{
  "index": "2006-B-4",
  "type": "COMB",
  "tag": [
    "COMB",
    "ALG"
  ],
  "difficulty": "",
  "question": "Let $Z$ denote the set of points in $\\mathbb{R}^n$ whose coordinates are 0\nor 1. (Thus $Z$ has $2^n$ elements, which are the vertices of a unit\nhypercube in $\\mathbb{R}^n$.) Given a vector subspace $V$\nof $\\mathbb{R}^n$, let $Z(V)$\ndenote the number of members of $Z$ that lie in $V$. Let $k$ be given,\n$0 \\leq k \\leq n$. Find the maximum, over all vector subspaces $V\n\\subseteq \\mathbb{R}^n$ of dimension $k$, of the number of points in\n$V \\cap Z$. [Editorial note: the proposers probably intended to write\n$Z(V)$ instead of\n``the number of points in $V \\cap Z$'', but this changes nothing.]",
  "solution": "The maximum is $2^k$, achieved for instance by the subspace\n\\[\n\\{(x_1, \\dots, x_n) \\in \\mathbb{R}^n: x_1 = \\cdots = x_{n-k} = 0\\}.\n\\]\n\n\\textbf{First solution:}\nMore generally, we show that any affine $k$-dimensional plane in\n$\\mathbb{R}^n$ can contain at most $2^k$ points in $Z$. The proof is by\ninduction on $k+n$; the case $k=n=0$ is clearly true.\n\nSuppose that $V$ is a $k$-plane in $\\mathbb{R}^n$. Denote the\nhyperplanes $\\{x_n = 0\\}$ and $\\{x_n = 1\\}$ by $V_0$ and $V_1$,\nrespectively. If $V\\cap V_0$ and $V\\cap V_1$ are each at most\n$(k-1)$-dimensional, then $V\\cap V_0\\cap Z$ and $V\\cap V_1 \\cap Z$ each\nhave cardinality at most $2^{k-1}$ by the induction assumption, and\nhence $V\\cap Z$ has at most $2^k$ elements. Otherwise, if $V\\cap V_0$ or\n$V\\cap V_1$ is $k$-dimensional, then $V \\subset V_0$ or $V\\subset V_1$;\nnow apply the induction hypothesis on $V$, viewed as a subset of\n$\\mathbb{R}^{n-1}$ by dropping the last coordinate.\n\n\\textbf{Second solution:}\nLet $S$ be a subset of $Z$ contained in a $k$-dimensional subspace of $V$.\nThis is equivalent to asking that any $t_1, \\dots, t_{k+1} \\in S$\nsatisfy a nontrivial linear dependence $c_1 t_1 + \\cdots + c_{k+1} t_{k+1} = 0$\nwith $c_1, \\dots, c_{k+1} \\in \\mathbb{R}$. Since $t_1, \\dots, t_{k+1} \\in\n\\mathbb{Q}^n$, given such a dependence we can always find another one with\n$c_1, \\dots, c_{k+1} \\in \\mathbb{Q}$; then by clearing denominators, we\ncan find one with $c_1, \\dots, c_{k+1} \\in \\mathbb{Z}$ and not all having a\ncommon factor.\n\nLet $\\mathbb{F}_2$ denote the field of two elements, and let\n$\\overline{S} \\subseteq \\mathbb{F}_2^n$ be the reductions modulo 2 of the points of\n$S$. Then any $t_1, \\dots, t_{k+1} \\in \\overline{S}$ satisfy a nontrivial\nlinear dependence, because we can take the dependence from the end of\nthe previous paragraph and reduce modulo 2. Hence $\\overline{S}$ is contained\nin a $k$-dimensional subspace of $\\mathbb{F}_{2^n}$, and the latter has cardinality\nexactly $2^k$. Thus $\\overline{S}$ has at most $2^k$ elements, as does\n$S$.\n\nVariant (suggested by David Savitt): if $\\overline{S}$ contained $k+1$\nlinearly independent elements, the $(k+1) \\times n$ matrix formed by these\nwould have a nonvanishing maximal minor. The lift of that minor back to $\\RR$\nwould also not vanish, so $S$ would contain $k+1$ linearly independent\nelements.\n\n\\textbf{Third solution:} (by Catalin Zara)\nLet $V$ be a $k$-dimensional subspace. Form the matrix whose rows are the elements\nof $V \\cap Z$; by construction, it has row rank at most $k$. It thus also has\ncolumn rank at most $k$; in particular, we can choose $k$ coordinates such that\neach point of $V \\cap Z$ is determined by those $k$ of its coordinates. Since\neach coordinate of a point in $Z$ can only take two values, $V \\cap Z$ can have\nat most $2^k$ elements.\n\n\\textbf{Remark:} The proposers probably did not realize that this problem appeared\nonline about three months before the exam, at\n\\texttt{http://www.artofproblemsolving.com/ Forum/viewtopic.php?t=105991}. (It\nmay very well have also appeared even earlier.)",
  "vars": [
    "Z",
    "V",
    "x_1",
    "x_n",
    "x_n-k",
    "V_0",
    "V_1",
    "S",
    "t_1",
    "t_k+1",
    "c_1",
    "c_k+1"
  ],
  "params": [
    "n",
    "k",
    "F_2",
    "R"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "Z": "vertexset",
        "V": "subspace",
        "x_1": "coordone",
        "x_n": "coordlast",
        "x_n-k": "coordgap",
        "V_0": "planezero",
        "V_1": "planeone",
        "S": "subsetpoints",
        "t_1": "vectorone",
        "t_k+1": "vectorkplus",
        "c_1": "coeffone",
        "c_k+1": "coeffkplus",
        "n": "dimcount",
        "k": "subdim",
        "F_2": "fieldtwo",
        "R": "realnums"
      },
      "question": "Let $vertexset$ denote the set of points in \\mathbb{realnums^{dimcount}} whose coordinates are 0\nor 1. (Thus $vertexset$ has $2^{dimcount}$ elements, which are the vertices of a unit\nhypercube in \\mathbb{realnums^{dimcount}}.) Given a vector subspace $subspace$\nof \\mathbb{realnums^{dimcount}}, let $vertexset(subspace)$\ndenote the number of members of $vertexset$ that lie in $subspace$. Let $subdim$ be given,\n$0 \\leq subdim \\leq dimcount$. Find the maximum, over all vector subspaces $subspace\n\\subseteq \\mathbb{realnums^{dimcount}}$ of dimension $subdim$, of the number of points in\n$subspace \\cap vertexset$. [Editorial note: the proposers probably intended to write\n$vertexset(subspace)$ instead of\n``the number of points in $subspace \\cap vertexset$'', but this changes nothing.]",
