{
  "index": "1959-A-2",
  "type": "GEO",
  "tag": [
    "GEO",
    "ALG"
  ],
  "difficulty": "",
  "question": "2. Prove that if the points in the complex plane corresponding to two distinct complex numbers \\( z_{1} \\) and \\( z_{2} \\) are two vertices of an equilateral triangle, then the third vertex corresponds to \\( -\\omega z_{1}-\\omega^{2} z_{2} \\), where \\( \\omega \\) is an imaginary cube root of unity.",
  "solution": "Solution. If the complex numbers \\( z_{1}, z_{2}, z_{3} \\) are the vertices of an equilateral triangle, then\n\\[\n\\frac{z_{3}-z_{1}}{z_{2}-z_{1}}=\\alpha=e^{ \\pm \\pi i / 3}\n\\]\n\nHence \\( z_{3}=(1-\\alpha) z_{1}+\\alpha z_{2}=-\\alpha^{2} z_{1}-\\alpha^{4} z_{2} \\), because \\( \\alpha^{2}-\\alpha+1=0 \\) and \\( \\alpha^{3}=-1 \\) for either determination of \\( \\alpha \\). Thus\n\\[\nz_{3}=-\\omega z_{1}-\\omega^{2} z_{2}\n\\]\nwhere \\( \\omega=\\alpha^{2}=e^{+2 \\pi i / 3} \\) is one of the two complex cube roots of unity.\nNote that given \\( z_{1} \\) and \\( z_{2} \\) there are two ways to complete an equilateral triangle, corresponding to the two choices for \\( \\omega \\).",
  "vars": [
    "z_1",
    "z_2",
    "z_3"
  ],
  "params": [
    "\\\\alpha",
    "\\\\omega"
  ],
  "sci_consts": [
    "e",
    "i"
  ],
  "variants": {
    "descriptive_long": {
      "map": {
        "z_1": "firstvertex",
        "z_2": "secondvertex",
        "z_3": "thirdvertex",
        "\\alpha": "rotationfactor",
        "\\omega": "cuberoot"
      },
      "question": "2. Prove that if the points in the complex plane corresponding to two distinct complex numbers \\( firstvertex \\) and \\( secondvertex \\) are two vertices of an equilateral triangle, then the third vertex corresponds to \\( -cuberoot firstvertex-cuberoot^{2} secondvertex \\), where \\( cuberoot \\) is an imaginary cube root of unity.",
      "solution": "Solution. If the complex numbers \\( firstvertex, secondvertex, thirdvertex \\) are the vertices of an equilateral triangle, then\n\\[\n\\frac{thirdvertex-firstvertex}{secondvertex-firstvertex}=rotationfactor=e^{ \\pm \\pi i / 3}\n\\]\n\nHence \\( thirdvertex=(1-rotationfactor) firstvertex+rotationfactor secondvertex=-rotationfactor^{2} firstvertex-rotationfactor^{4} secondvertex \\), because \\( rotationfactor^{2}-rotationfactor+1=0 \\) and \\( rotationfactor^{3}=-1 \\) for either determination of \\( rotationfactor \\). Thus\n\\[\nthirdvertex=-cuberoot firstvertex-cuberoot^{2} secondvertex\n\\]\nwhere \\( cuberoot=rotationfactor^{2}=e^{+2 \\pi i / 3} \\) is one of the two complex cube roots of unity.\nNote that given \\( firstvertex \\) and \\( secondvertex \\) there are two ways to complete an equilateral triangle, corresponding to the two choices for \\( cuberoot \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "z_1": "marshmallow",
        "z_2": "lighthouse",
        "z_3": "teacupset",
        "\\alpha": "thundersnow",
        "\\omega": "blueberries"
      },
      "question": "2. Prove that if the points in the complex plane corresponding to two distinct complex numbers \\( marshmallow \\) and \\( lighthouse \\) are two vertices of an equilateral triangle, then the third vertex corresponds to \\( -blueberries\\, marshmallow-blueberries^{2}\\, lighthouse \\), where blueberries is an imaginary cube root of unity.",
      "solution": "Solution. If the complex numbers \\( marshmallow, lighthouse, teacupset \\) are the vertices of an equilateral triangle, then\n\\[\n\\frac{teacupset-marshmallow}{lighthouse-marshmallow}=thundersnow=e^{ \\pm \\pi i / 3}\n\\]\nHence \\( teacupset=(1-thundersnow)\\, marshmallow+thundersnow\\, lighthouse=-thundersnow^{2}\\, marshmallow-thundersnow^{4}\\, lighthouse \\), because \\( thundersnow^{2}-thundersnow+1=0 \\) and \\( thundersnow^{3}=-1 \\) for either determination of thundersnow. Thus\n\\[\nteacupset=-blueberries\\, marshmallow-blueberries^{2}\\, lighthouse\n\\]\nwhere \\( blueberries=thundersnow^{2}=e^{+2 \\pi i / 3} \\) is one of the two complex cube roots of unity.\nNote that given marshmallow and lighthouse there are two ways to complete an equilateral triangle, corresponding to the two choices for blueberries."
