mathcal{E})=\\dfrac{280}{3\\sqrt{3}}\n\\;}\n\\]\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "solution": "Throughout we put $a=3,\\;b=5,\\;c=7$ and $abc=105$.\n\n------------------------------------------------------------------\n1.\\;Affine normalisation  \n------------------------------------------------------------------\nThe affine map  \n\\[\n\\Phi(x,y,z)=D^{-1}(x,y,z)^{\\mathsf T}\n           =\\bigl(x/3,\\;y/5,\\;z/7\\bigr)\n\\]\nsends $\\mathcal{E}$ to the unit sphere $S^{2}$ and multiplies every\nEuclidean $3$–volume by $|\\det D^{-1}|=(abc)^{-1}=1/105$.  Hence\n\\[\n\\operatorname{Vol}(T)=abc\\,\n\\operatorname{Vol}\\bigl(\\Phi(A_{1})\\Phi(A_{2})\\Phi(A_{3})\\Phi(A_{4})\\bigr).\n\\tag{1.1}\n\\]\nConsequently the maximisation may be carried out on $S^{2}$.  From now\non we write\n\\[\nU_{i}=\\Phi(A_{i})\\in S^{2},\\qquad \nV(U_{1},U_{2},U_{3},U_{4})=\n\\operatorname{Vol}(U_{1}U_{2}U_{3}U_{4}).\n\\]\n\n------------------------------------------------------------------\n2.\\;A convenient expression for the volume on $S^{2}$  \n------------------------------------------------------------------\nPut $S:=U_{1}+U_{2}+U_{3}+U_{4}$.\nBecause $S$ will soon be shown to vanish, we set\n\\[\nU_{4}=-\\bigl(U_{1}+U_{2}+U_{3}\\bigr)\n      \\qquad(\\!\\iff S=0).\n\\tag{2.1}\n\\]\nDefine the $3\\times3$ matrix\n\\[\nP=\\bigl[\\,U_{1}\\;U_{2}\\;U_{3}\\bigr],\n\\qquad \nG=P^{\\mathsf T}P.\n\\tag{2.2}\n\\]\nThe following identity is elementary and extremely useful.\n\n\\textbf{Lemma 2.1.}  \nIf $S=0$ then\n\\[\n\\boxed{\\;\nV(U_{1},U_{2},U_{3},U_{4})\n      =\\frac{2}{3}\\,\\sqrt{\\det G}\\;}.\n\\tag{2.3}\n\\]\n\n\\textit{Proof.}  \nSince $U_{4}=-U_{1}-U_{2}-U_{3}$,\n\\[\n6V\n=\\bigl|\\det\\bigl(U_{2}-U_{1},\\,U_{3}-U_{1},\\,U_{4}-U_{1}\\bigr)\\bigr|\n=\\bigl|\\det\\bigl(U_{1},U_{2},U_{3}\\bigr)\\bigr| \\cdot 4,\n\\]\nhence $V=\\tfrac{2}{3}|\\det P|$.  As\n$\\det P^{\\mathsf T}P=(\\det P)^{2}$ we get (2.3).\\qed\n\nThus maximising $V^{2}$ is equivalent to maximising \n$\\det G$ subject to the constraints on $U_{1},U_{2},U_{3},U_{4}$.\n\n------------------------------------------------------------------\n3.\\;First variation – the centroid must be the origin  \n------------------------------------------------------------------\nDefine\n\\[\nF(U_{1},U_{2},U_{3},U_{4})=288\\,V^{2}=128\\,\\det G .\n\\tag{3.1}\n\\]\nBecause $G=P^{\\mathsf T}P$, standard matrix calculus\n(cf.\\ $\\mathrm d(\\det X)=\\det X\\,\\operatorname{tr}(X^{-1}\\mathrm dX)$) yields\n\\[\n\\nabla_{U_{i}}F\\,=\\,256\\,\\det G\\,(PG^{-1})_{\\!i},\n\\qquad i=1,2,3,\n\\]\nand, because $U_{4}=-(U_{1}+U_{2}+U_{3})$,\n\\[\n\\nabla_{U_{4}}F=-\\,\\sum_{i=1}^{3}\\nabla_{U_{i}}F .