      "solution": "The maximum is $2^{subdim}$, achieved for instance by the subspace\n\\[\n\\{(coordone, \\dots, coordlast) \\in \\mathbb{realnums^{dimcount}}: coordone = \\cdots = coordgap = 0\\}.\n\\]\n\n\\textbf{First solution:}\nMore generally, we show that any affine $subdim$-dimensional plane in\n\\mathbb{realnums^{dimcount}} can contain at most $2^{subdim}$ points in $vertexset$. The proof is by\ninduction on $subdim+dimcount$; the case $subdim=dimcount=0$ is clearly true.\n\nSuppose that $subspace$ is a $subdim$-plane in \\mathbb{realnums^{dimcount}}. Denote the\nhyperplanes \\{coordlast = 0\\} and \\{coordlast = 1\\} by $planezero$ and $planeone$,\nrespectively. If $subspace\\cap planezero$ and $subspace\\cap planeone$ are each at most\n$(subdim-1)$-dimensional, then $subspace\\cap planezero\\cap vertexset$ and $subspace\\cap planeone \\cap vertexset$ each\nhave cardinality at most $2^{subdim-1}$ by the induction assumption, and\nhence $subspace\\cap vertexset$ has at most $2^{subdim}$ elements. Otherwise, if $subspace\\cap planezero$ or $subspace\\cap planeone$ is $subdim$-dimensional, then $subspace \\subset planezero$ or $subspace\\subset planeone$;\nnow apply the induction hypothesis on $subspace$, viewed as a subset of\n\\mathbb{realnums^{dimcount-1}} by dropping the last coordinate.\n\n\\textbf{Second solution:}\nLet $subsetpoints$ be a subset of $vertexset$ contained in a $subdim$-dimensional subspace of $subspace$.\nThis is equivalent to asking that any $vectorone, \\dots, vectorkplus \\in subsetpoints$\nsatisfy a nontrivial linear dependence $coeffone vectorone + \\cdots + coeffkplus vectorkplus = 0$\nwith $coeffone, \\dots, coeffkplus \\in \\mathbb{realnums}$. Since $vectorone, \\dots, vectorkplus \\in\n\\mathbb{Q^{dimcount}}$, given such a dependence we can always find another one with\n$coeffone, \\dots, coeffkplus \\in \\mathbb{Q}$; then by clearing denominators, we\ncan find one with $coeffone, \\dots, coeffkplus \\in \\mathbb{Z}$ and not all having a\ncommon factor.\n\nLet \\mathbb{fieldtwo denote the field of two elements, and let\n\\overline{subsetpoints} \\subseteq \\mathbb{fieldtwo^{dimcount}} be the reductions modulo 2 of the points of\n$subsetpoints$. Then any $vectorone, \\dots, vectorkplus \\in \\overline{subsetpoints}$ satisfy a nontrivial\nlinear dependence, because we can take the dependence from the end of\nthe previous paragraph and reduce modulo 2. Hence \\overline{subsetpoints} is contained\nin a $subdim$-dimensional subspace of \\mathbb{fieldtwo_{2^{dimcount}}}, and the latter has cardinality\nexactly $2^{subdim}$. Thus \\overline{subsetpoints} has at most $2^{subdim}$ elements, as does\n$subsetpoints$.\n\nVariant (suggested by David Savitt): if \\overline{subsetpoints} contained $subdim+1$\nlinearly independent elements, the $(subdim+1) \\times dimcount$ matrix formed by these\nwould have a nonvanishing maximal minor. The lift of that minor back to $\\realnums\\realnums$\nwould also not vanish, so $subsetpoints$ would contain $subdim+1$ linearly independent\nelements.\n\n\\textbf{Third solution:} (by Catalin Zara)\nLet $subspace$ be a $subdim$-dimensional subspace. Form the matrix whose rows are the elements\nof $subspace \\cap vertexset$; by construction, it has row rank at most $subdim$. It thus also has\ncolumn rank at most $subdim$; in particular, we can choose $subdim$ coordinates such that\neach point of $subspace \\cap vertexset$ is determined by those $subdim$ of its coordinates. Since\neach coordinate of a point in $vertexset$ can only take two values, $subspace \\cap vertexset$ can have\nat most $2^{subdim}$ elements.\n\n\\textbf{Remark:} The proposers probably did not realize that this problem appeared\nonline about three months before the exam, at\n\\texttt{http://www.artofproblemsolving.com/ Forum/viewtopic.php?t=105991}. (It\nmay very well have also appeared even earlier.)"
    },
    "descriptive_long_confusing": {
      "map": {
        "Z": "radiusval",
        "V": "heightset",
        "x_1": "pivotone",
        "x_n": "pivotend",
        "x_{n-k}": "pivotgap",
        "V_0": "heightzero",
        "V_1": "heightone",
        "S": "lengthbag",
        "t_1": "sampleone",
        "t_{k+1}": "sampleplus",
        "c_1": "coeffone",
        "c_{k+1}": "coeffplus",
        "n": "chipsize",
        "k": "stonepick",
        "F_2": "fieldpair",
        "R": "realzone"
      },
      "question": "Let $radiusval$ denote the set of points in $\\mathbb{realzone}^{chipsize}$ whose coordinates are 0\nor 1. (Thus $radiusval$ has $2^{chipsize}$ elements, which are the vertices of a unit\nhypercube in $\\mathbb{realzone}^{chipsize}$.) Given a vector subspace $heightset$\nof $\\mathbb{realzone}^{chipsize}$, let $radiusval(heightset)$\ndenote the number of members of $radiusval$ that lie in $heightset$. Let $stonepick$ be given,\n$0 \\leq stonepick \\leq chipsize$. Find the maximum, over all vector subspaces $heightset\n\\subseteq \\mathbb{realzone}^{chipsize}$ of dimension $stonepick$, of the number of points in\n$heightset \\cap radiusval$.",