    },
    "descriptive_long_misleading": {
      "map": {
        "z_1": "linepoint",
        "z_2": "planepoint",
        "z_3": "voidspace",
        "\\\\alpha": "stillfactor",
        "\\\\omega": "chaosfactor"
      },
      "question": "2. Prove that if the points in the complex plane corresponding to two distinct complex numbers \\( linepoint \\) and \\( planepoint \\) are two vertices of an equilateral triangle, then the third vertex corresponds to \\( -chaosfactor linepoint-chaosfactor^{2} planepoint \\), where \\( chaosfactor \\) is an imaginary cube root of unity.",
      "solution": "Solution. If the complex numbers \\( linepoint, planepoint, voidspace \\) are the vertices of an equilateral triangle, then\n\\[\n\\frac{voidspace-linepoint}{planepoint-linepoint}=stillfactor=e^{ \\pm \\pi i / 3}\n\\]\n\nHence \\( voidspace=(1-stillfactor) linepoint+stillfactor planepoint=-stillfactor^{2} linepoint-stillfactor^{4} planepoint \\), because \\( stillfactor^{2}-stillfactor+1=0 \\) and \\( stillfactor^{3}=-1 \\) for either determination of \\( stillfactor \\). Thus\n\\[\nvoidspace=-chaosfactor linepoint-chaosfactor^{2} planepoint\n\\]\nwhere \\( chaosfactor=stillfactor^{2}=e^{+2 \\pi i / 3} \\) is one of the two complex cube roots of unity.\nNote that given \\( linepoint \\) and \\( planepoint \\) there are two ways to complete an equilateral triangle, corresponding to the two choices for \\( chaosfactor \\)."
    },
    "garbled_string": {
      "map": {
        "z_1": "qzxwvtnp",
        "z_2": "hjgrksla",
        "z_3": "nbvcklqe",
        "\\alpha": "pwszmrtd",
        "\\omega": "fljgksnd"
      },
      "question": "2. Prove that if the points in the complex plane corresponding to two distinct complex numbers \\( qzxwvtnp \\) and \\( hjgrksla \\) are two vertices of an equilateral triangle, then the third vertex corresponds to \\( -fljgksnd qzxwvtnp-fljgksnd^{2} hjgrksla \\), where \\( fljgksnd \\) is an imaginary cube root of unity.",
      "solution": "Solution. If the complex numbers \\( qzxwvtnp, hjgrksla, nbvcklqe \\) are the vertices of an equilateral triangle, then\n\\[\n\\frac{nbvcklqe-qzxwvtnp}{hjgrksla-qzxwvtnp}=pwszmrtd=e^{ \\pm \\pi i / 3}\n\\]\nHence \\( nbvcklqe=(1-pwszmrtd) qzxwvtnp+pwszmrtd hjgrksla=-pwszmrtd^{2} qzxwvtnp-pwszmrtd^{4} hjgrksla \\), because \\( pwszmrtd^{2}-pwszmrtd+1=0 \\) and \\( pwszmrtd^{3}=-1 \\) for either determination of \\( pwszmrtd \\). Thus\n\\[\nnbvcklqe=-fljgksnd qzxwvtnp-fljgksnd^{2} hjgrksla\n\\]\nwhere \\( fljgksnd=pwszmrtd^{2}=e^{+2 \\pi i / 3} \\) is one of the two complex cube roots of unity.\nNote that given \\( qzxwvtnp \\) and \\( hjgrksla \\) there are two ways to complete an equilateral triangle, corresponding to the two choices for \\( fljgksnd \\)."
