{
  "index": "2015-B-3",
  "type": "ALG",
  "tag": [
    "ALG",
    "NT"
  ],
  "difficulty": "",
  "question": "Let $S$ be the set of all $2 \\times 2$ real matrices\n\\[\nM = \\begin{pmatrix} a & b \\\\\nc & d \\end{pmatrix}\n\\]\nwhose entries $a,b,c,d$ (in that order) form an arithmetic progression. Find all matrices $M$ in $S$ for which there is some integer $k>1$ such that $M^k$ is also in $S$.",
  "solution": "\\textbf{First solution:}\nAny element of $S$ can be written as $M = \\alpha A + \\beta B$, where $A = \\left( \\begin{smallmatrix} 1 & 1 \\\\ 1 & 1 \\end{smallmatrix} \\right)$, $B = \\left( \\begin{smallmatrix} -3 & -1 \\\\ 1 & 3 \\end{smallmatrix} \\right)$, and $\\alpha,\\beta \\in \\mathbb{R}$. Note that $A^2 = \\left( \\begin{smallmatrix} 4 & 4 \\\\ 4 & 4 \\end{smallmatrix} \\right)$ and\n$B^3 = \\left( \\begin{smallmatrix} -24 & -8 \\\\ 8 & 24 \\end{smallmatrix} \\right)$ are both in $S$, and so any matrix of the form $\\alpha A$ or $\\beta B$, $\\alpha,\\beta\\in\\mathbb{R}$, satisfies the given condition.\n\nWe claim that these are also the only matrices in $S$ satisfying the given condition. Indeed, suppose $M = \\alpha A + \\beta B$ where $\\alpha,\\beta \\neq 0$. Let $C = \\left( \\begin{smallmatrix} 1 & 1/\\sqrt{2} \\\\ -1 & 1/\\sqrt{2} \\end{smallmatrix} \\right)$ with inverse\n$C^{-1} = \\left( \\begin{smallmatrix} 1/2 & -1/2 \\\\ 1/\\sqrt{2} & 1/\\sqrt{2} \\end{smallmatrix} \\right)$. If we define $D = C^{-1}MC$, then $D = 2\\alpha \\left( \\begin{smallmatrix} 0 & \\gamma \\\\ \\gamma & 1 \\end{smallmatrix} \\right)$ where $\\gamma = -\\frac{\\beta\\sqrt{2}}{\\alpha}$. Now suppose that $M^k$ is in $S$ with $k\\geq 2$. Since $\\left( \\begin{smallmatrix} 1 & -1 \\end{smallmatrix} \\right) A \\left( \\begin{smallmatrix} 1 \\\\ -1 \\end{smallmatrix} \\right) = \\left( \\begin{smallmatrix} 1 & -1 \\end{smallmatrix} \\right) B \\left( \\begin{smallmatrix} 1 \\\\ -1 \\end{smallmatrix} \\right) = 0$,\nwe have $\\left( \\begin{smallmatrix} 1 & -1 \\end{smallmatrix} \\right) M^k \\left( \\begin{smallmatrix} 1 \\\\ -1 \\end{smallmatrix} \\right) = 0$, and so the upper left entry of $C^{-1} M^k C = D^k$ is $0$. On the other hand, from the expression for $D$, an easy induction on $k$ shows that\n$D^k = (2\\alpha)^k \\left( \\begin{smallmatrix} \\gamma^2 p_{k-1} & \\gamma p_k \\\\\n\\gamma p_k & p_{k+1} \\end{smallmatrix} \\right)$, where $p_k$ is defined inductively by $p_0 = 0$, $p_1 = 1$, $p_{k+2} = \\gamma^2 p_k + p_{k+1}$. In particular, it follows from the inductive definition that $p_k > 0$ when $k \\geq 1$, whence the upper left entry of $D^k$ is nonzero when $k \\geq 2$, a contradiction.\n\n\\noindent\n\\textbf{Remark:}\nA variant of this solution can be obtained by diagonalizing the matrix $M$.\n\n\\textbf{Second solution:}\nIf $a,b,c,d$ are in arithmetic progression, then we may write\n\\[\na = r-3s, b=r-s, c=r+s, d=r+3s\n\\]\nfor some $r,s$. If $s=0$, then clearly all powers of $M$ are in $xS$. Also, if $r=0$, then one easily checks that $M^3$ is in $S$.\n\nWe now assume $rs\\neq 0$, and show that in that case $M$ cannot be in $S$. First, note that the characteristic polynomial of $M$ is $x^2-2rx-8s^2$, and since $M$ is nonsingular (as $s\\neq 0$), this is also the minimal polynomial of $M$ by the Cayley-Hamilton theorem.  By repeatedly using the relation $M^2=2rM+8s^2I$, we see that for each positive integer, we have $M^k = t_k M + u_k I$ for unique real constants $t_k, u_k$ (uniqueness follows from the independence of $M$ and $I$).  Since $M$ is in $S$, we see that $M^k$ lies in $S$ only if $u_k=0$.\n\nOn the other hand, we claim that if $k>1$, then $rt_k>0$ and $u_k>0$ if $k$ is even, and $t_k>0$ and $ru_k>0$ if $k$ is odd (in particular, $u_k$ can never be zero).  The claim is true for $k=2$ by the relation $M^2=2rM+8s^2I$. Assuming the claim for $k$, and multiplying both sides of the relation $M^k = t_k M + u_k I$ by $M$, yields\n\\[\nM^{k+1} = t_k (2rM+8s^2I) + u_k M = (2rt_k+u_k) M + 8s^2t_k I,\n\\]\nimplying the claim for $k+1$.\n\n\\noindent\n\\textbf{Remark:}\n(from \\url{artofproblemsolving.com}, user \\texttt{hoeij})\nOnce one has $u_k = 0$, one can also finish using the relation $M \\cdot M^k = M^k \\cdot M$.",
  "vars": [
    "S",
    "M",
    "a",
    "b",
    "c",
    "d",
    "k",
    "p_k",
    "p_k-1",
    "p_k+1",
    "t_k",
    "u_k",
    "x",
    "D",
    "r",
    "s"
  ],
  "params": [
    "A",
    "B",
    "alpha",
    "beta",
    "gamma",
    "C",
    "I"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "S": "matrixset",
        "M": "matrixvar",
        "a": "firstterm",
        "b": "secondterm",
        "c": "thirdterm",
        "d": "fourthterm",
        "k": "powerindex",
        "p_k": "seqterm",
        "p_k-1": "seqprev",
        "p_k+1": "seqnext",
        "t_k": "coefft",
        "u_k": "coeffu",
        "x": "polyvar",
        "D": "transmat",
        "r": "arithcenter",
        "s": "arithstep",
        "A": "basisone",
        "B": "basistwo",
        "alpha": "alphacoef",
        "beta": "betacoef",
        "gamma": "gammacoef",
        "C": "changemat",
        "I": "identity"
      },