      "solution": "The maximum is $2^{stonepick}$, achieved for instance by the subspace\n\\[\n\\{(pivotone, \\dots, pivotend) \\in \\mathbb{realzone}^{chipsize}: pivotone = \\cdots = pivotgap = 0\\}.\n\\]\n\n\\textbf{First solution:}\nMore generally, we show that any affine $stonepick$-dimensional plane in\n$\\mathbb{realzone}^{chipsize}$ can contain at most $2^{stonepick}$ points in $radiusval$. The proof is by\ninduction on $stonepick+chipsize$; the case $stonepick=chipsize=0$ is clearly true.\n\nSuppose that $heightset$ is a $stonepick$-plane in $\\mathbb{realzone}^{chipsize}$. Denote the\nhyperplanes $\\{pivotend = 0\\}$ and $\\{pivotend = 1\\}$ by $heightzero$ and $heightone$,\nrespectively. If $heightset \\cap heightzero$ and $heightset \\cap heightone$ are each at most\n$(stonepick-1)$-dimensional, then $heightset \\cap heightzero \\cap radiusval$ and $heightset \\cap heightone \\cap radiusval$ each\nhave cardinality at most $2^{stonepick-1}$ by the induction assumption, and\nhence $heightset \\cap radiusval$ has at most $2^{stonepick}$ elements. Otherwise, if $heightset \\cap heightzero$ or\n$heightset \\cap heightone$ is $stonepick$-dimensional, then $heightset \\subset heightzero$ or $heightset \\subset heightone$;\nnow apply the induction hypothesis on $heightset$, viewed as a subset of\n$\\mathbb{realzone}^{chipsize-1}$ by dropping the last coordinate.\n\n\\textbf{Second solution:}\nLet $lengthbag$ be a subset of $radiusval$ contained in a $stonepick$-dimensional subspace of $heightset$.\nThis is equivalent to asking that any $sampleone, \\dots, sampleplus \\in lengthbag$\nsatisfy a nontrivial linear dependence $coeffone\\,sampleone + \\cdots + coeffplus\\,sampleplus = 0$\nwith $coeffone, \\dots, coeffplus \\in \\mathbb{realzone}$. Since $sampleone, \\dots, sampleplus \\in\n\\mathbb{Q}^{chipsize}$, given such a dependence we can always find another one with\n$coeffone, \\dots, coeffplus \\in \\mathbb{Q}$; then by clearing denominators, we\ncan find one with $coeffone, \\dots, coeffplus \\in \\mathbb{Z}$ and not all having a\ncommon factor.\n\nLet $\\mathbb{fieldpair}_2$ denote the field of two elements, and let\n$\\overline{lengthbag} \\subseteq \\mathbb{fieldpair}_2^{chipsize}$ be the reductions modulo 2 of the points of\n$lengthbag$. Then any $sampleone, \\dots, sampleplus \\in \\overline{lengthbag}$ satisfy a nontrivial\nlinear dependence, because we can take the dependence from the end of\nthe previous paragraph and reduce modulo 2. Hence $\\overline{lengthbag}$ is contained\nin a $stonepick$-dimensional subspace of $\\mathbb{fieldpair}_{2^{chipsize}}$, and the latter has cardinality\nexactly $2^{stonepick}$. Thus $\\overline{lengthbag}$ has at most $2^{stonepick}$ elements, as does\n$lengthbag$.\n\nVariant (suggested by David Savitt): if $\\overline{lengthbag}$ contained $stonepick+1$\nlinearly independent elements, the $(stonepick+1) \\times chipsize$ matrix formed by these\nwould have a nonvanishing maximal minor. The lift of that minor back to $\\RR$ would also not vanish, so $lengthbag$ would contain $stonepick+1$ linearly independent\nelements.\n\n\\textbf{Third solution:} (by Catalin Zara)\nLet $heightset$ be a $stonepick$-dimensional subspace. Form the matrix whose rows are the elements\nof $heightset \\cap radiusval$; by construction, it has row rank at most $stonepick$. It thus also has\ncolumn rank at most $stonepick$; in particular, we can choose $stonepick$ coordinates such that\neach point of $heightset \\cap radiusval$ is determined by those $stonepick$ of its coordinates. Since\neach coordinate of a point in $radiusval$ can only take two values, $heightset \\cap radiusval$ can have\nat most $2^{stonepick}$ elements.\n\n\\textbf{Remark:} The proposers probably did not realize that this problem appeared\nonline about three months before the exam, at\n\\texttt{http://www.artofproblemsolving.com/ Forum/viewtopic.php?t=105991}. (It\nmay very well have also appeared even earlier.)"
    },
    "descriptive_long_misleading": {
      "map": {
        "Z": "continuum",
        "V": "superspace",
        "x_1": "aggregateone",
        "x_n": "aggregatemany",
        "x_n-k": "aggregaterange",
        "V_0": "superspacezero",
        "V_1": "superspaceone",
        "S": "universe",
        "t_1": "elementone",
        "t_k+1": "elementplus",
        "c_1": "modifierone",
        "c_k+1": "modifierplus",
        "n": "emptiness",
        "k": "fullscale",
        "F_2": "skyriver",
        "R": "imaginary"
      },
      "question": "Let $continuum$ denote the set of points in \\mathbb{R}^{emptiness}$ whose coordinates are 0\nor 1. (Thus $continuum$ has $2^{emptiness}$ elements, which are the vertices of a unit\nhypercube in $\\mathbb{R}^{emptiness}$.) Given a vector subspace $superspace$\nof $\\mathbb{R}^{emptiness}$, let $continuum(superspace)$\ndenote the number of members of $continuum$ that lie in $superspace$. Let $fullscale$ be given,\n$0 \\leq fullscale \\leq emptiness$. Find the maximum, over all vector subspaces $superspace\n\\subseteq \\mathbb{R}^{emptiness}$ of dimension $fullscale$, of the number of points in\n$superspace \\cap continuum$. [Editorial note: the proposers probably intended to write\n$continuum(superspace)$ instead of\n``the number of points in $superspace \\cap continuum$'', but this changes nothing.]",
      "solution": "The maximum is $2^{fullscale}$, achieved for instance by the subspace\n\\[\n\\{(aggregateone, \\dots, aggregatemany) \\in \\mathbb{R}^{emptiness}: aggregateone = \\cdots = aggregaterange = 0\\}.\n\\]\n\n\\textbf{First solution:}\nMore generally, we show that any affine $fullscale$-dimensional plane in\n$\\mathbb{R}^{emptiness}$ can contain at most $2^{fullscale}$ points in $continuum$. The proof is by\ninduction on $fullscale+emptiness$; the case $fullscale=emptiness=0$ is clearly true.\n\nSuppose that $superspace$ is a $fullscale$-plane in $\\mathbb{R}^{emptiness}$. Denote the\nhyperplanes $\\{aggregatemany = 0\\}$ and $\\{aggregatemany = 1\\}$ by $superspacezero$ and $superspaceone$,\nrespectively. If $superspace\\cap superspacezero$ and $superspace\\cap superspaceone$ are each at most\n$(fullscale-1)$-dimensional, then $superspace\\cap superspacezero\\cap continuum$ and $superspace\\cap superspaceone \\cap continuum$ each\nhave cardinality at most $2^{fullscale-1}$ by the induction assumption, and\nhence $superspace\\cap continuum$ has at most $2^{fullscale}$ elements. Otherwise, if $superspace\\cap superspacezero$ or\n$superspace\\cap superspaceone$ is $fullscale$-dimensional, then $superspace \\subset superspacezero$ or $superspace\\subset superspaceone$;\nnow apply the induction hypothesis on $superspace$, viewed as a subset of\n$\\mathbb{R}^{emptiness-1}$ by dropping the last coordinate.