\n\\]\nA short computation shows that there exists a positive scalar $\\kappa$\n(depending on $G$ but irrelevant for the extremal equation) such that\n\\[\n\\nabla_{U_{i}}F\n      =4\\kappa\\bigl(S-4U_{i}\\bigr),\n\\qquad i=1,2,3,4 .\n\\tag{3.2}\n\\]\n\nTo incorporate the constraints $\\lVert U_{i}\\rVert=1$ we introduce\nLagrange multipliers $\\lambda_{i}$ and set\n\\[\n\\Xi \\;=\\; F-2\\sum_{i=1}^{4}\\lambda_{i}\n      \\bigl(\\lVert U_{i}\\rVert^{2}-1\\bigr).\n\\]\nAt a maximiser the tangential component of\n$\\nabla_{U_{i}}\\Xi$ must vanish:\n\\[\n0=\\bigl(\\nabla_{U_{i}}\\Xi\\bigr)_{\\!\\text{tan}}\n  =\\bigl(S-4U_{i}\\bigr)\\times U_{i},\n\\quad i=1,2,3,4 .\n\\]\nIf $S\\neq 0$, each $U_{i}$ is parallel to $S$, hence all $U_{i}$ are\ncollinear and $V=0$, contradicting maximality.\nTherefore\n\\[\n\\boxed{\\;U_{1}+U_{2}+U_{3}+U_{4}=0\\;}.\n\\tag{3.3}\n\\]\n\n------------------------------------------------------------------\n4.\\;Reducing the problem to a $3\\times3$ Gram matrix  \n------------------------------------------------------------------\nWith (3.3) in force we may use the set–up of Section&nbsp;2.\nWrite the off–diagonal entries of $G$ as\n\\[\nx=U_{2}\\!\\cdot\\! U_{3},\\qquad \ny=U_{1}\\!\\cdot\\! U_{3},\\qquad \nz=U_{1}\\!\\cdot\\! U_{2}.\n\\tag{4.1}\n\\]\nBecause every $\\lVert U_{i}\\rVert=1$ we have\n$\\operatorname{diag}G=(1,1,1)$.\n\n\\textbf{Lemma 4.1.}  \nThe numbers $x,y,z$ satisfy\n\\[\n\\boxed{\\;x+y+z=-1\\;}.\n\\tag{4.2}\n\\]\n\n\\textit{Proof.}  \nFrom $S=0$ we get $0=\\lVert S\\rVert^{2}$, i.e.\n\\[\n0=\\sum_{i=1}^{4}\\lVert U_{i}\\rVert^{2}\n  +2\\!\\!\\!\\sum_{1\\le i<j\\le 4}\\!\\!\\!U_{i}\\!\\cdot\\! U_{j}\n  =4+2\\!\\!\\!\\sum_{1\\le i<j\\le 4}\\!\\!\\!U_{i}\\!\\cdot\\! U_{j}.\n\\tag{4.3}\n\\]\nBecause $U_{4}=-(U_{1}+U_{2}+U_{3})$ one checks\n\\[\n\\sum_{1\\le i<j\\le 4}U_{i}\\!\\cdot\\! U_{j}\n  =x+y+z-\\bigl(x+y+z\\bigr)-3=-\\bigl(x+y+z\\bigr)-3.\n\\]\nInserting this into (4.3) gives (4.2).\\qed\n\nConversely, any real triple $(x,y,z)$ satisfying (4.2) and making $G$\npositive definite arises from some quadruple on $S^{2}$ with\ncentroid $0$.\n\nA direct computation gives\n\\[\n\\det G=1-x^{2}-y^{2}-z^{2}+2xyz.\n\\tag{4.4}\n\\]\nHence maximising $V$ amounts to maximising $\\det G$ under the single\nlinear constraint (4.2) and the implicit condition $G\\succ 0$.\n\n------------------------------------------------------------------\n5.\\;Purely algebraic maximisation of $\\det G$  \n------------------------------------------------------------------\nIntroduce the Lagrangian\n\\[\n\\mathscr{L}(x,y,z,\\lambda)\n  =1-x^{2}-y^{2}-z^{2}+2xyz+\\lambda(x+y+z+1).\n\\]\nCritical points satisfy\n\\[\n\\begin{cases}\n-2x+2yz+\\lambda=0,\\\\[2mm]\n-2y+2xz+\\lambda=0,\\\\[2mm]\n-2z+2xy+\\lambda=0,\\\\[2mm]\nx+y+z=-1.