{
  "index": "2019-A-3",
  "type": "ANA",
  "tag": [
    "ANA",
    "ALG"
  ],
  "difficulty": "",
  "question": "Given real numbers $b_0, b_1, \\dots, b_{2019}$ with $b_{2019} \\neq 0$, let $z_1,z_2,\\dots,z_{2019}$ be \nthe roots in the complex plane of the polynomial \n\\[\nP(z) = \\sum_{k=0}^{2019} b_k z^k.\n\\]\nLet $\\mu = (|z_1| + \\cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\\dots,z_{2019}$ to the origin. Determine the largest constant $M$ such that $\\mu \\geq M$ for all choices of $b_0,b_1,\\dots, b_{2019}$ that satisfy\n\\[\n1 \\leq b_0 < b_1 < b_2 < \\cdots < b_{2019} \\leq 2019.\n\\]",
  "solution": "The answer is $M = 2019^{-1/2019}$. For any choices of $b_0,\\ldots,b_{2019}$ as specified, AM-GM gives\n\\[\n\\mu \\geq |z_1\\cdots z_{2019}|^{1/2019} = |b_0/b_{2019}|^{1/2019} \\geq 2019^{-1/2019}.\n\\]\nTo see that this is best possible, consider $b_0,\\ldots,b_{2019}$ given by $b_k = 2019^{k/2019}$ for all $k$. Then \n\\[\nP(z/2019^{1/2019}) = \\sum_{k=0}^{2019} z^k = \\frac{z^{2020}-1}{z-1}\n\\]\nhas all of its roots on the unit circle. It follows that all of the roots of $P(z)$ have modulus $2019^{-1/2019}$, and so $\\mu = 2019^{-1/2019}$ in this case.",
  "vars": [
    "P",
    "k",
    "z",
    "z_1",
    "z_2019",
    "z_k",
    "\\\\mu"
  ],
  "params": [
    "M",
    "b_0",
    "b_1",
    "b_2019",
    "b_k"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "P": "polyfunc",
        "k": "indexk",
        "z": "complexvar",
        "z_1": "rootone",
        "z_2019": "rootlast",
        "z_k": "rootkth",
        "\\mu": "avgdist",
        "M": "maxconst",
        "b_0": "coefzero",
        "b_1": "coefone",
        "b_2019": "coeflast",
        "b_k": "coefkth"
      },
      "question": "Given real numbers $coefzero, coefone, \\dots, coeflast$ with $coeflast \\neq 0$, let $rootone,z_2,\\dots,rootlast$ be \nthe roots in the complex plane of the polynomial \n\\[\npolyfunc(complexvar) = \\sum_{indexk=0}^{2019} coefkth\\, complexvar^{indexk}.\n\\]\nLet $avgdist = (|rootone| + \\cdots + |rootlast|)/2019$ be the average of the distances from $rootone,z_2,\\dots,rootlast$ to the origin. Determine the largest constant $maxconst$ such that $avgdist \\geq maxconst$ for all choices of $coefzero,coefone,\\dots, coeflast$ that satisfy\n\\[\n1 \\leq coefzero < coefone < b_2 < \\cdots < coeflast \\leq 2019.\n\\]",
      "solution": "The answer is $maxconst = 2019^{-1/2019}$. For any choices of $coefzero,\\ldots,coeflast$ as specified, AM-GM gives\n\\[\navgdist \\geq |rootone\\cdots rootlast|^{1/2019} = |coefzero/coeflast|^{1/2019} \\geq 2019^{-1/2019}.\n\\]\nTo see that this is best possible, consider $coefzero,\\ldots,coeflast$ given by $coefkth = 2019^{indexk/2019}$ for all $indexk$. Then \n\\[\npolyfunc\\!\\bigl(complexvar/2019^{1/2019}\\bigr) = \\sum_{indexk=0}^{2019} complexvar^{indexk} = \\frac{complexvar^{2020}-1}{complexvar-1}\n\\]\nhas all of its roots on the unit circle. It follows that all of the roots of $polyfunc(complexvar)$ have modulus $2019^{-1/2019}$, and so $avgdist = 2019^{-1/2019}$ in this case."