      "question": "Let $matrixset$ be the set of all $2 \\times 2$ real matrices\n\\[\nmatrixvar = \\begin{pmatrix} firstterm & secondterm \\\\\nthirdterm & fourthterm \\end{pmatrix}\n\\]\nwhose entries $firstterm,secondterm,thirdterm,fourthterm$ (in that order) form an arithmetic progression. Find all matrices $matrixvar$ in $matrixset$ for which there is some integer $powerindex>1$ such that $matrixvar^{powerindex}$ is also in $matrixset$.",
      "solution": "\\textbf{First solution:}\nAny element of $matrixset$ can be written as $matrixvar = alphacoef basisone + betacoef basistwo$, where $basisone = \\left( \\begin{smallmatrix} 1 & 1 \\\\ 1 & 1 \\end{smallmatrix} \\right)$, $basistwo = \\left( \\begin{smallmatrix} -3 & -1 \\\\ 1 & 3 \\end{smallmatrix} \\right)$, and $alphacoef,betacoef \\in \\mathbb{R}$. Note that $basisone^2 = \\left( \\begin{smallmatrix} 4 & 4 \\\\ 4 & 4 \\end{smallmatrix} \\right)$ and\n$basistwo^3 = \\left( \\begin{smallmatrix} -24 & -8 \\\\ 8 & 24 \\end{smallmatrix} \\right)$ are both in $matrixset$, and so any matrix of the form $alphacoef basisone$ or $betacoef basistwo$, $alphacoef,betacoef\\in\\mathbb{R}$, satisfies the given condition.\n\nWe claim that these are also the only matrices in $matrixset$ satisfying the given condition. Indeed, suppose $matrixvar = alphacoef basisone + betacoef basistwo$ where $alphacoef,betacoef \\neq 0$. Let $changemat = \\left( \\begin{smallmatrix} 1 & 1/\\sqrt{2} \\\\ -1 & 1/\\sqrt{2} \\end{smallmatrix} \\right)$ with inverse\n$changemat^{-1} = \\left( \\begin{smallmatrix} 1/2 & -1/2 \\\\ 1/\\sqrt{2} & 1/\\sqrt{2} \\end{smallmatrix} \\right)$. If we define $transmat = changemat^{-1}matrixvar changemat$, then $transmat = 2alphacoef \\left( \\begin{smallmatrix} 0 & gammacoef \\\\ gammacoef & 1 \\end{smallmatrix} \\right)$ where $gammacoef = -\\frac{betacoef\\sqrt{2}}{alphacoef}$. Now suppose that $matrixvar^{powerindex}$ is in $matrixset$ with $powerindex\\geq 2$. Since $\\left( \\begin{smallmatrix} 1 & -1 \\end{smallmatrix} \\right) basisone \\left( \\begin{smallmatrix} 1 \\\\ -1 \\end{smallmatrix} \\right) = \\left( \\begin{smallmatrix} 1 & -1 \\end{smallmatrix} \\right) basistwo \\left( \\begin{smallmatrix} 1 \\\\ -1 \\end{smallmatrix} \\right) = 0$,\nwe have $\\left( \\begin{smallmatrix} 1 & -1 \\end{smallmatrix} \\right) matrixvar^{powerindex} \\left( \\begin{smallmatrix} 1 \\\\ -1 \\end{smallmatrix} \\right) = 0$, and so the upper left entry of $changemat^{-1} matrixvar^{powerindex} changemat = transmat^{powerindex}$ is $0$. On the other hand, from the expression for $transmat$, an easy induction on $powerindex$ shows that\n$transmat^{powerindex} = (2alphacoef)^{powerindex} \\left( \\begin{smallmatrix} gammacoef^2 seqprev & gammacoef seqterm \\\\\n gammacoef seqterm & seqnext \\end{smallmatrix} \\right)$, where $seqterm$ is defined inductively by $p_0 = 0$, $p_1 = 1$, $p_{powerindex+2} = gammacoef^2 p_{powerindex} + p_{powerindex+1}$. In particular, it follows from the inductive definition that $seqterm > 0$ when $powerindex \\geq 1$, whence the upper left entry of $transmat^{powerindex}$ is nonzero when $powerindex \\geq 2$, a contradiction.\n\n\\noindent\n\\textbf{Remark:}\nA variant of this solution can be obtained by diagonalizing the matrix $matrixvar$.\n\n\\textbf{Second solution:}\nIf $firstterm,secondterm,thirdterm,fourthterm$ are in arithmetic progression, then we may write\n\\[\nfirstterm = arithcenter-3arithstep,\\; secondterm=arithcenter-arithstep,\\; thirdterm=arithcenter+arithstep,\\; fourthterm=arithcenter+3arithstep\n\\]\nfor some $arithcenter,arithstep$. If $arithstep=0$, then clearly all powers of $matrixvar$ are in $xmatrixset$. Also, if $arithcenter=0$, then one easily checks that $matrixvar^3$ is in $matrixset$.\n\nWe now assume $arithcenterarithstep\\neq 0$, and show that in that case $matrixvar$ cannot be in $matrixset$. First, note that the characteristic polynomial of $matrixvar$ is $polyvar^2-2arithcenter polyvar-8arithstep^2$, and since $matrixvar$ is nonsingular (as $arithstep\\neq 0$), this is also the minimal polynomial of $matrixvar$ by the Cayley-Hamilton theorem.  By repeatedly using the relation $matrixvar^2=2arithcenter matrixvar+8arithstep^2 identity$, we see that for each positive integer, we have $matrixvar^{powerindex} = coefft matrixvar + coeffu identity$ for unique real constants $coefft, coeffu$ (uniqueness follows from the independence of $matrixvar$ and $identity$).  Since $matrixvar$ is in $matrixset$, we see that $matrixvar^{powerindex}$ lies in $matrixset$ only if $coeffu=0$.\n\nOn the other hand, we claim that if $powerindex>1$, then $arithcenter coefft>0$ and $coeffu>0$ if $powerindex$ is even, and $coefft>0$ and $arithcenter coeffu>0$ if $powerindex$ is odd (in particular, $coeffu$ can never be zero).  The claim is true for $powerindex=2$ by the relation $matrixvar^2=2arithcenter matrixvar+8arithstep^2 identity$. Assuming the claim for $powerindex$, and multiplying both sides of the relation $matrixvar^{powerindex} = coefft matrixvar + coeffu identity$ by $matrixvar$, yields\n\\[\nmatrixvar^{powerindex+1} = coefft (2arithcenter matrixvar+8arithstep^2 identity) + coeffu matrixvar = (2arithcenter coefft+coeffu) matrixvar + 8arithstep^2 coefft identity,\n\\]\nimplying the claim for $powerindex+1$.\n\n\\noindent\n\\textbf{Remark:}\n(from \\url{artofproblemsolving.com}, user \\texttt{hoeij})\nOnce one has $coeffu = 0$, one can also finish using the relation $matrixvar \\cdot matrixvar^{powerindex} = matrixvar^{powerindex} \\cdot matrixvar$."