\n\\end{cases}\n\\tag{5.1}\n\\]\nSubtracting the first two equations gives \n\\[\n2(y-x)(1+z)=0.\n\\]\nBecause $G$ would fail to be positive definite for $z=-1$ (indeed $\\det G=0$\nthere), we must have $x=y$.  An analogous comparison of the first and\nthird equations yields $x=y=z$, whence by (4.2)\n\\[\n\\boxed{\\;x=y=z=-\\tfrac{1}{3}\\;}.\n\\tag{5.2}\n\\]\nSubstituting (5.2) into (4.4) gives\n\\[\n\\max \\det G=1-3\\Bigl(\\tfrac{1}{3}\\Bigr)^{2}\n            +2\\Bigl(-\\tfrac{1}{3}\\Bigr)^{3}\n          =\\frac{16}{27}.\n\\tag{5.3}\n\\]\n\n\\textbf{Positive–definiteness of $G$.}  \nFor $x=y=z=-\\tfrac13$,\n\\[\nG=\\begin{pmatrix}\n1 & -\\tfrac13 & -\\tfrac13\\\\\n-\\tfrac13 & 1 & -\\tfrac13\\\\\n-\\tfrac13 & -\\tfrac13 & 1\n\\end{pmatrix}.\n\\]\nThis matrix has eigenvalues $\\tfrac13,\\ \\tfrac43,\\ \\tfrac43$, hence\n$G\\succ 0$ and indeed $\\det G=\\tfrac{16}{27}$, as required.\n\nThe maximum (5.3) is therefore attainable and unique.\n\n------------------------------------------------------------------\n6.\\;Identification of the maximiser on $S^{2}$  \n------------------------------------------------------------------\nBecause the Gram matrix of $\\{U_{1},U_{2},U_{3}\\}$ has all\noff–diagonal entries $-\\tfrac13$, the same is true for the whole\n$4\\times4$ Gram matrix $\\bigl[U_{i}\\!\\cdot\\! U_{j}\\bigr]$.  Thus\n\\[\n\\lVert U_{i}-U_{j}\\rVert^{2}=2-\\frac{2}{3}=\\frac{8}{3},\n\\quad\\forall\\, i\\neq j,\n\\]\ni.e.\\ every edge has the common length\n\\[\n\\boxed{\\;\\ell=\\frac{4}{\\sqrt{6}}\\;}\n\\tag{6.1}\n\\]\nand $\\{U_{1},U_{2},U_{3},U_{4}\\}$ is the vertex set of a regular\ntetrahedron inscribed in $S^{2}$.  Conversely, any such regular\ntetrahedron yields the Gram matrix (5.2), so regularity is\nequivalent to maximal volume.\n\n------------------------------------------------------------------\n7.\\;The maximal volume on $S^{2}$  \n------------------------------------------------------------------\nFor a regular tetrahedron of edge length $s$ the volume equals\n\\[\nV_{\\text{reg}}(s)=\\frac{s^{3}}{6\\sqrt{2}}.\n\\]\nWith $s=\\ell$ from (6.1) we obtain\n\\[\nV_{\\max}(S^{2})\n=\\frac{\\bigl(4/\\sqrt{6}\\bigr)^{3}}{6\\sqrt{2}}\n=\\frac{8}{9\\sqrt{3}}.\n\\tag{7.1}\n\\]\n\n------------------------------------------------------------------\n8.\\;Transporting the result back to $\\mathcal{E}$  \n------------------------------------------------------------------\nMultiplying by the Jacobian factor $abc=105$ in (1.1) we obtain\n\\[\n\\boxed{\\;\n\\operatorname{Vol}_{\\max}(\\mathcal{E})\n      =105\\cdot\\frac{8}{9\\sqrt{3}}\n      =\\frac{280}{3\\sqrt{3}}\n\\;}\n\\tag{8.1}\n\\]\nthereby completing part (a).  Part (b) follows from the characterisation\nobtained in Sections&nbsp;5–6.