    },
    "descriptive_long_confusing": {
      "map": {
        "P": "clocktower",
        "k": "lacemarker",
        "z": "snowglobe",
        "z_1": "papercrane",
        "z_2019": "coppervase",
        "z_k": "amberlight",
        "\\mu": "oceanfever",
        "M": "riverstone",
        "b_0": "dustymelon",
        "b_1": "ivorybeach",
        "b_2019": "silverquill",
        "b_k": "velvetbloom"
      },
      "question": "Given real numbers $dustymelon, ivorybeach, \\dots, silverquill$ with $silverquill \\neq 0$, let $papercrane,z_2,\\dots,coppervase$ be the roots in the complex plane of the polynomial\n\\[\nclocktower(snowglobe) = \\sum_{lacemarker=0}^{2019} velvetbloom\\, snowglobe^{lacemarker}.\n\\]\nLet $oceanfever = (|papercrane| + \\cdots + |coppervase|)/2019$ be the average of the distances from $papercrane,z_2,\\dots,coppervase$ to the origin. Determine the largest constant riverstone such that $oceanfever \\geq riverstone$ for all choices of $dustymelon,ivorybeach,\\dots, silverquill$ that satisfy\n\\[\n1 \\leq dustymelon < ivorybeach < b_2 < \\cdots < silverquill \\leq 2019.\n\\]",
      "solution": "The answer is $riverstone = 2019^{-1/2019}$. For any choices of $dustymelon,\\ldots,silverquill$ as specified, AM-GM gives\n\\[\noceanfever \\geq |papercrane\\cdots coppervase|^{1/2019} = |dustymelon/silverquill|^{1/2019} \\geq 2019^{-1/2019}.\n\\]\nTo see that this is best possible, consider $dustymelon,\\ldots,silverquill$ given by $velvetbloom = 2019^{lacemarker/2019}$ for all $lacemarker$. Then\n\\[\nclocktower\\bigl(snowglobe/2019^{1/2019}\\bigr) = \\sum_{lacemarker=0}^{2019} snowglobe^{lacemarker} = \\frac{snowglobe^{2020}-1}{snowglobe-1}\n\\]\nhas all of its roots on the unit circle. It follows that all of the roots of $clocktower(snowglobe)$ have modulus $2019^{-1/2019}$, and so $oceanfever = 2019^{-1/2019}$ in this case."
    },
    "descriptive_long_misleading": {
      "map": {
        "P": "constantpoly",
        "k": "invariable",
        "z": "realitynumber",
        "z_1": "realrootone",
        "z_2019": "realrootmax",
        "z_k": "realrootk",
        "\\\\mu": "extremevalue",
        "M": "tinyconstant",
        "b_0": "lastcoefficient",
        "b_1": "latecoefficient",
        "b_2019": "minorcoefficient",
        "b_k": "unfixedcoefficient"
      },
      "question": "Given real numbers $lastcoefficient, latecoefficient, \\dots, minorcoefficient$ with $minorcoefficient \\neq 0$, let $realrootone,z_2,\\dots,realrootmax$ be \nthe roots in the complex plane of the polynomial \n\\[\nconstantpoly(realitynumber) = \\sum_{invariable=0}^{2019} unfixedcoefficient realitynumber^{invariable}.\n\\]\nLet $extremevalue = (|realrootone| + \\cdots + |realrootmax|)/2019$ be the average of the distances from $realrootone,z_2,\\dots,realrootmax$ to the origin. Determine the largest constant $tinyconstant$ such that $extremevalue \\geq tinyconstant$ for all choices of $lastcoefficient,latecoefficient,\\dots, minorcoefficient$ that satisfy\n\\[\n1 \\leq lastcoefficient < latecoefficient < b_2 < \\cdots < minorcoefficient \\leq 2019.\n\\]",
      "solution": "The answer is $tinyconstant = 2019^{-1/2019}$. For any choices of $lastcoefficient,\\ldots,minorcoefficient$ as specified, AM-GM gives\n\\[\nextremevalue \\geq |realrootone\\cdots realrootmax|^{1/2019} = |lastcoefficient/minorcoefficient|^{1/2019} \\geq 2019^{-1/2019}.\n\\]\nTo see that this is best possible, consider $lastcoefficient,\\ldots,minorcoefficient$ given by $unfixedcoefficient = 2019^{invariable/2019}$ for all $invariable$. Then \n\\[\nconstantpoly\\left(realitynumber/2019^{1/2019}\\right) = \\sum_{invariable=0}^{2019} realitynumber^{invariable} = \\frac{realitynumber^{2020}-1}{realitynumber-1}\n\\]\nhas all of its roots on the unit circle. It follows that all of the roots of $constantpoly(realitynumber)$ have modulus $2019^{-1/2019}$, and so $extremevalue = 2019^{-1/2019}$ in this case."