    },
    "descriptive_long_confusing": {
      "map": {
        "S": "meadowland",
        "M": "stonepillar",
        "a": "whispering",
        "b": "drifting",
        "c": "lingering",
        "d": "thundering",
        "k": "moonlight",
        "p_k": "riverdance",
        "p_k-1": "riverdanceminus",
        "p_k+1": "riverdanceplus",
        "t_k": "lanternfire",
        "u_k": "shadowglow",
        "x": "frostglade",
        "D": "crystalveil",
        "r": "northwind",
        "s": "dawnchorus",
        "A": "sunflower",
        "B": "willowbark",
        "alpha": "silverbell",
        "beta": "coppersong",
        "gamma": "embermist",
        "C": "cloudanchor",
        "I": "starquartz"
      },
      "question": "Let $meadowland$ be the set of all $2 \\times 2$ real matrices\n\\[\nstonepillar = \\begin{pmatrix} whispering & drifting \\\\\nlingering & thundering \\end{pmatrix}\n\\]\nwhose entries $whispering,drifting,lingering,thundering$ (in that order) form an arithmetic progression. Find all matrices $stonepillar$ in $meadowland$ for which there is some integer $moonlight>1$ such that $stonepillar^{moonlight}$ is also in $meadowland$.",
      "solution": "\\textbf{First solution:}\nAny element of $meadowland$ can be written as $stonepillar = silverbell sunflower + coppersong willowbark$, where $sunflower = \\left( \\begin{smallmatrix} 1 & 1 \\\\ 1 & 1 \\end{smallmatrix} \\right)$, $willowbark = \\left( \\begin{smallmatrix} -3 & -1 \\\\ 1 & 3 \\end{smallmatrix} \\right)$, and $silverbell,coppersong \\in \\mathbb{R}$. Note that $sunflower^2 = \\left( \\begin{smallmatrix} 4 & 4 \\\\ 4 & 4 \\end{smallmatrix} \\right)$ and\n$willowbark^3 = \\left( \\begin{smallmatrix} -24 & -8 \\\\ 8 & 24 \\end{smallmatrix} \\right)$ are both in $meadowland$, and so any matrix of the form $silverbell sunflower$ or $coppersong willowbark$, $silverbell,coppersong\\in\\mathbb{R}$, satisfies the given condition.\n\nWe claim that these are also the only matrices in $meadowland$ satisfying the given condition. Indeed, suppose $stonepillar = silverbell sunflower + coppersong willowbark$ where $silverbell,coppersong \\neq 0$. Let $cloudanchor = \\left( \\begin{smallmatrix} 1 & 1/\\sqrt{2} \\\\ -1 & 1/\\sqrt{2} \\end{smallmatrix} \\right)$ with inverse\n$cloudanchor^{-1} = \\left( \\begin{smallmatrix} 1/2 & -1/2 \\\\ 1/\\sqrt{2} & 1/\\sqrt{2} \\end{smallmatrix} \\right)$. If we define $crystalveil = cloudanchor^{-1}stonepillar cloudanchor$, then $crystalveil = 2silverbell \\left( \\begin{smallmatrix} 0 & embermist \\\\ embermist & 1 \\end{smallmatrix} \\right)$ where $embermist = -\\frac{coppersong\\sqrt{2}}{silverbell}$. Now suppose that $stonepillar^{moonlight}$ is in $meadowland$ with $moonlight\\geq 2$. Since $\\left( \\begin{smallmatrix} 1 & -1 \\end{smallmatrix} \\right) sunflower \\left( \\begin{smallmatrix} 1 \\\\ -1 \\end{smallmatrix} \\right) = \\left( \\begin{smallmatrix} 1 & -1 \\end{smallmatrix} \\right) willowbark \\left( \\begin{smallmatrix} 1 \\\\ -1 \\end{smallmatrix} \\right) = 0$,\nwe have $\\left( \\begin{smallmatrix} 1 & -1 \\end{smallmatrix} \\right) stonepillar^{moonlight} \\left( \\begin{smallmatrix} 1 \\\\ -1 \\end{smallmatrix} \\right) = 0$, and so the upper left entry of $cloudanchor^{-1} stonepillar^{moonlight} cloudanchor = crystalveil^{moonlight}$ is $0$. On the other hand, from the expression for $crystalveil$, an easy induction on $moonlight$ shows that\n$crystalveil^{moonlight} = (2silverbell)^{moonlight} \\left( \\begin{smallmatrix} embermist^2 riverdanceminus & embermist riverdance \\\\\nembermist riverdance & riverdanceplus \\end{smallmatrix} \\right)$, where $riverdance$ is defined inductively by $riverdance_0 = 0$, $riverdance_1 = 1$, $riverdance_{moonlight+2} = embermist^2 riverdance_{moonlight} + riverdance_{moonlight+1}$. In particular, it follows from the inductive definition that $riverdance > 0$ when $moonlight \\geq 1$, whence the upper left entry of $crystalveil^{moonlight}$ is nonzero when $moonlight \\geq 2$, a contradiction.\n\n\\noindent\n\\textbf{Remark:}\nA variant of this solution can be obtained by diagonalizing the matrix $stonepillar$.\n\n\\textbf{Second solution:}\nIf $whispering,drifting,lingering,thundering$ are in arithmetic progression, then we may write\n\\[\nwhispering = northwind-3dawnchorus, \\quad drifting=northwind-dawnchorus, \\quad lingering=northwind+dawnchorus, \\quad thundering=northwind+3dawnchorus\n\\]\nfor some $northwind,dawnchorus$. If $dawnchorus=0$, then clearly all powers of $stonepillar$ are in $frostglademeadowland$. Also, if $northwind=0$, then one easily checks that $stonepillar^3$ is in $meadowland$.\n\nWe now assume $northwind dawnchorus\\neq 0$, and show that in that case $stonepillar$ cannot be in $meadowland$. First, note that the characteristic polynomial of $stonepillar$ is $frostglade^2-2northwind frostglade-8dawnchorus^2$, and since $stonepillar$ is nonsingular (as $dawnchorus\\neq 0$), this is also the minimal polynomial of $stonepillar$ by the Cayley-Hamilton theorem.  By repeatedly using the relation $stonepillar^2=2northwind stonepillar+8dawnchorus^2 starquartz$, we see that for each positive integer, we have $stonepillar^{moonlight} = lanternfire stonepillar + shadowglow starquartz$ for unique real constants $lanternfire, shadowglow$ (uniqueness follows from the independence of $stonepillar$ and $starquartz$).  Since $stonepillar$ is in $meadowland$, we see that $stonepillar^{moonlight}$ lies in $meadowland$ only if $shadowglow=0$.\n\nOn the other hand, we claim that if $moonlight>1$, then $northwind lanternfire>0$ and $shadowglow>0$ if $moonlight$ is even, and $lanternfire>0$ and $northwind shadowglow>0$ if $moonlight$ is odd (in particular, $shadowglow$ can never be zero).  The claim is true for $moonlight=2$ by the relation $stonepillar^2=2northwind stonepillar+8dawnchorus^2 starquartz$. Assuming the claim for $moonlight$, and multiplying both sides of the relation $stonepillar^{moonlight} = lanternfire stonepillar + shadowglow starquartz$ by $stonepillar$, yields\n\\[\nstonepillar^{moonlight+1} = lanternfire (2northwind stonepillar+8dawnchorus^2 starquartz) + shadowglow stonepillar = (2northwind lanternfire+shadowglow) stonepillar + 8dawnchorus^2 lanternfire starquartz,\n\\]\nimplying the claim for $moonlight+1$.\n\n\\noindent\n\\textbf{Remark:}\n(from \\url{artofproblemsolving.com}, user \\texttt{hoeij})\nOnce one has $shadowglow = 0$, one can also finish using the relation $stonepillar \\cdot stonepillar^{moonlight} = stonepillar^{moonlight} \\cdot stonepillar$.",