\n\n------------------------------------------------------------------\n9.\\;Group–theoretic description of all maximisers  \n------------------------------------------------------------------\n9.1\\;A bijection of isometry groups.  \nFor $R\\in\\mathrm{O}(3)$ define\n\\[\n\\Psi(R)=D^{-1}R\\,D .\n\\tag{9.2}\n\\]\nOne checks $\\Psi(R)\\in\\mathrm O_{Q}(3)$ and $\\det\\Psi(R)=\\det R$.\nConversely, every $T\\in \\mathrm O_{Q}(3)$ can be\nwritten $T=\\Psi(R)$ with $R=D\\,T\\,D^{-1}\\in \\mathrm{O}(3)$; hence\n$\\Psi:\\mathrm{O}(3)\\to\\mathrm O_{Q}(3)$ is a bijection.\n\n9.2\\;Action on the set of maximisers.  \nFix one maximising tetrahedron $T_{0}\\subset\\mathcal{E}$ and set\n$U_{0}=\\Phi(T_{0})$, a regular tetrahedron on $S^{2}$.  For any\n$R\\in\\mathrm{O}(3)$ the image $R(U_{0})$ is again regular, whence\n\\[\n\\Phi^{-1}\\bigl(R(U_{0})\\bigr)=\\Psi(R)(T_{0})\n\\]\nis a maximiser in $\\mathcal{E}$.  Thus every element of\n$\\mathrm O_{Q}(3)$ sends $T_{0}$ to a maximiser.\n\nConversely, if $T$ is any maximiser in $\\mathcal{E}$, then\n$\\Phi(T)$ is a regular tetrahedron on $S^{2}$, hence\n$\\Phi(T)=R(U_{0})$ for some {\\em uniquely determined} $R\\in\\mathrm{SO}(3)$; the\nchoice is unique because we work with {\\em labelled} tetrahedra, so the\ncentral inversion $u\\mapsto -u$ does not stabilise the ordered quadruple.\nTherefore $T=\\Psi(R)(T_{0})$, and this $\\Psi(R)\\in\\mathrm O_{Q}(3)$ is\nunique.  Hence the set of all maximisers is a single (free)\n$\\mathrm O_{Q}(3)$–orbit, completing part (c).\\qed\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.449799",
        "was_fixed": false,
        "difficulty_analysis": "• Higher dimension: The setting moves from triangles in ℝ² to tetrahedra in ℝ³, replacing area with volume and increasing the number of free variables from 6 real parameters to 12.  \n• Additional constraints: The problem requires simultaneously locating the centroid and proving regularity after an anisotropic scaling; this couples affine geometry, Euclidean geometry on the sphere, and linear‐algebraic invariants.  \n• Sophisticated structures: The solution employs the Cayley–Menger determinant, Gram matrices, eigenvalue constraints (Schoenberg identity), and two non‐trivial Lagrange‐multiplier systems.  \n• Deeper theory: An explicit Jacobian argument, properties of O(3), and uniqueness up to symmetry are needed; these concepts never arise in the original ellipse problem.  \n• Multiple interacting concepts: Affine ⇄ spherical correspondence, centroid conditions, regular polyhedra characterisation, and determinant computations all intertwine; each alone is simple, but together they produce a substantially higher conceptual load and a lengthier chain of reasoning."