\n\n\\textbf{Second solution:}\nLet $universe$ be a subset of $continuum$ contained in a $fullscale$-dimensional subspace of $superspace$.\nThis is equivalent to asking that any $elementone, \\dots, elementplus \\in universe$\nsatisfy a nontrivial linear dependence $modifierone\\, elementone + \\cdots + modifierplus\\, elementplus = 0$\nwith $modifierone, \\dots, modifierplus \\in \\mathbb{imaginary}$. Since $elementone, \\dots, elementplus \\in\n\\mathbb{Q}^{emptiness}$, given such a dependence we can always find another one with\n$modifierone, \\dots, modifierplus \\in \\mathbb{Q}$; then by clearing denominators, we\ncan find one with $modifierone, \\dots, modifierplus \\in \\mathbb{Z}$ and not all having a\ncommon factor.\n\nLet $\\mathbb{skyriver}_2$ denote the field of two elements, and let\n$\\overline{universe} \\subseteq \\mathbb{skyriver}_2^{emptiness}$ be the reductions modulo 2 of the points of\n$universe$. Then any $elementone, \\dots, elementplus \\in \\overline{universe}$ satisfy a nontrivial\nlinear dependence, because we can take the dependence from the end of\nthe previous paragraph and reduce modulo 2. Hence $\\overline{universe}$ is contained\nin a $fullscale$-dimensional subspace of $\\mathbb{skyriver}_{2^{emptiness}}$, and the latter has cardinality\nexactly $2^{fullscale}$. Thus $\\overline{universe}$ has at most $2^{fullscale}$ elements, as does\n$universe$.\n\nVariant (suggested by David Savitt): if $\\overline{universe}$ contained $fullscale+1$\nlinearly independent elements, the $(fullscale+1) \\times emptiness$ matrix formed by these\nwould have a nonvanishing maximal minor. The lift of that minor back to $\\RR$\nwould also not vanish, so $universe$ would contain $fullscale+1$ linearly independent\nelements.\n\n\\textbf{Third solution:} (by Catalin Zara)\nLet $superspace$ be a $fullscale$-dimensional subspace. Form the matrix whose rows are the elements\nof $superspace \\cap continuum$; by construction, it has row rank at most $fullscale$. It thus also has\ncolumn rank at most $fullscale$; in particular, we can choose $fullscale$ coordinates such that\neach point of $superspace \\cap continuum$ is determined by those $fullscale$ of its coordinates. Since\neach coordinate of a point in $continuum$ can only take two values, $superspace \\cap continuum$ can have\nat most $2^{fullscale}$ elements.\n\n\\textbf{Remark:} The proposers probably did not realize that this problem appeared\nonline about three months before the exam, at\n\\texttt{http://www.artofproblemsolving.com/ Forum/viewtopic.php?t=105991}. (It\nmay very well have also appeared even earlier.)"
    },
    "garbled_string": {
      "map": {
        "Z": "yqzbmnara",
        "V": "zjxfkptui",
        "x_1": "bqvmslnei",
        "x_n": "gjdwuabci",
        "x_n-k": "juzrpytom",
        "V_0": "pkyozlena",
        "V_1": "hfqernubi",
        "S": "kscivolme",
        "t_1": "velsmizop",
        "t_k+1": "ncqdbfria",
        "c_1": "whusdzepa",
        "c_k+1": "rtamvleqo",
        "n": "zlemivqra",
        "k": "udxferlot",
        "F_2": "krivapoms",
        "R": "xvucnsepa"
      },
      "question": "Let $yqzbmnara$ denote the set of points in $\\mathbb{xvucnsepa}^{zlemivqra}$ whose coordinates are 0\nor 1. (Thus $yqzbmnara$ has $2^{zlemivqra}$ elements, which are the vertices of a unit\nhypercube in $\\mathbb{xvucnsepa}^{zlemivqra}$.) Given a vector subspace $zjxfkptui$\nof $\\mathbb{xvucnsepa}^{zlemivqra}$, let $yqzbmnara(zjxfkptui)$\ndenote the number of members of $yqzbmnara$ that lie in $zjxfkptui$. Let $udxferlot$ be given,\n$0 \\leq udxferlot \\leq zlemivqra$. Find the maximum, over all vector subspaces $zjxfkptui\n\\subseteq \\mathbb{xvucnsepa}^{zlemivqra}$ of dimension $udxferlot$, of the number of points in\n$zjxfkptui \\cap yqzbmnara$. [Editorial note: the proposers probably intended to write\n$yqzbmnara(zjxfkptui)$ instead of\n``the number of points in $zjxfkptui \\cap yqzbmnara$'', but this changes nothing.]",
      "solution": "The maximum is $2^{udxferlot}$, achieved for instance by the subspace\n\\[\n\\{(bqvmslnei, \\dots, gjdwuabci) \\in \\mathbb{xvucnsepa}^{zlemivqra}: bqvmslnei = \\cdots = juzrpytom = 0\\}.\n\\]\n\n\\textbf{First solution:}\nMore generally, we show that any affine $udxferlot$-dimensional plane in\n$\\mathbb{xvucnsepa}^{zlemivqra}$ can contain at most $2^{udxferlot}$ points in $yqzbmnara$. The proof is by\ninduction on $udxferlot+zlemivqra$; the case $udxferlot=zlemivqra=0$ is clearly true.\n\nSuppose that $zjxfkptui$ is a $udxferlot$-plane in $\\mathbb{xvucnsepa}^{zlemivqra}$. Denote the\nhyperplanes $\\{gjdwuabci = 0\\}$ and $\\{gjdwuabci = 1\\}$ by $pkyozlena$ and $hfqernubi$,\nrespectively. If $zjxfkptui\\cap pkyozlena$ and $zjxfkptui\\cap hfqernubi$ are each at most\n$(udxferlot-1)$-dimensional, then $zjxfkptui\\cap pkyozlena\\cap yqzbmnara$ and $zjxfkptui\\cap hfqernubi \\cap yqzbmnara$ each\nhave cardinality at most $2^{udxferlot-1}$ by the induction assumption, and\nhence $zjxfkptui\\cap yqzbmnara$ has at most $2^{udxferlot}$ elements. Otherwise, if $zjxfkptui\\cap pkyozlena$ or\n$zjxfkptui\\cap hfqernubi$ is $udxferlot$-dimensional, then $zjxfkptui \\subset pkyozlena$ or $zjxfkptui\\subset hfqernubi$;\nnow apply the induction hypothesis on $zjxfkptui$, viewed as a subset of\n$\\mathbb{xvucnsepa}^{zlemivqra-1}$ by dropping the last coordinate.