      "}woju": ""
    },
    "descriptive_long_misleading": {
      "map": {
        "S": "disjointcollection",
        "M": "singledigit",
        "a": "lastterm",
        "b": "penultimate",
        "c": "firstelement",
        "d": "zerothpart",
        "k": "fractional",
        "p_k": "constantvalue",
        "p_k-1": "futurevalue",
        "p_k+1": "pastvalue",
        "t_k": "denominator",
        "u_k": "numerator",
        "x": "fixedconstant",
        "D": "staticvector",
        "r": "imaginarypart",
        "s": "randomgap",
        "A": "variablematrix",
        "B": "changematrix",
        "alpha": "endingvalue",
        "beta": "startingvalue",
        "gamma": "positivesum",
        "C": "staticbasis",
        "I": "zerooperator"
      },
      "question": "<<<\nLet $disjointcollection$ be the set of all $2 \\times 2$ real matrices\n\\[\nsingledigit = \\begin{pmatrix} lastterm & penultimate \\\\\nfirstelement & zerothpart \\end{pmatrix}\n\\]\nwhose entries $lastterm,penultimate,firstelement,zerothpart$ (in that order) form an arithmetic progression. Find all matrices $singledigit$ in $disjointcollection$ for which there is some integer $fractional>1$ such that $singledigit^{fractional}$ is also in $disjointcollection$.\n>>>",
      "solution": "<<<\n\\textbf{First solution:}\nAny element of $disjointcollection$ can be written as\n\\[\nsingledigit = endingvalue\\,variablematrix + startingvalue\\,changematrix,\n\\]\nwhere\n\\[\nvariablematrix = \\left( \\begin{smallmatrix} 1 & 1 \\\\ 1 & 1 \\end{smallmatrix} \\right),\\qquad\nchangematrix = \\left( \\begin{smallmatrix} -3 & -1 \\\\ 1 & 3 \\end{smallmatrix} \\right),\n\\]\nand $endingvalue,startingvalue\\in\\mathbb{R}$.  Note that\n\\[\nvariablematrix^{2}= \\left( \\begin{smallmatrix} 4 & 4 \\\\ 4 & 4 \\end{smallmatrix} \\right),\\qquad\nchangematrix^{3}= \\left( \\begin{smallmatrix} -24 & -8 \\\\ 8 & 24 \\end{smallmatrix} \\right)\n\\]\nare both in $disjointcollection$, and so any matrix of the form $endingvalue\\,variablematrix$ or $startingvalue\\,changematrix$, $endingvalue,startingvalue\\in\\mathbb{R}$, satisfies the given condition.\n\nWe claim that these are also the only matrices in $disjointcollection$ satisfying the given condition.  Indeed, suppose\n\\[\nsingledigit = endingvalue\\,variablematrix + startingvalue\\,changematrix\n\\]\nwith $endingvalue,startingvalue\\neq 0$.  Let\n\\[\nstaticbasis = \\begin{pmatrix} 1 & 1/\\sqrt{2} \\\\ -1 & 1/\\sqrt{2} \\end{pmatrix},\\qquad\nstaticbasis^{-1}= \\begin{pmatrix} 1/2 & -1/2 \\\\ 1/\\sqrt{2} & 1/\\sqrt{2} \\end{pmatrix}.\n\\]\nIf we define\n\\[\nstaticvector = staticbasis^{-1} \\, singledigit \\, staticbasis,\n\\]\nthen\n\\[\nstaticvector = 2\\,endingvalue!\n\\begin{pmatrix} 0 & positivesum \\\\ positivesum & 1 \\end{pmatrix},\n\\qquad\\text{where } positivesum = -\\frac{startingvalue\\sqrt{2}}{endingvalue}.\n\\]\n\nNow suppose that $singledigit^{fractional}$ is in $disjointcollection$ with $fractional\\ge 2$.\nSince\n\\[\n\\begin{pmatrix} 1 & -1 \\end{pmatrix} variablematrix \\begin{pmatrix} 1 \\\\ -1 \\end{pmatrix}\n= \\begin{pmatrix} 1 & -1 \\end{pmatrix} changematrix \\begin{pmatrix} 1 \\\\ -1 \\end{pmatrix}\n=0,\n\\]\nwe have\n\\[\n\\begin{pmatrix} 1 & -1 \\end{pmatrix} singledigit^{fractional} \\begin{pmatrix} 1 \\\\ -1 \\end{pmatrix}=0,\n\\]\nand so the upper–left entry of\n\\[\nstaticbasis^{-1} \\, singledigit^{fractional} \\, staticbasis\n           = staticvector^{fractional}\n\\]\nis $0$.  On the other hand, from the expression for $staticvector$, an easy induction on $fractional$ shows that\n\\[\nstaticvector^{fractional}\n= (2\\,endingvalue)^{\\!fractional}\n\\begin{pmatrix}\npositivesum^{2}\\,futurevalue & positivesum\\,constantvalue \\\\\npositivesum\\,constantvalue & pastvalue\n\\end{pmatrix},\n\\]\nwhere $constantvalue$ is defined inductively by\n\\[\nconstantvalue_{0}=0,\\qquad\nconstantvalue_{1}=1,\\qquad\nconstantvalue_{m+2}=positivesum^{2}\\,constantvalue_{m}+constantvalue_{m+1}.\n\\]\nIn particular, it follows from the inductive definition that\n$constantvalue_{m}>0$ when $m\\ge 1$, whence the upper–left entry of\n$staticvector^{\\!fractional}$ is non-zero when $fractional\\ge 2$, a contradiction.\n\n\\noindent\n\\textbf{Remark:}\nA variant of this solution can be obtained by diagonalizing the matrix $singledigit$.\n\n\\textbf{Second solution:}\nIf $lastterm,penultimate,firstelement,zerothpart$ are in arithmetic progression, then we may write\n\\[\nlastterm = imaginarypart-3\\,randomgap,\\quad\npenultimate = imaginarypart-randomgap,\\quad\nfirstelement = imaginarypart+randomgap,\\quad\nzerothpart = imaginarypart+3\\,randomgap\n\\]\nfor some $imaginarypart,randomgap$.  If $randomgap=0$, then clearly all powers of $singledigit$ are in $fixedconstant\\,disjointcollection$.  Also, if $imaginarypart=0$, then one easily checks that $singledigit^{3}$ is in $disjointcollection$.\n\nWe now assume $imaginarypart\\,randomgap\\neq 0$, and show that in that case $singledigit$ cannot be in $disjointcollection$.  First, note that the characteristic polynomial of $singledigit$ is\n\\[\nfixedconstant^{2}-2\\,imaginarypart\\,fixedconstant-8\\,randomgap^{2},\n\\]\nand since $singledigit$ is nonsingular (as $randomgap\\neq 0$), this is also the minimal polynomial of $singledigit$ by the Cayley–Hamilton theorem.  