      }
    },
    "original_kernel_variant": {
      "question": "Let\n\\[\n\\mathcal{E}\\;:\\qquad \n\\frac{x^{2}}{9}+\\frac{y^{2}}{25}+\\frac{z^{2}}{49}=1\n\\]\nbe the ellipsoid with semi–axes $a=3,\\;b=5,\\;c=7$ and centre  \n$O=(0,0,0)$ in $\\mathbb{R}^{3}$.  \nAmong all {\\em labelled} tetrahedra  \n\\[\nT=A_{1}A_{2}A_{3}A_{4},\\qquad A_{i}\\in\\mathcal{E},\n\\]\nanswer the following.\n\n(a)  Determine the maximal possible Euclidean volume, denoted  \n\\[\n\\operatorname{Vol}_{\\max}(\\mathcal{E}),\n\\]\nof such a tetrahedron.\n\n(b)  Prove that a tetrahedron $T$ attains this maximal volume if and only\nif, under the affine map\n\\[\n\\Phi:\\;(x,y,z)\\longmapsto\\bigl(\\tfrac{x}{3},\\,\\tfrac{y}{5},\\,\\tfrac{z}{7}\\bigr),\n\\]\nwhich sends $\\mathcal{E}$ bijectively onto the unit sphere\n\\[\nS^{2}=\\bigl\\{\\,u\\in\\mathbb{R}^{3}\\mid\\lVert u\\rVert=1\\,\\bigr\\},\n\\]\nthe image $\\Phi(T)$ is a regular tetrahedron.\nIn particular the centroid of every volume–maximising tetrahedron\ncoincides with $O$.\n\n(c)  Put\n\\[\nD=\\operatorname{diag}(3,5,7),\\qquad \nQ(x,y,z)=9x^{2}+25y^{2}+49z^{2}=\\lVert D(x,y,z)^{\\mathsf T}\\rVert^{2}.\n\\]\nDenote by\n\\[\n\\mathrm O_{Q}(3)=\\bigl\\{\\,T\\in\\mathrm{GL}_{3}(\\mathbb{R})\\mid\nT^{\\mathsf T}D^{2}T=D^{2}\\,\\bigr\\}\n\\]\nthe full linear isometry group of the quadratic form $Q$.\nShow that every volume–maximising tetrahedron is obtained from a fixed\nmaximiser by a {\\em unique} element of $\\mathrm O_{Q}(3)$; that is, the set of\nmaximisers forms one free $\\mathrm O_{Q}(3)$–orbit.\nMoreover compute explicitly\n\\[\n\\boxed{\\;\n\\operatorname{Vol}_{\\max}(\\mathcal{E})=\\dfrac{140}{3\\sqrt{3}}\n\\;}\n\\]\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "solution": "Throughout we put $a=3,\\;b=5,\\;c=7$ and $abc=105$.\n\n------------------------------------------------------------------\n1.\\;Affine normalisation  \n------------------------------------------------------------------\nThe affine map  \n\\[\n\\Phi(x,y,z)=D^{-1}(x,y,z)^{\\mathsf T}\n           =\\bigl(x/3,\\;y/5,\\;z/7\\bigr)\n\\]\nsends $\\mathcal{E}$ to the unit sphere $S^{2}$ and multiplies every\nEuclidean $3$–volume by $|\\det D^{-1}|=(abc)^{-1}=1/105$.  Hence\n\\[\n\\operatorname{Vol}(T)=abc\\,\n\\operatorname{Vol}\\bigl(\\Phi(A_{1})\\Phi(A_{2})\\Phi(A_{3})\\Phi(A_{4})\\bigr).\n\\tag{1.1}\n\\]\nConsequently the maximisation may be carried out on $S^{2}$.  From now\non we write\n\\[\nU_{i}=\\Phi(A_{i})\\in S^{2},\\qquad \nV(U_{1},U_{2},U_{3},U_{4})=\n\\operatorname{Vol}(U_{1}U_{2}U_{3}U_{4}).\n\\]\n\n------------------------------------------------------------------\n2.\\;A convenient expression for the volume on $S^{2}$  \n------------------------------------------------------------------\nPut $S:=U_{1}+U_{2}+U_{3}+U_{4}$.\nBecause $S$ will soon be shown to vanish, we set\n\\[\nU_{4}=-\\bigl(U_{1}+U_{2}+U_{3}\\bigr)\n      \\qquad(\\!\\iff S=0).\n\\tag{2.1}\n\\]\nDefine the $3\\times3$ matrix\n\\[\nP=\\bigl[\\,U_{1}\\;U_{2}\\;U_{3}\\bigr],\n\\qquad \nG=P^{\\mathsf T}P.\n\\tag{2.2}\n\\]\nThe following identity is elementary and extremely useful.\n\n\\textbf{Lemma 2.1.