    },
    "kernel_variant": {
      "question": "Let  \n\\[\n\\varepsilon:=e^{-\\pi i/3}= \\tfrac12-\\tfrac{\\sqrt3}{2}i,\n\\qquad\n\\rho:=\\varepsilon-1=e^{-2\\pi i/3},\n\\]\nso that $\\rho^{3}=1$ and $\\rho\\neq1$.  \nFor two distinct complex numbers $(z_{0},z_{1})$ define the (clockwise)  \nequilateral–completion map  \n\\[\nE(z_{0},z_{1})\\;:=\\;(1-\\varepsilon)\\,z_{0}\\;+\\;\\varepsilon\\,z_{1}.\\tag{$\\star$}\n\\]\n\n(The point $E(z_{0},z_{1})$ is the unique third vertex that completes\nthe clockwise–oriented equilateral triangle on the ordered side\n$z_{0}z_{1}$.)  \n\nIntroduce the linear transformation  \n\\[\n\\Phi:\\Bbb C^{2}\\longrightarrow\\Bbb C^{2},\\qquad\n\\Phi(z_{0},z_{1})=(z_{1},E(z_{0},z_{1})).\n\\]\n\nStarting from a non-degenerate ordered edge $(z_{0},z_{1})$ consider the\nsequence of vertices  \n\\[\nz_{n+1}\\;=\\;E(z_{\\,n-1},z_{\\,n})\\qquad(n\\ge 1).\\tag{$\\clubsuit$}\n\\]\n(The ordered pairs $(z_{n-1},z_{n})$ arise by iterating $\\Phi$.)\n\nAnswer the following questions.\n\n1.  Algebraic background  \n    (a)  Prove that $E$ is $\\Bbb C$–linear in the ordered pair\n         $(z_{0},z_{1})$, and determine the matrix $A$ of $\\Phi$ with\n         respect to the standard basis of $\\Bbb C^{2}$.  \n    (b)  Compute the characteristic and the minimal polynomial of $A$,\n         prove that $A$ has the distinct eigenvalues $1$ and $\\rho$,\n         deduce that $A$ is diagonalisable and satisfies $A^{3}=I$\n         while $A\\neq I$ and $A^{2}\\neq I$.  \n         Conclude that $\\Phi$ has exact order $3$.\n\n2.  Explicit dynamics  \n    (a)  Put $w:=z_{1}-z_{0}$.  Show that for every $n\\ge0$\n         \\[\n           z_{n}\n           \\;=\\;\n           z_{0}\\;+\\;\\frac{1-\\rho^{\\,n}}{1-\\rho}\\,w.\n         \\]\n    (b)  Conclude that $(z_{n})$ is $3$-periodic and that the orbit is\n         constant if and only if $z_{1}=z_{0}$.\n\n3.  Geometry  \n    (a)  Verify directly from $(\\star)$ that\n         $\\Delta_{0}:=(z_{0},z_{1},z_{2})$ is a clockwise–oriented\n         equilateral triangle.  \n    (b)  Let\n         \\[\n           G\\;:=\\;\\frac{z_{0}+z_{1}+z_{2}}{3}\n         \\]\n         be its centroid.  Prove the fundamental rotation identity\n         \\[\n           z_{n+1}-G\n           \\;=\\;\n           \\rho\\,(z_{n}-G)\\qquad(n\\ge0),\n         \\]\n         and deduce that $\\Phi$ acts, on single vertices, as the\n         rotation of order $3$ about $G$ through the angle\n         $\\arg\\rho=-\\tfrac{2\\pi}{3}$.  Determine, in terms of $|w|$,  \n           • the circum-radius $R$, and  \n           • the in-radius $r$  \n         of $\\Delta_{0}$.  (Show in particular that $R\\neq r$ although\n         both centres coincide at $G$.)  \n    (c)  Conversely, show that every clockwise-oriented equilateral\n         triangle in $\\Bbb C$ arises from a unique ordered side\n         $(z_{0},z_{1})$ via the scheme $(\\clubsuit)$.