By repeatedly using the relation\n\\[\nsingledigit^{2}=2\\,imaginarypart\\,singledigit+8\\,randomgap^{2}\\,zerooperator,\n\\]\nwe see that for each positive integer we have\n\\[\nsingledigit^{fractional}=denominator\\,singledigit+numerator\\,zerooperator\n\\]\nfor unique real constants $denominator,numerator$ (uniqueness follows from the independence of $singledigit$ and $zerooperator$).  Since $singledigit$ is in $disjointcollection$, we see that $singledigit^{fractional}$ lies in $disjointcollection$ only if $numerator=0$.\n\nOn the other hand, we claim that if $fractional>1$, then $imaginarypart\\,denominator>0$ and $numerator>0$ if $fractional$ is even, and $denominator>0$ and $imaginarypart\\,numerator>0$ if $fractional$ is odd (in particular, $numerator$ can never be zero).  The claim is true for $fractional=2$ by the relation $singledigit^{2}=2\\,imaginarypart\\,singledigit+8\\,randomgap^{2}\\,zerooperator$.  Assuming the claim for $fractional$, and multiplying both sides of the relation $singledigit^{fractional}=denominator\\,singledigit+numerator\\,zerooperator$ by $singledigit$, yields\n\\[\nsingledigit^{fractional+1}\n= denominator\\,(2\\,imaginarypart\\,singledigit+8\\,randomgap^{2}\\,zerooperator) + numerator\\,singledigit\n= (2\\,imaginarypart\\,denominator+numerator)\\,singledigit + 8\\,randomgap^{2}\\,denominator\\,zerooperator,\n\\]\nimplying the claim for $fractional+1$.\n\n\\noindent\n\\textbf{Remark:}\n(from \\url{artofproblemsolving.com}, user \\texttt{hoeij})\nOnce one has $numerator = 0$, one can also finish using the relation\n\\[\nsingledigit \\cdot singledigit^{fractional}=singledigit^{fractional}\\cdot singledigit.\n\\]\n>>>"
    },
    "garbled_string": {
      "map": {
        "S": "mvrgoqza",
        "M": "lupqtsni",
        "a": "dixefnba",
        "b": "hrviycso",
        "c": "tewkamzg",
        "d": "plohxqre",
        "k": "yfmndvsc",
        "p_k": "roqlnmbv",
        "p_k-1": "jszpwiqe",
        "p_k+1": "aghcfklt",
        "t_k": "vmjqferu",
        "u_k": "lxikzhab",
        "x": "qpsendjw",
        "D": "kdhvonau",
        "r": "tlswemvz",
        "s": "gicyorpa",
        "A": "opzkyfjd",
        "B": "mwnapslr",
        "alpha": "uqsfczni",
        "beta": "hfrxoadm",
        "gamma": "dbwzlepr",
        "C": "yuxofnie",
        "I": "fjpsezrt"
      },
      "question": "Let $mvrgoqza$ be the set of all $2 \\times 2$ real matrices\n\\[\nlupqtsni = \\begin{pmatrix} dixefnba & hrviycso \\\\\ntewkamzg & plohxqre \\end{pmatrix}\n\\]\nwhose entries $dixefnba,hrviycso,tewkamzg,plohxqre$ (in that order) form an arithmetic progression. Find all matrices $lupqtsni$ in $mvrgoqza$ for which there is some integer $yfmndvsc>1$ such that $lupqtsni^{yfmndvsc}$ is also in $mvrgoqza$.",
      "solution": "\\textbf{First solution:}\nAny element of $mvrgoqza$ can be written as $lupqtsni = uqsfczni opzkyfjd + hfrxoadm mwnapslr$, where $opzkyfjd = \\left( \\begin{smallmatrix} 1 & 1 \\\\ 1 & 1 \\end{smallmatrix} \\right)$, $mwnapslr = \\left( \\begin{smallmatrix} -3 & -1 \\\\ 1 & 3 \\end{smallmatrix} \\right)$, and $uqsfczni,hfrxoadm \\in \\mathbb{R}$. Note that $opzkyfjd^{2} = \\left( \\begin{smallmatrix} 4 & 4 \\\\ 4 & 4 \\end{smallmatrix} \\right)$ and\n$mwnapslr^{3} = \\left( \\begin{smallmatrix} -24 & -8 \\\\ 8 & 24 \\end{smallmatrix} \\right)$ are both in $mvrgoqza$, and so any matrix of the form $uqsfczni opzkyfjd$ or $hfrxoadm mwnapslr$, $uqsfczni,hfrxoadm\\in\\mathbb{R}$, satisfies the given condition.\n\nWe claim that these are also the only matrices in $mvrgoqza$ satisfying the given condition. Indeed, suppose $lupqtsni = uqsfczni opzkyfjd + hfrxoadm mwnapslr$ where $uqsfczni,hfrxoadm \\neq 0$. Let $yuxofnie = \\left( \\begin{smallmatrix} 1 & 1/\\sqrt{2} \\\\ -1 & 1/\\sqrt{2} \\end{smallmatrix} \\right)$ with inverse\n$yuxofnie^{-1} = \\left( \\begin{smallmatrix} 1/2 & -1/2 \\\\ 1/\\sqrt{2} & 1/\\sqrt{2} \\end{smallmatrix} \\right)$. If we define $kdhvonau = yuxofnie^{-1}lupqtsni yuxofnie$, then $kdhvonau = 2uqsfczni \\left( \\begin{smallmatrix} 0 & dbwzlepr \\\\ dbwzlepr & 1 \\end{smallmatrix} \\right)$ where $dbwzlepr = -\\frac{hfrxoadm\\sqrt{2}}{uqsfczni}$. Now suppose that $lupqtsni^{yfmndvsc}$ is in $mvrgoqza$ with $yfmndvsc\\geq 2$. Since $\\left( \\begin{smallmatrix} 1 & -1 \\end{smallmatrix} \\right) opzkyfjd \\left( \\begin{smallmatrix} 1 \\\\ -1 \\end{smallmatrix} \\right) = \\left( \\begin{smallmatrix} 1 & -1 \\end{smallmatrix} \\right) mwnapslr \\left( \\begin{smallmatrix} 1 \\\\ -1 \\end{smallmatrix} \\right) = 0$,\nwe have $\\left( \\begin{smallmatrix} 1 & -1 \\end{smallmatrix} \\right) lupqtsni^{yfmndvsc} \\left( \\begin{smallmatrix} 1 \\\\ -1 \\end{smallmatrix} \\right) = 0$, and so the upper left entry of $yuxofnie^{-1} lupqtsni^{yfmndvsc} yuxofnie = kdhvonau^{yfmndvsc}$ is $0$. On the other hand, from the expression for $kdhvonau$, an easy induction on $yfmndvsc$ shows that\n$kdhvonau^{yfmndvsc} = (2uqsfczni)^{yfmndvsc} \\left( \\begin{smallmatrix} dbwzlepr^2 jszpwiqe & dbwzlepr roqlnmbv \\\\\n dbwzlepr roqlnmbv & aghcfklt \\end{smallmatrix} \\right)$, where $roqlnmbv$ is defined inductively by $p_0 = 0$, $p_1 = 1$, $p_{yfmndvsc+2} = dbwzlepr^2 roqlnmbv + aghcfklt$. In particular, it follows from the inductive definition that $roqlnmbv > 0$ when $yfmndvsc \\geq 1$, whence the upper left entry of $kdhvonau^{yfmndvsc}$ is nonzero when $yfmndvsc \\geq 2$, a contradiction.\n\n\\noindent\n\\textbf{Remark:}\nA variant of this solution can be obtained by diagonalizing the matrix $lupqtsni$.