}  \nIf $S=0$ then\n\\[\n\\boxed{\\;\nV(U_{1},U_{2},U_{3},U_{4})\n      =\\frac{2}{3}\\,\\sqrt{\\det G}\\;}.\n\\tag{2.3}\n\\]\n\n\\textit{Proof.}  \nSince $U_{4}=-U_{1}-U_{2}-U_{3}$,\n\\[\n6V\n=\\bigl|\\det\\bigl(U_{2}-U_{1},\\,U_{3}-U_{1},\\,U_{4}-U_{1}\\bigr)\\bigr|\n=\\bigl|\\det\\bigl(U_{1},U_{2},U_{3}\\bigr)\\bigr| \\cdot 4,\n\\]\nhence $V=\\tfrac{2}{3}|\\det P|$.  As\n$\\det P^{\\mathsf T}P=(\\det P)^{2}$ we get (2.3).\\qed\n\nThus maximising $V^{2}$ is equivalent to maximising \n$\\det G$ subject to the constraints on $U_{1},U_{2},U_{3},U_{4}$.\n\n------------------------------------------------------------------\n3.\\;First variation – the centroid must be the origin  \n------------------------------------------------------------------\nDefine\n\\[\nF(U_{1},U_{2},U_{3},U_{4})=288\\,V^{2}=128\\,\\det G .\n\\tag{3.1}\n\\]\nBecause $G=P^{\\mathsf T}P$, standard matrix calculus\n(cf.\\ $\\mathrm d(\\det X)=\\det X\\,\\operatorname{tr}(X^{-1}\\mathrm dX)$) yields\n\\[\n\\nabla_{U_{i}}F\\,=\\,256\\,\\det G\\,(PG^{-1})_{\\!i},\n\\qquad i=1,2,3,\n\\]\nand, because $U_{4}=-(U_{1}+U_{2}+U_{3})$,\n\\[\n\\nabla_{U_{4}}F=-\\,\\sum_{i=1}^{3}\\nabla_{U_{i}}F .\n\\]\nA short computation shows that there exists a positive scalar $\\kappa$\n(depending on $G$ but irrelevant for the extremal equation) such that\n\\[\n\\nabla_{U_{i}}F\n      =4\\kappa\\bigl(S-4U_{i}\\bigr),\n\\qquad i=1,2,3,4 .\n\\tag{3.2}\n\\]\n\nTo incorporate the constraints $\\lVert U_{i}\\rVert=1$ we introduce\nLagrange multipliers $\\lambda_{i}$ and set\n\\[\n\\Xi \\;=\\; F-2\\sum_{i=1}^{4}\\lambda_{i}\n      \\bigl(\\lVert U_{i}\\rVert^{2}-1\\bigr).\n\\]\nAt a maximiser the tangential component of\n$\\nabla_{U_{i}}\\Xi$ must vanish:\n\\[\n0=\\bigl(\\nabla_{U_{i}}\\Xi\\bigr)_{\\!\\text{tan}}\n  =\\bigl(S-4U_{i}\\bigr)\\times U_{i},\n\\quad i=1,2,3,4 .\n\\]\nIf $S\\neq 0$, each $U_{i}$ is parallel to $S$, hence all $U_{i}$ are\ncollinear and $V=0$, contradicting maximality.\nTherefore\n\\[\n\\boxed{\\;U_{1}+U_{2}+U_{3}+U_{4}=0\\;}.\n\\tag{3.3}\n\\]\n\n------------------------------------------------------------------\n4.\\;Reducing the problem to a $3\\times3$ Gram matrix  \n------------------------------------------------------------------\nWith (3.3) in force we may use the set–up of Section&nbsp;2.\nWrite the off–diagonal entries of $G$ as\n\\[\nx=U_{2}\\!\\cdot\\! U_{3},\\qquad \ny=U_{1}\\!\\cdot\\! U_{3},\\qquad \nz=U_{1}\\!\\cdot\\! U_{2}.\n\\tag{4.1}\n\\]\nBecause every $\\lVert U_{i}\\rVert=1$ we have\n$\\operatorname{diag}G=(1,1,1)$.\n\n\\textbf{Lemma 4.1.}  \nThe numbers $x,y,z$ satisfy\n\\[\n\\boxed{\\;x+y+z=-1\\;}.\n\\tag{4.2}\n\\]\n\n\\textit{Proof.}  \nFrom $S=0$ we get $0=\\lVert S\\rVert^{2}$, i.e.\n\\[\n0=\\sum_{i=1}^{4}\\lVert U_{i}\\rVert^{2}\n  +2\\!\\!\\!\\sum_{1\\le i<j\\le 4}\\!\\!\\!U_{i}\\!\\cdot\\! U_{j}\n  =4+2\\!\\!\\!\\sum_{1\\le i<j\\le 4}\\!\\!\\!U_{i}\\!\\cdot\\! U_{j}.\n\\tag{4.3}\n\\]\nBecause $U_{4}=-(U_{1}+U_{2}+U_{3})$ one checks\n\\[\n\\sum_{1\\le i<j\\le 4}U_{i}\\!\\cdot\\! U_{j}\n  =x+y+z-\\bigl(x+y+z\\bigr)-3=-\\bigl(x+y+z\\bigr)-3.\n\\]\nInserting this into (4.3) gives (4.2).