\n\n\\textbf{Second solution:}\nLet $kscivolme$ be a subset of $yqzbmnara$ contained in a $udxferlot$-dimensional subspace of $zjxfkptui$.\nThis is equivalent to asking that any $velsmizop, \\dots, ncqdbfria \\in kscivolme$\nsatisfy a nontrivial linear dependence $whusdzepa\\, velsmizop + \\cdots + rtamvleqo\\, ncqdbfria = 0$\nwith $whusdzepa, \\dots, rtamvleqo \\in \\mathbb{xvucnsepa}$. Since $velsmizop, \\dots, ncqdbfria \\in\n\\mathbb{Q}^{zlemivqra}$, given such a dependence we can always find another one with\n$whusdzepa, \\dots, rtamvleqo \\in \\mathbb{Q}$; then by clearing denominators, we\ncan find one with $whusdzepa, \\dots, rtamvleqo \\in \\mathbb{Z}$ and not all having a\ncommon factor.\n\nLet $\\mathbb{krivapoms}$ denote the field of two elements, and let\n$\\overline{kscivolme} \\subseteq \\mathbb{krivapoms}^{zlemivqra}$ be the reductions modulo 2 of the points of\n$kscivolme$. Then any $velsmizop, \\dots, ncqdbfria \\in \\overline{kscivolme}$ satisfy a nontrivial\nlinear dependence, because we can take the dependence from the end of\nthe previous paragraph and reduce modulo 2. Hence $\\overline{kscivolme}$ is contained\nin a $udxferlot$-dimensional subspace of $\\mathbb{krivapoms}_{2^{zlemivqra}}$, and the latter has cardinality\nexactly $2^{udxferlot}$. Thus $\\overline{kscivolme}$ has at most $2^{udxferlot}$ elements, as does\n$kscivolme$.\n\nVariant (suggested by David Savitt): if $\\overline{kscivolme}$ contained $udxferlot+1$\nlinearly independent elements, the $(udxferlot+1) \\times zlemivqra$ matrix formed by these\nwould have a nonvanishing maximal minor. The lift of that minor back to $\\RR$\nwould also not vanish, so $kscivolme$ would contain $udxferlot+1$ linearly independent\nelements.\n\n\\textbf{Third solution:} (by Catalin Zara)\nLet $zjxfkptui$ be a $udxferlot$-dimensional subspace. Form the matrix whose rows are the elements\nof $zjxfkptui \\cap yqzbmnara$; by construction, it has row rank at most $udxferlot$. It thus also has\ncolumn rank at most $udxferlot$; in particular, we can choose $udxferlot$ coordinates such that\neach point of $zjxfkptui \\cap yqzbmnara$ is determined by those $udxferlot$ of its coordinates. Since\neach coordinate of a point in $yqzbmnara$ can only take two values, $zjxfkptui \\cap yqzbmnara$ can have\nat most $2^{udxferlot}$ elements.\n\n\\textbf{Remark:} The proposers probably did not realize that this problem appeared\nonline about three months before the exam, at\n\\texttt{http://www.artofproblemsolving.com/ Forum/viewtopic.php?t=105991}. (It\nmay very well have also appeared even earlier.)"
    },
    "kernel_variant": {
      "question": "Let $m\\ge 2$ be a fixed integer and, for every $d\\ge 1$, put \n\\[\n\\mu_{d}:=\\bigl\\{\\,e^{2\\pi i t/d}\\mid t=0,1,\\dots ,d-1\\bigr\\}\\subset\\Bbb C^{\\times}.\n\\]\nFor $n\\ge 1$ define \n\\[\nP_{m}^{\\,n}:=\\underbrace{\\mu_{m}\\times\\mu_{m}\\times\\cdots\\times\\mu_{m}}_{n\\text{ factors}}\n              \\;\\;\\subset\\; \\bigl(\\Bbb C^{\\times}\\bigr)^{n}.\n\\]\n\nGiven a complex vector subspace $V\\le\\Bbb C^{\\,n}$ we set \n\\[\nZ_{m}(V):=\\bigl\\lvert V\\cap P_{m}^{\\,n}\\bigr\\rvert,\n\\qquad \nM(n,k;m):=\\max_{\\substack{V\\le\\Bbb C^{\\,n}\\\\ \\dim_{\\Bbb C}V=k}} Z_{m}(V),\n\\qquad 1\\le k\\le n.\n\\]\n\n1.  Prove the sharp bound\n\\[\n\\boxed{\\;M(n,k;m)=m^{\\,k}\\;}.\n\\tag{$\\ast$}\n\\]\n\n2.  Show that a $k$–plane $V\\le\\Bbb C^{\\,n}$ satisfies $Z_{m}(V)=m^{\\,k}$ if and only if, after a suitable permutation of \\emph{all} $n$ coordinates, $V$ is the graph of a diagonal linear map\n\\[\nL:\\Bbb C^{\\,k}\\longrightarrow\\Bbb C^{\\,n-k},\n\\qquad\nL(z_{1},\\dots ,z_{k})=\n      \\bigl(\\lambda_{k+1}\\,z_{\\sigma(k+1)},\\dots ,\n            \\lambda_{n}\\,z_{\\sigma(n)}\\bigr),\n\\tag{$\\dagger$}\n\\]\nwhere  \n\n(a) $\\sigma:\\{k+1,\\dots ,n\\}\\longrightarrow\\{1,\\dots ,k\\}$ is an arbitrary map;  \n\n(b) each coefficient $\\lambda_{j}$ lies in $\\mu_{m}$.\n\n3.  Construct explicitly at least one extremal $k$–plane $V$ (that is, a plane with $Z_{m}(V)=m^{\\,k}$) and describe the complete family of all such extremal planes.\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "solution": "Fix integers $n\\ge 1$, $1\\le k\\le n$ and $m\\ge 2$.  Write\n\\[\nP:=P_{m}^{\\,n}=\\mu_{m}^{\\,n}.\n\\]\n\n--------------------------------------------------------------------\n1.\\;The upper bound $Z_{m}(V)\\le m^{\\,k}$.\n\nLet $V\\le\\Bbb C^{\\,n}$ be a $k$–dimensional subspace.  \nChoose a $\\Bbb C$–basis $\\mathbf v_{1},\\dots ,\\mathbf v_{k}$ of $V$ and form the $n\\times k$ matrix\n\\[\nM:=\\bigl(\\mathbf v_{1}\\;\\mathbf v_{2}\\;\\cdots\\;\\mathbf v_{k}\\bigr),\n\\qquad\n\\text{columns }\\mathbf v_{j}.\n\\]\n\nBecause $\\operatorname{rank}M=k$, the matrix contains a $k\\times k$ minor of full rank.  \nAfter\n\n(i) a simultaneous permutation of the \\emph{rows} of $M$ (i.e.\\ of the ambient coordinates),  \n\n(ii) an invertible linear recombination of the \\emph{columns} (i.e.\\ replacing the basis of $V$ by another basis obtained through an element of $\\operatorname{GL}_{k}(\\Bbb C)$),\n\nwe may and shall assume that\n\\[\nM=\\begin{pmatrix}I_{k}\\\\[2pt] A\\end{pmatrix},\n\\qquad\nA\\in\\Bbb C^{\\,(n-k)\\times k},\n\\tag{1}\n\\]\nwhere $I_{k}$ denotes the $k\\times k$ identity matrix.\n\nThus the projection\n\\[\n\\pi:\\Bbb C^{\\,n}\\longrightarrow\\Bbb C^{\\,k},\n\\qquad\n\\pi(z_{1},\\dots ,z_{n})=(z_{1},\\dots ,z_{k}),\n\\]\nrestricts to an \\emph{isomorphism} $\\pi|_{V}:V\\!\\xrightarrow{\\;\\cong\\;}\\Bbb C^{\\,k}$.  In particular,\n\\[\n\\pi\\text{ is injective on }V.