\n\n\\textbf{Second solution:}\nIf $dixefnba,hrviycso,tewkamzg,plohxqre$ are in arithmetic progression, then we may write\n\\[\ndixefnba = tlswemvz-3gicyorpa, \\; hrviycso=tlswemvz-gicyorpa, \\; tewkamzg=tlswemvz+gicyorpa, \\; plohxqre=tlswemvz+3gicyorpa\n\\]\nfor some $tlswemvz,gicyorpa$. If $gicyorpa=0$, then clearly all powers of $lupqtsni$ are in $qpsendjwmvrgoqza$. Also, if $tlswemvz=0$, then one easily checks that $lupqtsni^{3}$ is in $mvrgoqza$.\n\nWe now assume $tlswemvz gicyorpa\\neq 0$, and show that in that case $lupqtsni$ cannot be in $mvrgoqza$. First, note that the characteristic polynomial of $lupqtsni$ is $qpsendjw^2-2tlswemvz qpsendjw-8gicyorpa^2$, and since $lupqtsni$ is nonsingular (as $gicyorpa\\neq 0$), this is also the minimal polynomial of $lupqtsni$ by the Cayley–Hamilton theorem.  By repeatedly using the relation $lupqtsni^{2}=2tlswemvz lupqtsni+8gicyorpa^{2}fjpsezrt$, we see that for each positive integer, we have $lupqtsni^{yfmndvsc} = vmjqferu lupqtsni + lxikzhab fjpsezrt$ for unique real constants $vmjqferu, lxikzhab$ (uniqueness follows from the independence of $lupqtsni$ and $fjpsezrt$).  Since $lupqtsni$ is in $mvrgoqza$, we see that $lupqtsni^{yfmndvsc}$ lies in $mvrgoqza$ only if $lxikzhab=0$.\n\nOn the other hand, we claim that if $yfmndvsc>1$, then $tlswemvz vmjqferu>0$ and $lxikzhab>0$ if $yfmndvsc$ is even, and $vmjqferu>0$ and $tlswemvz lxikzhab>0$ if $yfmndvsc$ is odd (in particular, $lxikzhab$ can never be zero).  The claim is true for $yfmndvsc=2$ by the relation $lupqtsni^{2}=2tlswemvz lupqtsni+8gicyorpa^{2}fjpsezrt$. Assuming the claim for $yfmndvsc$, and multiplying both sides of the relation $lupqtsni^{yfmndvsc} = vmjqferu lupqtsni + lxikzhab fjpsezrt$ by $lupqtsni$, yields\n\\[\nlupqtsni^{yfmndvsc+1} = vmjqferu (2tlswemvz lupqtsni+8gicyorpa^{2}fjpsezrt) + lxikzhab lupqtsni = (2tlswemvz vmjqferu+lxikzhab) lupqtsni + 8gicyorpa^{2}vmjqferu fjpsezrt,\n\\]\nimplying the claim for $yfmndvsc+1$.\n\n\\noindent\n\\textbf{Remark:}\n(from \\url{artofproblemsolving.com}, user \\texttt{hoeij})\nOnce one has $lxikzhab = 0$, one can also finish using the relation $lupqtsni \\cdot lupqtsni^{yfmndvsc} = lupqtsni^{yfmndvsc} \\cdot lupqtsni$.",
      "solution_notes": ""
    },
    "kernel_variant": {
      "question": "Let an integer $n\\ge 3$ be fixed and define  \n\\[\n\\mathcal S_n \\;=\\;\n\\Bigl\\{\nC(r,s):=\\operatorname{circ}\\bigl(r,\\;r+s,\\;r+2s,\\dots ,r+(n-1)s\\bigr)\\;:\\;r,s\\in\\mathbb R\n\\Bigr\\},\n\\]\nthe $2$–dimensional real vector space of all real $n\\times n$ circulant matrices whose first row forms an arithmetic progression.  \n\nDetermine \\emph{all} matrices $M\\in\\mathcal S_n$ for which there exists an integer exponent $k>1$ such that the power $M^{\\,k}$ again lies in $\\mathcal S_n$.\n\n--------------------------------------------------------------------",
      "solution": "Throughout we fix an integer $n\\ge 3$.\n\n\\textbf{1.\\ Algebraic set–up}  \nLet  \n\\[\nP:=(e_{2},e_{3},\\dots ,e_{n},e_{1}),\n\\]\nthe cyclic right–shift permutation matrix.  Any real circulant matrix is a real polynomial in $P$.  \nSet  \n\\[\nU:=\\sum_{j=0}^{n-1}P^{\\,j},\\qquad \nV:=\\sum_{j=0}^{n-1}j\\,P^{\\,j}\\quad(j\\text{ understood modulo }n).\n\\]\nThe first rows of $U$ and $V$ are $(1,1,\\dots ,1)$ and $(0,1,\\dots ,n-1)$, respectively, whence  \n\\[\n\\mathcal S_n \\;=\\;\\{\\,rU+sV:r,s\\in\\mathbb R\\,\\}.\n\\]\nWriting $e:=(1,\\dots ,1)^{\\top}$ we have $U=ee^{\\top}$ and, by direct multiplication,\n\\[\nU^{2}=nU,\\qquad UV=VU=\\frac{n(n-1)}{2}\\,U. \\tag{1}\n\\]\n\n\\textbf{2.\\ Simultaneous diagonalisation}  \nPut $\\omega:=\\mathrm e^{2\\pi\\mathrm i/n}$ and  \n\\[\nF:=(\\omega^{ij})_{0\\le i,j\\le n-1},\\qquad \nF^{-1}=\\frac1n\\overline F^{\\,\\top}.\n\\]\nThen  \n\\[\nF^{-1}P\\,F=\\operatorname{diag}\\bigl(1,\\omega,\\dots ,\\omega^{\\,n-1}\\bigr),\\qquad\nF^{-1}U\\,F=nE_{00}.   \\tag{2}\n\\]\nFor $1\\le m\\le n-1$ set  \n\\[\nS_{m}:=\\sum_{j=0}^{n-1}\\!j\\,\\omega^{jm}\n       =\\frac{n}{\\omega^{m}-1}=-\\frac{n}{1-\\omega^{m}}. \\tag{3}\n\\]\nConsequently  \n\\[\nF^{-1}V\\,F=\\operatorname{diag}\\!\\bigl(\\tfrac{n(n-1)}{2},\\,S_{1},\\dots ,S_{n-1}\\bigr). \\tag{4}\n\\]\n\n\\textbf{3.\\ Eigenvalues of $M=rU+sV$}  \nCombining (2)–(4) we obtain  \n\\[\n\\lambda_{0}=nr+\\tfrac{n(n-1)}{2}\\,s,\\qquad\n\\lambda_{m}=sS_{m}\\quad(1\\le m\\le n-1). \\tag{5}\n\\]\n\n\\textbf{4.\\ A necessary condition for a power lying again in $\\mathcal S_n$}  \nAssume an integer $k>1$ satisfies  \n\\[\nM^{\\,k}=r_{k}U+s_{k}V\\qquad(r_{k},s_{k}\\in\\mathbb R). \\tag{6}\n\\]\nComparing the eigenvalues for $1\\le m\\le n-1$ yields  \n\\[\n(sS_{m})^{k}=s_{k}S_{m}. \\tag{7}\n\\]\nIf $s\\neq 0$ we may divide by $s^{\\,k}S_{m}$ and arrive at  \n\\[\nS_{m}^{\\,k-1}\\;=\\;\\frac{s_{k}}{s^{\\,k}}\\quad(1\\le m\\le n-1). \\tag{8}\n\\]\nHence \\emph{all} numbers $S_{1}^{\\,d},\\dots ,S_{n-1}^{\\,d}$ with $d:=k-1$ must be equal.\n\n\\textbf{5.\\ When can $S_{1}^{\\,d},\\dots ,S_{n-1}^{\\,d}$ coincide?}\n\n\\emph{Lemma 1 (non–coincidence for $n\\ge 4$).}  \nIf $n\\ge 4$ and $d\\ge 1$, the multiset $\\{S_{1}^{\\,d},\\dots ,S_{n-1}^{\\,d}\\}$ contains at least two distinct elements.\n\n\\emph{Proof.}  \nBy (3)\n\\[\n|S_{m}|=\\frac{n}{2\\sin(\\pi m/n)}\\qquad(1\\le m\\le n-1).\n\\]\nFor $n\\ge 4$ we have $\\sin(\\pi/n)\\neq\\sin(2\\pi/n)$ and therefore $|S_{1}|\\neq |S_{2}|$, so $S_{1}^{\\,d}\\neq S_{2}^{\\,d}$ for every $d\\ge 1$.\\hfill$\\square$\n\n\\medskip\n\\emph{Lemma 2 (coincidence for $n=3$).}  \nFor $n=3$ and $d\\ge 1$,\n\\[\nS_{1}^{\\,d}=S_{2}^{\\,d}\n\\quad\\Longleftrightarrow\\quad\nd\\equiv 0\\pmod{6}.\n\\]\n\n\\emph{Proof.}  \nHere $\\omega=\\mathrm e^{2\\pi\\mathrm i/3}$ and $S_{2}=\\overline{S_{1}}$.  From (3)\n\\[\n\\frac{S_{1}}{S_{2}}\n=\\frac{\\omega^{2}-1}{\\omega-1}=1+\\omega\n=\\mathrm e^{\\pi\\mathrm i/3},\n\\]\na primitive sixth root of unity.  Thus\n\\[\nS_{1}^{\\,d}=S_{2}^{\\,d}\\;\\Longleftrightarrow\\;\n\\bigl( \\mathrm e^{\\pi\\mathrm i/3}\\bigr)^{d}=1\n\\;\\Longleftrightarrow\\;\nd\\equiv 0\\pmod{6}.\n\\]\nIn particular $S_{1}^{\\,d}$ is then real (as it equals its own conjugate), completing the proof.