\n\n4.  Arithmetic refinement over the Eisenstein integers  \n   Let $\\omega:=e^{2\\pi i/3}$ and $\\mathfrak E:=\\Bbb Z[\\omega]$.\n\n   (a)  Prove that $\\varepsilon=-\\omega$ and $1-\\varepsilon=1+\\omega$\n        both lie in $\\mathfrak E$ and deduce that\n        $E(\\mathfrak E,\\mathfrak E)\\subset\\mathfrak E$; hence for\n        $z_{0},z_{1}\\in\\mathfrak E$ the whole $3$-periodic orbit\n        $(z_{0},z_{1},z_{2})$ lies in $\\mathfrak E$.  Exhibit an explicit\n        example with exactly one of $z_{0},z_{1}$ in $\\mathfrak E$\n        for which $z_{2}\\notin\\mathfrak E$.  \n\n   (b)  From now on assume that the three vertices themselves lie in\n        $\\mathfrak E$.  Fix a non-zero $d\\in\\mathfrak E$.  \n        Up to translation by elements of $\\mathfrak E$, determine and\n        count all clockwise-oriented equilateral triangles whose first\n        side-vector equals $d$.  Explain how the unit group\n        $\\{\\pm1,\\pm\\omega,\\pm\\omega^{2}\\}$ acts on this family.\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "solution": "We keep the notation  \n\\[\n\\varepsilon=e^{-\\pi i/3},\\qquad\n\\rho=\\varepsilon-1=e^{-2\\pi i/3},\\qquad\nw:=z_{1}-z_{0}.\n\\]\n\n\\textbf{1. Algebraic background}\n\n(a)  For $(z_{0},z_{1}),(u_{0},u_{1})\\in\\Bbb C^{2}$ and\n$\\lambda\\in\\Bbb C$ we have\n\\[\nE(\\lambda z_{0}+u_{0},\\lambda z_{1}+u_{1})\n=(1-\\varepsilon)(\\lambda z_{0}+u_{0})\n+\\varepsilon(\\lambda z_{1}+u_{1})\n=\\lambda E(z_{0},z_{1})+E(u_{0},u_{1}),\n\\]\nso $E$ is \\emph{linear} (not bilinear) in the ordered pair.  \nWriting column vectors $v=(z_{0},z_{1})^{\\mathsf T}$ we obtain\n\\[\n\\Phi(v)=\\bigl(z_{1},(1-\\varepsilon)z_{0}+\\varepsilon z_{1}\\bigr)^{\\mathsf T}\n=A\\,v\n\\quad\\text{with}\\quad\nA=\\begin{bmatrix}0&1\\\\ 1-\\varepsilon&\\varepsilon\\end{bmatrix}.\n\\]\n\n(b)  The characteristic polynomial is\n\\[\n\\chi_{A}(\\lambda)=\n\\det(\\lambda I-A)=\\lambda^{2}-\\varepsilon\\lambda-(1-\\varepsilon).\n\\]\nSince $\\chi_{A}(1)=0$ and $\\chi_{A}(\\rho)=0$, we factor\n\\[\n\\chi_{A}(\\lambda)=(\\lambda-1)(\\lambda-\\rho).\n\\]\nThe two roots are distinct, hence $A$ is diagonalisable and the minimal\npolynomial equals $\\chi_{A}$.  Because $\\rho^{3}=1$ we have\n$\\lambda^{3}-1=(\\lambda-1)(\\lambda-\\rho)(\\lambda-\\rho^{2})$, thus\n$\\chi_{A}\\mid(\\lambda^{3}-1)$ and $A^{3}=I$.  Since $\\rho\\neq\\pm1$,\nneither $A$ nor $A^{2}$ equals $I$, so $\\Phi$ has exact order $3$.\n\n\\medskip\n\\textbf{2. Explicit dynamics}\n\nLet $v_{n}:=(z_{\\,n-1},z_{\\,n})^{\\mathsf T}$ for $n\\ge1$.  Then\n$v_{n}=A^{\\,n-1}v_{1}$.  Diagonalising $A$ gives\n\\[\nA^{n}=P_{1}+\\rho^{\\,n}P_{\\rho},\n\\qquad\nP_{1}+P_{\\rho}=I,\n\\qquad\nP_{1}P_{\\rho}=0,\n\\]\nwhere $P_{1},P_{\\rho}$ are the eigen-projectors.\nApplying this to $v_{1}$ and taking the second coordinate yields\n\\[\nz_{n}=z_{0}+\\frac{1-\\rho^{\\,n}}{1-\\rho}\\,w\\qquad(n\\ge0).