\n\\tag{2}\n\\]\n\nNow let $(z_{1},\\dots ,z_{n})\\in V\\cap P$.  Then $(z_{1},\\dots ,z_{k})=\\pi(z_{1},\\dots ,z_{n})\\in\\mu_{m}^{\\,k}$.  \nBecause of the injectivity in (2), the first $k$ coordinates \\emph{determine the whole point}.  \nSince each of the first $k$ coordinates can take exactly $m$ different values, one has\n\\[\nZ_{m}(V)=\\lvert V\\cap P\\rvert\\le\\lvert\\mu_{m}^{\\,k}\\rvert=m^{k}.\n\\]\nHence $M(n,k;m)\\le m^{k}$.\n\n--------------------------------------------------------------------\n2.\\;A family of planes achieving the bound.\n\nFor arbitrary data\n\\[\n\\sigma:\\{k+1,\\dots ,n\\}\\longrightarrow\\{1,\\dots ,k\\},\n\\qquad\n\\lambda_{j}\\in\\mu_{m}\\quad(j>k),\n\\]\ndefine the linear map\n\\[\nL:\\Bbb C^{\\,k}\\longrightarrow\\Bbb C^{\\,n-k},\n\\qquad\nL(z_{1},\\dots ,z_{k})\n     =\\bigl(\\lambda_{k+1}z_{\\sigma(k+1)},\\dots ,\n            \\lambda_{n}z_{\\sigma(n)}\\bigr)\n\\]\nand the corresponding $k$–plane\n\\[\nV(\\sigma;\\lambda)\n:=\\bigl\\{(z_{1},\\dots ,z_{n})\\in\\Bbb C^{\\,n}\\mid\n        (z_{k+1},\\dots ,z_{n})=L(z_{1},\\dots ,z_{k})\\bigr\\}.\n\\]\n\nBecause every basic coordinate $z_{s}$ with $1\\le s\\le k$ can be chosen freely in $\\mu_{m}$ and the remaining coordinates are determined by $L$, the map\n\\[\n\\mu_{m}^{\\,k}\\longrightarrow V(\\sigma;\\lambda)\\cap P,\n\\qquad\n(z_{1},\\dots ,z_{k})\\longmapsto\n\\bigl(z_{1},\\dots ,z_{k},\n      \\lambda_{k+1}z_{\\sigma(k+1)},\\dots ,\n      \\lambda_{n}z_{\\sigma(n)}\\bigr)\n\\]\nis bijective.  Hence\n\\[\nZ_{m}\\bigl(V(\\sigma;\\lambda)\\bigr)=m^{k},\n\\tag{3}\n\\]\nso these planes are extremal.  Consequently $M(n,k;m)\\ge m^{k}$, and together with the upper bound we obtain\n\\[\nM(n,k;m)=m^{k}.\n\\]\n\n--------------------------------------------------------------------\n3.\\;Characterisation of \\emph{all} extremal planes.\n\nLet $V$ be any $k$–plane with $Z_{m}(V)=m^{k}$.  \nApply the construction of Section~1 to $V$; after a permutation of coordinates (to obtain the block form (1)) we again have the injection (2).\nSince $\\pi$ maps $V\\cap P$ into $\\mu_{m}^{\\,k}$ and both sets have the same size $m^{k}$, $\\pi$ actually provides a \\emph{bijection}\n\\[\n\\pi:V\\cap P\\longrightarrow\\mu_{m}^{\\,k}.\n\\tag{4}\n\\]\n\nBecause $V$ is the graph of some linear map\n\\[\nL=(L_{k+1},\\dots ,L_{n}):\\Bbb C^{\\,k}\\longrightarrow\\Bbb C^{\\,n-k}\n\\]\n(cf.\\ (1)), equality (4) is equivalent to\n\\[\nL\\bigl(\\mu_{m}^{\\,k}\\bigr)\\subset\\mu_{m}^{\\,n-k}.\n\\tag{5}\n\\]\n\nWrite each coordinate of $L$ as\n\\[\nL_{j}(z_{1},\\dots ,z_{k})\n     =a_{j1}z_{1}+a_{j2}z_{2}+\\cdots +a_{jk}z_{k},\n\\qquad\nj=k+1,\\dots ,n.\n\\]\n\nLemma 3.1.  \nLet $m\\ge 2$ and $a_{1},\\dots ,a_{k}\\in\\Bbb C$.  \nThe linear form $F(z_{1},\\dots ,z_{k})=\\sum_{r=1}^{k}a_{r}z_{r}$ satisfies\n\\[\nF\\bigl(\\mu_{m}^{\\,k}\\bigr)\\subset\\mu_{m}\n\\]\nif and only if exactly one coefficient $a_{r}$ is non–zero and that coefficient lies in $\\mu_{m}$.\n\nProof.  \nAssume at least two coefficients are non–zero, say $a_{1},a_{2}\\neq 0$.  \nFix $(z_{2},\\dots ,z_{k})\\in\\mu_{m}^{\\,k-1}$ arbitrarily and set\n$c:=a_{2}z_{2}+\\cdots +a_{k}z_{k}$.  Varying $z_{1}$ through $\\mu_{m}$ yields the set\n\\[\nS:=\\{a_{1}z_{1}+c\\mid z_{1}\\in\\mu_{m}\\},\n\\]\nan $m$–gon centred at $c$ of radius $\\lvert a_{1}\\rvert$.  \nIf $c\\neq 0$, then $S$ is translated off the unit circle; if $c=0$ but $\\lvert a_{1}\\rvert\\neq 1$, then $S$ again misses the unit circle.  \nHence the only viable situation is $c=0$ and $\\lvert a_{1}\\rvert=1$, forcing $a_{2}=\\dots =a_{k}=0$.  \nFinally $\\lvert a_{1}\\rvert=1$ and $F(\\mu_{m}^{\\,k})=\\mu_{m}$ imply $a_{1}\\in\\mu_{m}$.  \nThe converse is immediate. $\\square$\n\nApplying Lemma 3.1 to each $L_{j}$ we obtain\n\\[\nL_{j}(z_{1},\\dots ,z_{k})=\\lambda_{j}\\,z_{\\sigma(j)},\n\\qquad\n\\lambda_{j}\\in\\mu_{m},\n\\]\nfor some index $\\sigma(j)\\in\\{1,\\dots ,k\\}$.  \nThus $V$ is precisely of the form $(\\dagger)$ in the statement.\nConversely, planes of the form $(\\dagger)$ were shown to be extremal in Section~2.  \nTherefore the characterisation is complete.\n\n--------------------------------------------------------------------\n4.\\;Conclusion.\n\nSections 1 and 2 establish the bound\n\\[\nM(n,k;m)=m^{k}.\n\\]\nSections 2 and 3 provide the full description of extremal $k$–planes, and any choice of the data $\\sigma,\\lambda_{k+1},\\dots ,\\lambda_{n}$ produces an explicit maximiser, completing Tasks 2 and 3.\n\n\\hfill$\\square$\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.801984",
        "was_fixed": false,
        "difficulty_analysis": "• Variable alphabet sizes m₁,…,mₙ force the competitor to optimise over unequal factors, not just count powers; the answer depends on an ordered product of the greatest mⱼ, unlike the uniform cases (2ⁿ, 3ⁿ) of the earlier problems.  \n• Proving optimality now requires a non-trivial column-span argument followed by a delicate counting step using cyclic-group homomorphisms; obtaining equality conditions demands analysing when those homomorphisms collapse to constants, giving the rigidity part of the statement.  \n• The problem insists on classifying ALL extremal subspaces, not merely producing one.  This necessitates an extra “if and only if’’ layer beyond the original task.  \n• Multiple interacting ideas are involved: rank considerations from linear algebra, direct-product structure of finite cyclic groups, and optimisation over heterogeneous parameters.  \n• Because the mⱼ are arbitrary, simple pattern recognition from the binary or ternary cases fails; the solver must devise an argument that works in complete generality."