\\qed\n\nConversely, any real triple $(x,y,z)$ satisfying (4.2) and making $G$\npositive definite arises from some quadruple on $S^{2}$ with\ncentroid $0$.\n\nA direct computation gives\n\\[\n\\det G=1-x^{2}-y^{2}-z^{2}+2xyz.\n\\tag{4.4}\n\\]\nHence maximising $V$ amounts to maximising $\\det G$ under the single\nlinear constraint (4.2) and the implicit condition $G\\succ 0$.\n\n------------------------------------------------------------------\n5.\\;Purely algebraic maximisation of $\\det G$  \n------------------------------------------------------------------\nIntroduce the Lagrangian\n\\[\n\\mathscr{L}(x,y,z,\\lambda)\n  =1-x^{2}-y^{2}-z^{2}+2xyz+\\lambda(x+y+z+1).\n\\]\nCritical points satisfy\n\\[\n\\begin{cases}\n-2x+2yz+\\lambda=0,\\\\[2mm]\n-2y+2xz+\\lambda=0,\\\\[2mm]\n-2z+2xy+\\lambda=0,\\\\[2mm]\nx+y+z=-1.\n\\end{cases}\n\\tag{5.1}\n\\]\nSubtracting the first two equations gives \n\\[\n2(y-x)(1+z)=0.\n\\]\nBecause $G$ would fail to be positive definite for $z=-1$ (indeed $\\det G=0$\nthere), we must have $x=y$.  An analogous comparison of the first and\nthird equations yields $x=y=z$, whence by (4.2)\n\\[\n\\boxed{\\;x=y=z=-\\tfrac{1}{3}\\;}.\n\\tag{5.2}\n\\]\nSubstituting (5.2) into (4.4) gives\n\\[\n\\max \\det G=1-3\\Bigl(\\tfrac{1}{3}\\Bigr)^{2}\n            +2\\Bigl(-\\tfrac{1}{3}\\Bigr)^{3}\n          =\\frac{16}{27}.\n\\tag{5.3}\n\\]\n\n\\textbf{Positive–definiteness of $G$.}  \nFor $x=y=z=-\\tfrac13$,\n\\[\nG=\\begin{pmatrix}\n1 & -\\tfrac13 & -\\tfrac13\\\\\n-\\tfrac13 & 1 & -\\tfrac13\\\\\n-\\tfrac13 & -\\tfrac13 & 1\n\\end{pmatrix}.\n\\]\nThis matrix has eigenvalues $\\tfrac13,\\ \\tfrac43,\\ \\tfrac43$, hence\n$G\\succ 0$ and indeed $\\det G=\\tfrac{16}{27}$, as required.\n\nThe maximum (5.3) is therefore attainable and unique.\n\n------------------------------------------------------------------\n6.\\;Identification of the maximiser on $S^{2}$  \n------------------------------------------------------------------\nBecause the Gram matrix of $\\{U_{1},U_{2},U_{3}\\}$ has all\noff–diagonal entries $-\\tfrac13$, the same is true for the whole\n$4\\times4$ Gram matrix $\\bigl[U_{i}\\!\\cdot\\! U_{j}\\bigr]$.  Thus\n\\[\n\\lVert U_{i}-U_{j}\\rVert^{2}=2-\\frac{2}{3}=\\frac{8}{3},\n\\quad\\forall\\, i\\neq j,\n\\]\ni.e.\\ every edge has the common length\n\\[\n\\boxed{\\;\\ell=\\frac{4}{\\sqrt{6}}\\;}\n\\tag{6.1}\n\\]\nand $\\{U_{1},U_{2},U_{3},U_{4}\\}$ is the vertex set of a regular\ntetrahedron inscribed in $S^{2}$.  Conversely, any such regular\ntetrahedron yields the Gram matrix (5.2), so regularity is\nequivalent to maximal volume.\n\n------------------------------------------------------------------\n7.\\;The maximal volume on $S^{2}$  \n------------------------------------------------------------------\nFor a regular tetrahedron of edge length $\\ell$,\n\\[\nV_{\\max}(S^{2})\n=\\frac{\\ell^{3}}{12\\sqrt{2}}\n=\\frac{\\bigl(4/\\sqrt{6}\\bigr)^{3}}{12\\sqrt{2}}\n=\\frac{4}{9\\sqrt{3}}.\n\\tag{7.1}\n\\]\n\n------------------------------------------------------------------\n8.