\\hfill$\\square$\n\n\\textbf{6.\\ Consequences for the existence of $k$}\n\n$\\bullet$ If $n\\ge 4$ and $s\\neq 0$, Lemma 1 contradicts (8); hence $s=0$.  \nTherefore, for $n\\ge 4$ the only admissible matrices are the constant–row circulants $M=rU$ (and they work for \\emph{all} $k$ by (1)).  \n\n$\\bullet$ Let $n=3$ and $s\\neq 0$.  Lemma 2 shows that (8) can hold only when\n\\[\nk-1\\equiv 0\\pmod{6}\\quad\\Longleftrightarrow\\quad k\\equiv 1\\pmod{6}. \\tag{9}\n\\]\nPut $d:=k-1=6\\ell$ and define  \n\\[\ns_{k}:=s^{\\,k}S_{1}^{\\,d}=s^{\\,k}S_{1}^{\\,k-1}\\in\\mathbb R,\\qquad\nr_{k}:=\\frac{\\lambda_{0}^{\\,k}-3s_{k}}{3}\\in\\mathbb R, \\tag{10}\n\\]\nwhere $\\lambda_{0}$ is given in (5).  \nBecause (5) implies  \n\\[\n3r_{k}+3s_{k}=\\lambda_{0}^{\\,k},\\qquad\ns_{k}S_{1}=(sS_{1})^{k},\\qquad\ns_{k}S_{2}=(sS_{2})^{k},\n\\]\nthe matrices $M^{\\,k}$ and $r_{k}U+s_{k}V$ share all eigenvalues.  As $F$ simultaneously diagonalises $U$ and $V$, this forces  \n\\[\nM^{\\,k}=r_{k}U+s_{k}V\\in\\mathcal S_{3}.\n\\]\nHence for $n=3$ every $k$ satisfying (9) works.\n\n\\textbf{7.\\ Verification of the remaining cases}\n\n$\\bullet$ If $s=0$ then $M=rU$ and (1) gives  \n\\[\nM^{\\,k}=r^{\\,k}n^{\\,k-1}U\\in\\mathcal S_{n}\\qquad\\forall\\,k\\ge 1.\n\\]\n\n$\\bullet$ If $n=3$, $s\\neq 0$ and $k$ obeys (9), Step 6 provides real numbers $r_{k},s_{k}$ with $M^{\\,k}=r_{k}U+s_{k}V\\in\\mathcal S_{3}$.\n\nThe necessity of $s=0$ for $n\\ge 4$ and of $k\\equiv1\\pmod 6$ for $n=3,\\;s\\neq 0$ has already been established, so the classification is complete.\n\n\\textbf{8.\\ Final answer}  \nFor $M=C(r,s)=rU+sV\\in\\mathcal S_{n}$ the existence of an exponent $k>1$ with $M^{\\,k}\\in\\mathcal S_{n}$ is characterised as follows:\n\n(i)  If $n\\ge 4$, such a $k$ exists \\emph{iff} $s=0$\n     (and then every $k\\ge 1$ works).\n\n(ii) If $n=3$, such a $k$ exists \\emph{iff}\n     either $s=0$ (any $k$) or $s\\neq 0$ and  \n     \\[\n       k\\equiv 1\\pmod{6}.\n     \\]\n\nEquivalently  \n\\[\n\\boxed{\n  \\begin{aligned}\n    &n\\ge 4:\\;M\\text{ qualifies }\\Longleftrightarrow s=0;\\\\[2pt]\n    &n=3:\\;M\\text{ qualifies }\\Longleftrightarrow\n        \\bigl(s=0\\bigr)\\ \\text{or}\\\n        \\bigl(s\\neq 0\\text{ and }k\\equiv 1\\pmod{6}\\bigr).\n  \\end{aligned}}\n\\]\n\n--------------------------------------------------------------------",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.838225",
        "was_fixed": false,
        "difficulty_analysis": "• Higher dimension & more variables:\n  The ambient space is now the \\(2\\)-parameter family of all\n  \\(n\\times n\\) circulant matrices with \\(n\\ge3\\) (instead of a single\n  \\(2\\times2\\) family).\n\n• Sophisticated structures:\n  The solution exploits the representation of circulant matrices as\n  polynomials in the cyclic shift, diagonalisation by the discrete\n  Fourier transform, and complex–analytic arguments on roots of unity.\n\n• Deeper theoretical requirements:\n  One must understand how spectra behave under matrix powers,\n  determine when a vector of eigenvalues can again correspond to an\n  arithmetic–progression circulant, and use injectivity properties of\n  power maps on complex lines.\n\n• Multiple interacting concepts:\n  Linear algebra (circulant matrices, rank-1 projectors, commuting\n  operators), algebraic number theory (sums of roots of unity),\n  complex analysis (argument considerations), and combinatorial\n  identities all appear simultaneously.\n\n• Increased proof length & subtlety:\n  Establishing necessity involves a non-trivial spectral equation and a\n  “no–coalescence” lemma for the numbers \\(S_m\\); sufficiency requires\n  controlling powers via rank-one projector relations.  \n  None of these issues arise in the original \\(2\\times2\\) setting,\n  making the enhanced variant markedly harder."
      }
    },
    "original_kernel_variant": {
      "question": "Let an integer $n\\ge 3$ be fixed and define  \n\\[\n\\mathcal S_n \\;=\\;\n\\Bigl\\{\nC(r,s):=\\operatorname{circ}\\bigl(r,\\;r+s,\\;r+2s,\\dots ,r+(n-1)s\\bigr)\\;:\\;r,s\\in\\mathbb R\n\\Bigr\\},\n\\]\nthe $2$–dimensional real vector space of all real $n\\times n$ circulant matrices whose first row forms an arithmetic progression.  \n\nDetermine \\emph{all} matrices $M\\in\\mathcal S_n$ for which there exists an integer exponent $k>1$ such that the power $M^{\\,k}$ again lies in $\\mathcal S_n$.\n\n--------------------------------------------------------------------",
      "solution": "Throughout we fix an integer $n\\ge 3$.\n\n\\textbf{1.\\ Algebraic set–up}  \nLet  \n\\[\nP:=(e_{2},e_{3},\\dots ,e_{n},e_{1}),\n\\]\nthe cyclic right–shift permutation matrix.  Any real circulant matrix is a real polynomial in $P$.  \nSet  \n\\[\nU:=\\sum_{j=0}^{n-1}P^{\\,j},\\qquad \nV:=\\sum_{j=0}^{n-1}j\\,P^{\\,j}\\quad(j\\text{ understood modulo }n).\n\\]\nThe first rows of $U$ and $V$ are $(1,1,\\dots ,1)$ and $(0,1,\\dots ,n-1)$, respectively, whence  \n\\[\n\\mathcal S_n \\;=\\;\\{\\,rU+sV:r,s\\in\\mathbb R\\,\\}.\n\\]\nWriting $e:=(1,\\dots ,1)^{\\top}$ we have $U=ee^{\\top}$ and, by direct multiplication,\n\\[\nU^{2}=nU,\\qquad UV=VU=\\frac{n(n-1)}{2}\\,U. \\tag{1}\n\\]\n\n\\textbf{2.\\ Simultaneous diagonalisation}  \nPut $\\omega:=\\mathrm e^{2\\pi\\mathrm i/n}$ and  \n\\[\nF:=(\\omega^{ij})_{0\\le i,j\\le n-1},\\qquad \nF^{-1}=\\frac1n\\overline F^{\\,\\top}.\n\\]\nThen  \n\\[\nF^{-1}P\\,F=\\operatorname{diag}\\bigl(1,\\omega,\\dots ,\\omega^{\\,n-1}\\bigr),\\qquad\nF^{-1}U\\,F=nE_{00}.   \\tag{2}\n\\]\nFor $1\\le m\\le n-1$ set  \n\\[\nS_{m}:=\\sum_{j=0}^{n-1}\\!j\\,\\omega^{jm}\n       =\\frac{n}{\\omega^{m}-1}=-\\frac{n}{1-\\omega^{m}}. \\tag{3}\n\\]\nConsequently  \n\\[\nF^{-1}V\\,F=\\operatorname{diag}\\!\\bigl(\\tfrac{n(n-1)}{2},\\,S_{1},\\dots ,S_{n-1}\\bigr). \\tag{4}\n\\]\n\n\\textbf{3.