\n\\]\nBecause $\\rho^{3}=1$, the sequence $(z_{n})$ is $3$-periodic.  It is\nconstant iff $w=0$, i.e. $z_{1}=z_{0}$.\n\n\\medskip\n\\textbf{3. Geometry}\n\n(a)  From $(\\star)$ we compute\n\\[\nz_{2}-z_{0}=E(z_{0},z_{1})-z_{0}=\\varepsilon w,\\qquad\nz_{2}-z_{1}=\\rho w.\n\\]\nHence all three side lengths equal $|w|$ and the oriented angle from\n$z_{1}-z_{0}$ to $z_{2}-z_{1}$ equals $\\arg\\rho=-\\tfrac{2\\pi}{3}$,\nso $\\Delta_{0}$ is clockwise equilateral.\n\n(b)  Put $G:=(z_{0}+z_{1}+z_{2})/3$.  Using the explicit formula for\n$z_{n}$ with $n=0,1,2$ we find\n\\[\nz_{1}-G=\\rho\\,(z_{0}-G),\\qquad\nz_{2}-G=\\rho\\,(z_{1}-G),\n\\]\nand by induction\n$z_{n+1}-G=\\rho\\,(z_{n}-G)$ for all $n\\ge0$.  Hence $\\Phi$\nacts as the rotation about $G$ through the angle $-120^{\\circ}$.\n\nBecause every vertex is at distance\n\\[\nR=|z_{0}-G|=\\frac{|w|}{\\sqrt3}\n\\]\nfrom $G$, this distance is the circum-radius.  In an equilateral\ntriangle the in-radius is $r=R/2$, i.e.\n\\[\nr=\\frac{|w|}{2\\sqrt3},\n\\qquad\n\\text{so}\\quad R\\neq r.\n\\]\n\n(c)  Conversely, given a clockwise equilateral triangle\n$T=\\{p,q,r\\}$ with ordered side $(p,q)$, the definition of $E$\nforces $r=E(p,q)$; iteration of $(\\clubsuit)$ therefore reproduces\n$T$.  Different ordered sides give different triangles, establishing\na bijection.\n\n\\medskip\n\\textbf{4. Arithmetic refinement over $\\mathfrak E=\\Bbb Z[\\omega]$}\n\n(a)  We have $\\varepsilon=-\\omega$ and $1-\\varepsilon=1+\\omega\\in\\mathfrak\nE$, hence $E(\\mathfrak E,\\mathfrak E)\\subset\\mathfrak E$ and any orbit\nstarting in $\\mathfrak E^{2}$ remains in $\\mathfrak E$ by\n$3$-periodicity.  As a counter-example with only one entry in\n$\\mathfrak E$ take\n\\[\nz_{0}=\\tfrac12\\notin\\mathfrak E,\\qquad\nz_{1}=0\\in\\mathfrak E.\n\\]\nThen\n\\[\nz_{2}=E(z_{0},z_{1})=(1-\\varepsilon)\\tfrac12\n=\\tfrac12(1+\\omega)\\notin\\mathfrak E.\n\\]\n\n(b)  \\emph{Now assume that all three vertices of the triangle are in\n$\\mathfrak E$.}  After translating by $-z_{0}\\in\\mathfrak E$ we may\nsuppose the first vertex is $0$.  If the first side-vector equals\n$d\\in\\mathfrak E\\setminus\\{0\\}$, the other vertices are\n\\[\n0,\\ d,\\ \\varepsilon d=-\\omega d.\n\\]\nConversely any translated triangle in $\\mathfrak E^{2}$ with first\nside-vector $d$ must coincide with this one.  Hence, \\emph{modulo\ntranslations by $\\mathfrak E$}, there is exactly one clockwise\nequilateral triangle with prescribed non-zero\n$d\\in\\mathfrak E$.\n\nLet $U=\\{\\pm1,\\pm\\omega,\\pm\\omega^{2}\\}$ be the unit group of\n$\\mathfrak E$.  Multiplication by $u\\in U$ sends the representative\n$\\{0,d,\\varepsilon d\\}$ to $u\\cdot\\{0,d,\\varepsilon d\\}$, thereby\nacting freely and transitively on the six possible first side-vectors\nof the same $\\mathfrak E$-norm.\n\n\\hfill$\\square$\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.510197",
        "was_fixed": false,