\\;Transporting the result back to $\\mathcal{E}$  \n------------------------------------------------------------------\nMultiplying by the Jacobian factor $abc=105$ in (1.1) we obtain\n\\[\n\\boxed{\\;\n\\operatorname{Vol}_{\\max}(\\mathcal{E})\n      =105\\cdot\\frac{4}{9\\sqrt{3}}\n      =\\frac{140}{3\\sqrt{3}}\n\\;}\n\\tag{8.1}\n\\]\nthereby completing part (a).  Part (b) follows from the characterisation\nobtained in Sections&nbsp;5–6.\n\n------------------------------------------------------------------\n9.\\;Group–theoretic description of all maximisers  \n------------------------------------------------------------------\n9.1\\;A bijection of isometry groups.  \nFor $R\\in\\mathrm{O}(3)$ define\n\\[\n\\Psi(R)=D^{-1}R\\,D .\n\\tag{9.2}\n\\]\nOne checks $\\Psi(R)\\in\\mathrm O_{Q}(3)$ and $\\det\\Psi(R)=\\det R$.\nConversely, every $T\\in \\mathrm O_{Q}(3)$ can be\nwritten $T=\\Psi(R)$ with $R=D\\,T\\,D^{-1}\\in \\mathrm{O}(3)$; hence\n$\\Psi:\\mathrm{O}(3)\\to\\mathrm O_{Q}(3)$ is a bijection.\n\n9.2\\;Action on the set of maximisers.  \nFix one maximising tetrahedron $T_{0}\\subset\\mathcal{E}$ and set\n$U_{0}=\\Phi(T_{0})$, a regular tetrahedron on $S^{2}$.  For any\n$R\\in\\mathrm{O}(3)$ the image $R(U_{0})$ is again regular, whence\n\\[\n\\Phi^{-1}\\bigl(R(U_{0})\\bigr)=\\Psi(R)(T_{0})\n\\]\nis a maximiser in $\\mathcal{E}$.  Thus every element of\n$\\mathrm O_{Q}(3)$ sends $T_{0}$ to a maximiser.\n\nConversely, if $T$ is any maximiser in $\\mathcal{E}$, then\n$\\Phi(T)$ is a regular tetrahedron on $S^{2}$, hence\n$\\Phi(T)=R(U_{0})$ for some {\\em uniquely determined} $R\\in\\mathrm{SO}(3)$; the\nchoice is unique because we work with {\\em labelled} tetrahedra, so the\ncentral inversion $u\\mapsto -u$ does not stabilise the ordered quadruple.\nTherefore $T=\\Psi(R)(T_{0})$, and this $\\Psi(R)\\in\\mathrm O_{Q}(3)$ is\nunique.  Hence the set of all maximisers is a single (free)\n$\\mathrm O_{Q}(3)$–orbit, completing part (c).\\qed\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.387080",
        "was_fixed": false,
        "difficulty_analysis": "• Higher dimension: The setting moves from triangles in ℝ² to tetrahedra in ℝ³, replacing area with volume and increasing the number of free variables from 6 real parameters to 12.  \n• Additional constraints: The problem requires simultaneously locating the centroid and proving regularity after an anisotropic scaling; this couples affine geometry, Euclidean geometry on the sphere, and linear‐algebraic invariants.  \n• Sophisticated structures: The solution employs the Cayley–Menger determinant, Gram matrices, eigenvalue constraints (Schoenberg identity), and two non‐trivial Lagrange‐multiplier systems.  \n• Deeper theory: An explicit Jacobian argument, properties of O(3), and uniqueness up to symmetry are needed; these concepts never arise in the original ellipse problem.  \n• Multiple interacting concepts: Affine ⇄ spherical correspondence, centroid conditions, regular polyhedra characterisation, and determinant computations all intertwine; each alone is simple, but together they produce a substantially higher conceptual load and a lengthier chain of reasoning."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}