\\ Eigenvalues of $M=rU+sV$}  \nCombining (2)–(4) we obtain  \n\\[\n\\lambda_{0}=nr+\\tfrac{n(n-1)}{2}\\,s,\\qquad\n\\lambda_{m}=sS_{m}\\quad(1\\le m\\le n-1). \\tag{5}\n\\]\n\n\\textbf{4.\\ A necessary condition for a power lying again in $\\mathcal S_n$}  \nAssume an integer $k>1$ satisfies  \n\\[\nM^{\\,k}=r_{k}U+s_{k}V\\qquad(r_{k},s_{k}\\in\\mathbb R). \\tag{6}\n\\]\nComparing the eigenvalues for $1\\le m\\le n-1$ yields  \n\\[\n(sS_{m})^{k}=s_{k}S_{m}. \\tag{7}\n\\]\nIf $s\\neq 0$ we may divide by $s^{\\,k}S_{m}$ and arrive at  \n\\[\nS_{m}^{\\,k-1}\\;=\\;\\frac{s_{k}}{s^{\\,k}}\\quad(1\\le m\\le n-1). \\tag{8}\n\\]\nHence \\emph{all} numbers $S_{1}^{\\,d},\\dots ,S_{n-1}^{\\,d}$ with $d:=k-1$ must be equal.\n\n\\textbf{5.\\ When can $S_{1}^{\\,d},\\dots ,S_{n-1}^{\\,d}$ coincide?}\n\n\\emph{Lemma 1 (non–coincidence for $n\\ge 4$).}  \nIf $n\\ge 4$ and $d\\ge 1$, the multiset $\\{S_{1}^{\\,d},\\dots ,S_{n-1}^{\\,d}\\}$ contains at least two distinct elements.\n\n\\emph{Proof.}  \nBy (3)\n\\[\n|S_{m}|=\\frac{n}{2\\sin(\\pi m/n)}\\qquad(1\\le m\\le n-1).\n\\]\nFor $n\\ge 4$ we have $\\sin(\\pi/n)\\neq\\sin(2\\pi/n)$ and therefore $|S_{1}|\\neq |S_{2}|$, so $S_{1}^{\\,d}\\neq S_{2}^{\\,d}$ for every $d\\ge 1$.\\hfill$\\square$\n\n\\medskip\n\\emph{Lemma 2 (coincidence for $n=3$).}  \nFor $n=3$ and $d\\ge 1$,\n\\[\nS_{1}^{\\,d}=S_{2}^{\\,d}\n\\quad\\Longleftrightarrow\\quad\nd\\equiv 0\\pmod{6}.\n\\]\n\n\\emph{Proof.}  \nHere $\\omega=\\mathrm e^{2\\pi\\mathrm i/3}$ and $S_{2}=\\overline{S_{1}}$.  From (3)\n\\[\n\\frac{S_{1}}{S_{2}}\n=\\frac{\\omega^{2}-1}{\\omega-1}=1+\\omega\n=\\mathrm e^{\\pi\\mathrm i/3},\n\\]\na primitive sixth root of unity.  Thus\n\\[\nS_{1}^{\\,d}=S_{2}^{\\,d}\\;\\Longleftrightarrow\\;\n\\bigl( \\mathrm e^{\\pi\\mathrm i/3}\\bigr)^{d}=1\n\\;\\Longleftrightarrow\\;\nd\\equiv 0\\pmod{6}.\n\\]\nIn particular $S_{1}^{\\,d}$ is then real (as it equals its own conjugate), completing the proof.\\hfill$\\square$\n\n\\textbf{6.\\ Consequences for the existence of $k$}\n\n$\\bullet$ If $n\\ge 4$ and $s\\neq 0$, Lemma 1 contradicts (8); hence $s=0$.  \nTherefore, for $n\\ge 4$ the only admissible matrices are the constant–row circulants $M=rU$ (and they work for \\emph{all} $k$ by (1)).  \n\n$\\bullet$ Let $n=3$ and $s\\neq 0$.  Lemma 2 shows that (8) can hold only when\n\\[\nk-1\\equiv 0\\pmod{6}\\quad\\Longleftrightarrow\\quad k\\equiv 1\\pmod{6}. \\tag{9}\n\\]\nPut $d:=k-1=6\\ell$ and define  \n\\[\ns_{k}:=s^{\\,k}S_{1}^{\\,d}=s^{\\,k}S_{1}^{\\,k-1}\\in\\mathbb R,\\qquad\nr_{k}:=\\frac{\\lambda_{0}^{\\,k}-3s_{k}}{3}\\in\\mathbb R, \\tag{10}\n\\]\nwhere $\\lambda_{0}$ is given in (5).  \nBecause (5) implies  \n\\[\n3r_{k}+3s_{k}=\\lambda_{0}^{\\,k},\\qquad\ns_{k}S_{1}=(sS_{1})^{k},\\qquad\ns_{k}S_{2}=(sS_{2})^{k},\n\\]\nthe matrices $M^{\\,k}$ and $r_{k}U+s_{k}V$ share all eigenvalues.  As $F$ simultaneously diagonalises $U$ and $V$, this forces  \n\\[\nM^{\\,k}=r_{k}U+s_{k}V\\in\\mathcal S_{3}.\n\\]\nHence for $n=3$ every $k$ satisfying (9) works.\n\n\\textbf{7.\\ Verification of the remaining cases}\n\n$\\bullet$ If $s=0$ then $M=rU$ and (1) gives  \n\\[\nM^{\\,k}=r^{\\,k}n^{\\,k-1}U\\in\\mathcal S_{n}\\qquad\\forall\\,k\\ge 1.\n\\]\n\n$\\bullet$ If $n=3$, $s\\neq 0$ and $k$ obeys (9), Step 6 provides real numbers $r_{k},s_{k}$ with $M^{\\,k}=r_{k}U+s_{k}V\\in\\mathcal S_{3}$.\n\nThe necessity of $s=0$ for $n\\ge 4$ and of $k\\equiv1\\pmod 6$ for $n=3,\\;s\\neq 0$ has already been established, so the classification is complete.\n\n\\textbf{8.\\ Final answer}  \nFor $M=C(r,s)=rU+sV\\in\\mathcal S_{n}$ the existence of an exponent $k>1$ with $M^{\\,k}\\in\\mathcal S_{n}$ is characterised as follows:\n\n(i)  If $n\\ge 4$, such a $k$ exists \\emph{iff} $s=0$\n     (and then every $k\\ge 1$ works).\n\n(ii) If $n=3$, such a $k$ exists \\emph{iff}\n     either $s=0$ (any $k$) or $s\\neq 0$ and  \n     \\[\n       k\\equiv 1\\pmod{6}.\n     \\]\n\nEquivalently  \n\\[\n\\boxed{\n  \\begin{aligned}\n    &n\\ge 4:\\;M\\text{ qualifies }\\Longleftrightarrow s=0;\\\\[2pt]\n    &n=3:\\;M\\text{ qualifies }\\Longleftrightarrow\n        \\bigl(s=0\\bigr)\\ \\text{or}\\\n        \\bigl(s\\neq 0\\text{ and }k\\equiv 1\\pmod{6}\\bigr).\n  \\end{aligned}}\n\\]\n\n--------------------------------------------------------------------",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.640850",
        "was_fixed": false,
        "difficulty_analysis": "• Higher dimension & more variables:\n  The ambient space is now the \\(2\\)-parameter family of all\n  \\(n\\times n\\) circulant matrices with \\(n\\ge3\\) (instead of a single\n  \\(2\\times2\\) family).\n\n• Sophisticated structures:\n  The solution exploits the representation of circulant matrices as\n  polynomials in the cyclic shift, diagonalisation by the discrete\n  Fourier transform, and complex–analytic arguments on roots of unity.\n\n• Deeper theoretical requirements:\n  One must understand how spectra behave under matrix powers,\n  determine when a vector of eigenvalues can again correspond to an\n  arithmetic–progression circulant, and use injectivity properties of\n  power maps on complex lines.\n\n• Multiple interacting concepts:\n  Linear algebra (circulant matrices, rank-1 projectors, commuting\n  operators), algebraic number theory (sums of roots of unity),\n  complex analysis (argument considerations), and combinatorial\n  identities all appear simultaneously.\n\n• Increased proof length & subtlety:\n  Establishing necessity involves a non-trivial spectral equation and a\n  “no–coalescence” lemma for the numbers \\(S_m\\); sufficiency requires\n  controlling powers via rank-one projector relations.  \n  None of these issues arise in the original \\(2\\times2\\) setting,\n  making the enhanced variant markedly harder."
      }
    }
  },
  "checked": true,
  "problem_type": "proof",
  "iteratively_fixed": true
}