      }
    },
    "original_kernel_variant": {
      "question": "Let $m\\ge 2$ be a fixed integer and, for every $d\\ge 1$, put \n\\[\n\\mu_{d}:=\\bigl\\{\\,e^{2\\pi i t/d}\\mid t=0,1,\\dots ,d-1\\bigr\\}\\subset\\Bbb C^{\\times}.\n\\]\nFor $n\\ge 1$ define \n\\[\nP_{m}^{\\,n}:=\\underbrace{\\mu_{m}\\times\\mu_{m}\\times\\cdots\\times\\mu_{m}}_{n\\text{ factors}}\n              \\;\\;\\subset\\; \\bigl(\\Bbb C^{\\times}\\bigr)^{n}.\n\\]\n\nGiven a complex vector subspace $V\\le\\Bbb C^{\\,n}$ we set \n\\[\nZ_{m}(V):=\\bigl\\lvert V\\cap P_{m}^{\\,n}\\bigr\\rvert,\n\\qquad \nM(n,k;m):=\\max_{\\substack{V\\le\\Bbb C^{\\,n}\\\\ \\dim_{\\Bbb C}V=k}} Z_{m}(V),\n\\qquad 1\\le k\\le n.\n\\]\n\n1.  Prove the sharp bound\n\\[\n\\boxed{\\;M(n,k;m)=m^{\\,k}\\;}.\n\\tag{$\\ast$}\n\\]\n\n2.  Show that a $k$–plane $V\\le\\Bbb C^{\\,n}$ satisfies $Z_{m}(V)=m^{\\,k}$ if and only if, after a suitable permutation of \\emph{all} $n$ coordinates, $V$ is the graph of a diagonal linear map\n\\[\nL:\\Bbb C^{\\,k}\\longrightarrow\\Bbb C^{\\,n-k},\n\\qquad\nL(z_{1},\\dots ,z_{k})=\n      \\bigl(\\lambda_{k+1}\\,z_{\\sigma(k+1)},\\dots ,\n            \\lambda_{n}\\,z_{\\sigma(n)}\\bigr),\n\\tag{$\\dagger$}\n\\]\nwhere  \n\n(a) $\\sigma:\\{k+1,\\dots ,n\\}\\longrightarrow\\{1,\\dots ,k\\}$ is an arbitrary map;  \n\n(b) each coefficient $\\lambda_{j}$ lies in $\\mu_{m}$.\n\n3.  Construct explicitly at least one extremal $k$–plane $V$ (that is, a plane with $Z_{m}(V)=m^{\\,k}$) and describe the complete family of all such extremal planes.\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "solution": "Fix integers $n\\ge 1$, $1\\le k\\le n$ and $m\\ge 2$.  Write\n\\[\nP:=P_{m}^{\\,n}=\\mu_{m}^{\\,n}.\n\\]\n\n--------------------------------------------------------------------\n1.\\;The upper bound $Z_{m}(V)\\le m^{\\,k}$.\n\nLet $V\\le\\Bbb C^{\\,n}$ be a $k$–dimensional subspace.  \nChoose a $\\Bbb C$–basis $\\mathbf v_{1},\\dots ,\\mathbf v_{k}$ of $V$ and form the $n\\times k$ matrix\n\\[\nM:=\\bigl(\\mathbf v_{1}\\;\\mathbf v_{2}\\;\\cdots\\;\\mathbf v_{k}\\bigr),\n\\qquad\n\\text{columns }\\mathbf v_{j}.\n\\]\n\nBecause $\\operatorname{rank}M=k$, the matrix contains a $k\\times k$ minor of full rank.  \nAfter\n\n(i) a simultaneous permutation of the \\emph{rows} of $M$ (i.e.\\ of the ambient coordinates),  \n\n(ii) an invertible linear recombination of the \\emph{columns} (i.e.\\ replacing the basis of $V$ by another basis obtained through an element of $\\operatorname{GL}_{k}(\\Bbb C)$),\n\nwe may and shall assume that\n\\[\nM=\\begin{pmatrix}I_{k}\\\\[2pt] A\\end{pmatrix},\n\\qquad\nA\\in\\Bbb C^{\\,(n-k)\\times k},\n\\tag{1}\n\\]\nwhere $I_{k}$ denotes the $k\\times k$ identity matrix.\n\nThus the projection\n\\[\n\\pi:\\Bbb C^{\\,n}\\longrightarrow\\Bbb C^{\\,k},\n\\qquad\n\\pi(z_{1},\\dots ,z_{n})=(z_{1},\\dots ,z_{k}),\n\\]\nrestricts to an \\emph{isomorphism} $\\pi|_{V}:V\\!\\xrightarrow{\\;\\cong\\;}\\Bbb C^{\\,k}$.  In particular,\n\\[\n\\pi\\text{ is injective on }V.\n\\tag{2}\n\\]\n\nNow let $(z_{1},\\dots ,z_{n})\\in V\\cap P$.  Then $(z_{1},\\dots ,z_{k})=\\pi(z_{1},\\dots ,z_{n})\\in\\mu_{m}^{\\,k}$.  \nBecause of the injectivity in (2), the first $k$ coordinates \\emph{determine the whole point}.  \nSince each of the first $k$ coordinates can take exactly $m$ different values, one has\n\\[\nZ_{m}(V)=\\lvert V\\cap P\\rvert\\le\\lvert\\mu_{m}^{\\,k}\\rvert=m^{k}.\n\\]\nHence $M(n,k;m)\\le m^{k}$.\n\n--------------------------------------------------------------------\n2.\\;A family of planes achieving the bound.\n\nFor arbitrary data\n\\[\n\\sigma:\\{k+1,\\dots ,n\\}\\longrightarrow\\{1,\\dots ,k\\},\n\\qquad\n\\lambda_{j}\\in\\mu_{m}\\quad(j>k),\n\\]\ndefine the linear map\n\\[\nL:\\Bbb C^{\\,k}\\longrightarrow\\Bbb C^{\\,n-k},\n\\qquad\nL(z_{1},\\dots ,z_{k})\n     =\\bigl(\\lambda_{k+1}z_{\\sigma(k+1)},\\dots ,\n            \\lambda_{n}z_{\\sigma(n)}\\bigr)\n\\]\nand the corresponding $k$–plane\n\\[\nV(\\sigma;\\lambda)\n:=\\bigl\\{(z_{1},\\dots ,z_{n})\\in\\Bbb C^{\\,n}\\mid\n        (z_{k+1},\\dots ,z_{n})=L(z_{1},\\dots ,z_{k})\\bigr\\}.\n\\]\n\nBecause every basic coordinate $z_{s}$ with $1\\le s\\le k$ can be chosen freely in $\\mu_{m}$ and the remaining coordinates are determined by $L$, the map\n\\[\n\\mu_{m}^{\\,k}\\longrightarrow V(\\sigma;\\lambda)\\cap P,\n\\qquad\n(z_{1},\\dots ,z_{k})\\longmapsto\n\\bigl(z_{1},\\dots ,z_{k},\n      \\lambda_{k+1}z_{\\sigma(k+1)},\\dots ,\n      \\lambda_{n}z_{\\sigma(n)}\\bigr)\n\\]\nis bijective.  