        "difficulty_analysis": "Compared with the original “find the third vertex’’ question, the\nenhanced variant\n\n• elevates the geometry from a *single* equilateral triangle to the\n   iterative dynamics that *generates an entire regular hexagon*;\n\n• introduces linear–algebraic machinery: one must build and analyse a\n   2×2 complex matrix, find its minimal polynomial, eigenvalues and\n   order;\n\n• requires bridging algebra and geometry (matrix powers ↔ rotations of\n   edge-vectors) to prove regularity and locate the centre;\n\n• adds a number-theoretic layer over the Gaussian integers, demanding a\n   lattice/ideal test for integrality of all six vertices and a counting\n   argument for equivalence classes of hexagons;\n\n• compels the solver to juggle several interacting concepts—complex\n   affine maps, root-of-unity rotations, linear recurrence, group order,\n   lattice arithmetic—making it substantially more intricate than both\n   the original problem and the current kernel variant."
      }
    },
    "original_kernel_variant": {
      "question": "Let  \n      ε := e^{−π i /3}= ½ −(√3 /2)i ,   ρ := ε−1 = e^{−2π i /3}  \n(so ρ³ = 1, ρ ≠ 1).  \nFor two distinct complex numbers (z₀ ,z₁) define the clockwise\nequilateral transform  \n\n  E(z₀ ,z₁) := (1−ε) z₀ + ε z₁.  (★)\n\n(The point E(z₀ ,z₁) is the unique third vertex that completes the\nclockwise-oriented equilateral triangle on the ordered side z₀z₁.)\n\nIntroduce the linear map  \n\n  Φ : ℂ² → ℂ²,  Φ(z₀ ,z₁) = (z₁ , E(z₀ ,z₁)).\n\nStarting from a non-degenerate ordered edge (z₀ ,z₁) consider the\nsequence of vertices  \n\n  z_{n+1} = E(z_{\\,n−1}, z_{\\,n})  (n ≥ 1).  (♣)\n\n(The ordered pairs (z_{n−1},z_{n}) are obtained by iterating Φ.)\n\nAnswer the following questions.\n\n1.  Algebraic background  \n    (a)  Prove that Φ is ℂ–linear and determine the matrix A of Φ in the\n         standard basis of ℂ².  \n    (b)  Compute the characteristic and the minimal polynomial of A,\n         prove that A has the distinct eigenvalues 1 and ρ, deduce that\n         A is diagonalisable and satisfies A³ = I while\n         A ≠ I, A² ≠ I.  Show that Φ therefore has exact order 3.\n\n2.  Explicit dynamics  \n    (a)  Put w := z₁−z₀.  Show that for every n ≥ 0  \n              z_n = z₀ + [(1−ρⁿ)/(1−ρ)] w.  \n    (b)  Conclude that (z_n) is 3-periodic and that the orbit is\n         constant iff z₁ = z₀.\n\n3.  Geometry  \n    (a)  Verify directly from (★) that  \n         Δ₀ := (z₀ ,z₁ ,z₂) is an equilateral triangle oriented\n         clockwise.  \n    (b)  Let  \n             G := (z₀+z₁+z₂)/3  \n         be its centroid.  Prove the fundamental rotation identity  \n\n             z_{n+1} − G = ρ ( z_n − G )  (n ≥ 0),\n\n         and deduce that Φ acts, on the level of single vertices, as the\n         order-3 rotation about G through the angle arg ρ = −120°.  \n         Determine, in terms of |w|,  \n             • the circum-radius  R  and  \n             • the in-radius     r  \n         of Δ₀.  (Show in particular that R ≠ r, although the two\n         centres coincide.)  \n    (c)  Conversely, show that every clockwise-oriented equilateral\n         triangle in ℂ arises from a unique ordered side (z₀ ,z₁) via\n         the scheme (♣).\n\n4.  Arithmetic refinement over the Eisenstein integers  \n   Let ω := e^{2π i /3} and 𝔈 := ℤ[ω].\n\n   (a)  Prove that ε = −ω and 1−ε = 1+ω both belong to 𝔈 and deduce\n        that E(𝔈,𝔈) ⊂ 𝔈; hence for z₀ ,z₁ ∈ 𝔈 the whole 3-periodic\n        orbit (z₀ ,z₁ ,z₂) lies in 𝔈.  Exhibit an explicit example\n        with exactly one of z₀ ,z₁ in 𝔈 for which z₂ ∉ 𝔈.  \n\n   (b)  Fix a non-zero d ∈ 𝔈.  Up to translation by elements of 𝔈,\n        determine and count all clockwise-oriented equilateral triangles\n        whose first side-vector equals d.  Explain how the unit group\n        {±1, ±ω, ±ω²} acts on this family.\n\n\n\n────────────────────────────────────────────────────────────────────────",
      "solution": "Notation.  ε = e^{−π i /3}, ρ := ε−1 = e^{−2π i /3} (ρ³ = 1, ρ ≠ 1).\n\n────────────────────────────────\n1.  Algebraic background  \n\n(a)  The map (★) is ℂ-linear in each argument and homogeneous of degree\none, hence E itself is ℂ-bilinear; consequently Φ, defined by\nΦ(z₀ ,z₁) = (z₁ ,E(z₀ ,z₁)), is ℂ-linear.  With column vectors\nv = (z₀ ,z₁)ᵗ we get  \n\n  Φ(v) = ( z₁ ,(1−ε)z₀ + ε z₁ ) = A v with  \n\n  A = [ 0 1; 1−ε ε ].  (1)\n\n(b)  Characteristic polynomial  \n\n χ_A(λ) = det(λI − A) = λ² − ε λ − (1−ε). (2)\n\nPlugging λ = 1 and λ = ρ gives χ_A(1)=χ_A(ρ)=0, so\n\n χ_A(λ)= (λ−1)(λ−ρ).  \n\nThe two roots are distinct; hence A is diagonalisable and its minimal\npolynomial equals χ_A.  As ρ³=1 we have χ_A | (λ³−1), therefore A³=I.\nBecause ρ ≠ ±1, neither A nor A² is the identity, so Φ has exact\norder 3.\n\n────────────────────────────────\n2.  Explicit dynamics  \n\nLet w := z₁−z₀ and v_n := (z_{\\,n−1}, z_{\\,n})ᵗ (n ≥ 1).\nThen v_{n}=A^{n−1}v_1.  The spectral resolution of A is  \n\n A^{n}= P₁ + ρ^{\\,n} P_ρ,  P₁+P_ρ=I, P₁P_ρ=0, (3)\n\nwhere P₁, P_ρ are the eigen-projectors.  Applying (3) to v_1 and taking\nthe second coordinate yields  \n\n z_n = z₀ + [(1−ρ^{\\,n})/(1−ρ)] w,  n ≥ 0. (4)\n\n(b)  Because ρ³ = 1, the right–hand side is 3-periodic.  If some\nz_{n+1}=z_n then the multiplier of w in (4) vanishes, forcing w=0,\ni.e. z₁=z₀.  Thus the orbit is constant exactly in the degenerate case;\notherwise it has minimal period 3.\n\n────────────────────────────────\n3.  Geometry  \n\n(a)  From (★)\n\n z₂−z₀ = ε w,  z₂−z₁ = ρ w,\n\nhence all sides have length |w| and the oriented angle from z₁−z₀ to\nz₂−z₁ equals arg ρ = −120°, i.e. clockwise.  Thus Δ₀ is equilateral.\n\n(b)  Put G := (z₀+z₁+z₂)/3.  \nApplying (4) for n = 0,1,2 we get  \n\n z₁−G = ρ (z₀−G),  z₂−G = ρ (z₁−G),\n\nand multiplying by ρ iteratively gives for all n ≥ 0  \n\n z_{n+1} − G = ρ ( z_n − G ). (5)\n\nHence Φ acts as the rotation R_{G,−120°}.\n\nCircum- and in-radii.  \nBecause all vertices are at the same distance from G, the circum-radius\nis  \n\n R = |z₀−G| = |w|/√3.  \n\nFor an equilateral triangle, the in-radius satisfies r = R/2, so  \n\n r = |w|/(2√3).  \n\nThus R ≠ r, although both centres coincide at G.\n\n(c)  Conversely, let T be any clockwise equilateral triangle with\nordered side (p,q).  By construction E(p,q) is the unique vertex r that\ncompletes T, and iterating (♣) reproduces T.  Different ordered sides\ngive different triangles, so the correspondence is bijective.\n\n────────────────────────────────\n4.  Arithmetic refinement over 𝔈 = ℤ[ω]  \n\n(a)  We have ε = −ω and 1−ε = 1+ω ∈ 𝔈, so E(𝔈,𝔈) ⊂ 𝔈 and any orbit\nissued from z₀ ,z₁ ∈ 𝔈 stays in 𝔈 by 3-periodicity.\nCounter-example with only one entry in 𝔈: take  \n z₀ = ½ ∉ 𝔈,  z₁ = 0 ∈ 𝔈.  \nThen  \n\n z₂ = E(z₀ ,z₁) = (1−ε)·½ + ε·0 = (1+ω)/2 ∉ 𝔈,\n\nso the orbit leaves 𝔈.\n\n(b)  Translate so that the first vertex is 0.  With side-vector d ≠ 0\nthe triangle is {0, d, ε d}= {0, d, −ω d}, which lies in 𝔈 by (a) and\nis clockwise.  Any other translate differs by an element of 𝔈, hence\nmodulo translations there is exactly one such triangle.\n\nLet U = {±1, ±ω, ±ω²} be the unit group of 𝔈.  Multiplying the fixed\nside-vector d by a unit u sends the triangle {0,d,ε d} to\n{0,ud, ε(ud)} = u·{0,d,ε d}.  Thus U acts freely and transitively on\nthe family of admissible side-vectors of the same norm, while the set\nof triangles themselves consists of a single orbit modulo translations.\n\n\n\n────────────────────────────────────────────────────────────────────────",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.426933",
        "was_fixed": false,
        "difficulty_analysis": "Compared with the original “find the third vertex’’ question, the\nenhanced variant\n\n• elevates the geometry from a *single* equilateral triangle to the\n   iterative dynamics that *generates an entire regular hexagon*;\n\n• introduces linear–algebraic machinery: one must build and analyse a\n   2×2 complex matrix, find its minimal polynomial, eigenvalues and\n   order;\n\n• requires bridging algebra and geometry (matrix powers ↔ rotations of\n   edge-vectors) to prove regularity and locate the centre;\n\n• adds a number-theoretic layer over the Gaussian integers, demanding a\n   lattice/ideal test for integrality of all six vertices and a counting\n   argument for equivalence classes of hexagons;\n\n• compels the solver to juggle several interacting concepts—complex\n   affine maps, root-of-unity rotations, linear recurrence, group order,\n   lattice arithmetic—making it substantially more intricate than both\n   the original problem and the current kernel variant."
      }
    }
  },
  "checked": true,
  "problem_type": "proof",
  "iteratively_fixed": true
}