Hence\n\\[\nZ_{m}\\bigl(V(\\sigma;\\lambda)\\bigr)=m^{k},\n\\tag{3}\n\\]\nso these planes are extremal.  Consequently $M(n,k;m)\\ge m^{k}$, and together with the upper bound we obtain\n\\[\nM(n,k;m)=m^{k}.\n\\]\n\n--------------------------------------------------------------------\n3.\\;Characterisation of \\emph{all} extremal planes.\n\nLet $V$ be any $k$–plane with $Z_{m}(V)=m^{k}$.  \nApply the construction of Section~1 to $V$; after a permutation of coordinates (to obtain the block form (1)) we again have the injection (2).\nSince $\\pi$ maps $V\\cap P$ into $\\mu_{m}^{\\,k}$ and both sets have the same size $m^{k}$, $\\pi$ actually provides a \\emph{bijection}\n\\[\n\\pi:V\\cap P\\longrightarrow\\mu_{m}^{\\,k}.\n\\tag{4}\n\\]\n\nBecause $V$ is the graph of some linear map\n\\[\nL=(L_{k+1},\\dots ,L_{n}):\\Bbb C^{\\,k}\\longrightarrow\\Bbb C^{\\,n-k}\n\\]\n(cf.\\ (1)), equality (4) is equivalent to\n\\[\nL\\bigl(\\mu_{m}^{\\,k}\\bigr)\\subset\\mu_{m}^{\\,n-k}.\n\\tag{5}\n\\]\n\nWrite each coordinate of $L$ as\n\\[\nL_{j}(z_{1},\\dots ,z_{k})\n     =a_{j1}z_{1}+a_{j2}z_{2}+\\cdots +a_{jk}z_{k},\n\\qquad\nj=k+1,\\dots ,n.\n\\]\n\nLemma 3.1.  \nLet $m\\ge 2$ and $a_{1},\\dots ,a_{k}\\in\\Bbb C$.  \nThe linear form $F(z_{1},\\dots ,z_{k})=\\sum_{r=1}^{k}a_{r}z_{r}$ satisfies\n\\[\nF\\bigl(\\mu_{m}^{\\,k}\\bigr)\\subset\\mu_{m}\n\\]\nif and only if exactly one coefficient $a_{r}$ is non–zero and that coefficient lies in $\\mu_{m}$.\n\nProof.  \nAssume at least two coefficients are non–zero, say $a_{1},a_{2}\\neq 0$.  \nFix $(z_{2},\\dots ,z_{k})\\in\\mu_{m}^{\\,k-1}$ arbitrarily and set\n$c:=a_{2}z_{2}+\\cdots +a_{k}z_{k}$.  Varying $z_{1}$ through $\\mu_{m}$ yields the set\n\\[\nS:=\\{a_{1}z_{1}+c\\mid z_{1}\\in\\mu_{m}\\},\n\\]\nan $m$–gon centred at $c$ of radius $\\lvert a_{1}\\rvert$.  \nIf $c\\neq 0$, then $S$ is translated off the unit circle; if $c=0$ but $\\lvert a_{1}\\rvert\\neq 1$, then $S$ again misses the unit circle.  \nHence the only viable situation is $c=0$ and $\\lvert a_{1}\\rvert=1$, forcing $a_{2}=\\dots =a_{k}=0$.  \nFinally $\\lvert a_{1}\\rvert=1$ and $F(\\mu_{m}^{\\,k})=\\mu_{m}$ imply $a_{1}\\in\\mu_{m}$.  \nThe converse is immediate. $\\square$\n\nApplying Lemma 3.1 to each $L_{j}$ we obtain\n\\[\nL_{j}(z_{1},\\dots ,z_{k})=\\lambda_{j}\\,z_{\\sigma(j)},\n\\qquad\n\\lambda_{j}\\in\\mu_{m},\n\\]\nfor some index $\\sigma(j)\\in\\{1,\\dots ,k\\}$.  \nThus $V$ is precisely of the form $(\\dagger)$ in the statement.\nConversely, planes of the form $(\\dagger)$ were shown to be extremal in Section~2.  \nTherefore the characterisation is complete.\n\n--------------------------------------------------------------------\n4.\\;Conclusion.\n\nSections 1 and 2 establish the bound\n\\[\nM(n,k;m)=m^{k}.\n\\]\nSections 2 and 3 provide the full description of extremal $k$–planes, and any choice of the data $\\sigma,\\lambda_{k+1},\\dots ,\\lambda_{n}$ produces an explicit maximiser, completing Tasks 2 and 3.\n\n\\hfill$\\square$\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.612673",
        "was_fixed": false,
        "difficulty_analysis": "• Variable alphabet sizes m₁,…,mₙ force the competitor to optimise over unequal factors, not just count powers; the answer depends on an ordered product of the greatest mⱼ, unlike the uniform cases (2ⁿ, 3ⁿ) of the earlier problems.  \n• Proving optimality now requires a non-trivial column-span argument followed by a delicate counting step using cyclic-group homomorphisms; obtaining equality conditions demands analysing when those homomorphisms collapse to constants, giving the rigidity part of the statement.  \n• The problem insists on classifying ALL extremal subspaces, not merely producing one.  This necessitates an extra “if and only if’’ layer beyond the original task.  \n• Multiple interacting ideas are involved: rank considerations from linear algebra, direct-product structure of finite cyclic groups, and optimisation over heterogeneous parameters.  \n• Because the mⱼ are arbitrary, simple pattern recognition from the binary or ternary cases fails; the solver must devise an argument that works in complete generality."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}