    },
    "garbled_string": {
      "map": {
        "P": "qzxwvtnp",
        "k": "hjgrksla",
        "z": "pxoriflm",
        "z_1": "mcdhsgal",
        "z_2019": "yajvfncq",
        "z_k": "olwsnxzu",
        "\\mu": "tewqbzhn",
        "M": "pfjsoerw",
        "b_0": "qcivgnlm",
        "b_1": "rpkhqzad",
        "b_2019": "tmlghowe",
        "b_k": "vnusbqea"
      },
      "question": "Given real numbers $qcivgnlm, rpkhqzad, \\dots, tmlghowe$ with $tmlghowe \\neq 0$, let $mcdhsgal,z_2,\\dots,yajvfncq$ be \nthe roots in the complex plane of the polynomial \n\\[\nqzxwvtnp(pxoriflm) = \\sum_{hjgrksla=0}^{2019} vnusbqea pxoriflm^{hjgrksla}.\n\\]\nLet $tewqbzhn = (|mcdhsgal| + \\cdots + |yajvfncq|)/2019$ be the average of the distances from $mcdhsgal,z_2,\\dots,yajvfncq$ to the origin. Determine the largest constant $pfjsoerw$ such that $tewqbzhn \\geq pfjsoerw$ for all choices of $qcivgnlm,rpkhqzad,\\dots, tmlghowe$ that satisfy\n\\[\n1 \\leq qcivgnlm < rpkhqzad < b_2 < \\cdots < tmlghowe \\leq 2019.\n\\]",
      "solution": "The answer is $pfjsoerw = 2019^{-1/2019}$. For any choices of $qcivgnlm,\\ldots,tmlghowe$ as specified, AM-GM gives\n\\[\ntewqbzhn \\geq |mcdhsgal\\cdots yajvfncq|^{1/2019} = |qcivgnlm/tmlghowe|^{1/2019} \\geq 2019^{-1/2019}.\n\\]\nTo see that this is best possible, consider $qcivgnlm,\\ldots,tmlghowe$ given by $vnusbqea = 2019^{hjgrksla/2019}$ for all $hjgrksla$. Then \n\\[\nqzxwvtnp(pxoriflm/2019^{1/2019}) = \\sum_{hjgrksla=0}^{2019} pxoriflm^{hjgrksla} = \\frac{pxoriflm^{2020}-1}{pxoriflm-1}\n\\]\nhas all of its roots on the unit circle. It follows that all of the roots of $qzxwvtnp(pxoriflm)$ have modulus $2019^{-1/2019}$, and so $tewqbzhn = 2019^{-1/2019}$ in this case."
    },
    "kernel_variant": {
      "question": "Fix an integer $N\\ge 2$ and a real parameter $\\beta>0$.  \nLet  \n\\[\nQ(z)=\\sum_{k=0}^{N}c_k z^{k},\\qquad c_N\\ne 0 ,\n\\]\nbe a complex polynomial whose \\emph{real, positive} coefficients satisfy  \n\n(I) (upper–lower strip) $3\\le c_0\\le c_1\\le\\cdots\\le c_N\\le 3N$;  \n\n(II) (log–convexity)  \n\\[\n\\frac{c_{k+1}}{c_k}\\le\\frac{c_{k+2}}{c_{k+1}},\\qquad 0\\le k\\le N-2;\n\\]\n\n(III) (prescribed logarithmic mean)  \n\\[\n\\frac{1}{N+1}\\sum_{k=0}^{N}\\ln c_k=\\Lambda,\\qquad \n\\ln(3\\sqrt N)\\le \\Lambda\\le \\ln(3N).\n\\]\n\nDenote the (complex) roots of $Q$ by $w_1,\\dots ,w_N$ and introduce the\norder–$\\beta$ power mean of their moduli\n\\[\n\\mu_\\beta=\\Bigl(\\tfrac{|w_1|^\\beta+\\dots+|w_N|^\\beta}{N}\\Bigr)^{1/\\beta},\n\\qquad \\beta>0.\n\\]\n\n(a)  Determine, in closed form, the largest constant\n\\[\nM_{N,\\beta}=M_{N,\\beta}(3,3N,\\Lambda)\n\\]\nsuch that\n\\[\n\\mu_\\beta\\ge M_{N,\\beta}\n\\]\nfor \\emph{every} polynomial whose coefficients obey\n\\textup{(I)}, \\textup{(II)} and \\textup{(III)}.\n\n(b)  For all admissible triples $(N,\\beta,\\Lambda)$ describe, \\emph{up\nto permutations of the roots}, all coefficient families for which the\nequality $\\mu_\\beta=M_{N,\\beta}$ is attained.",
      "solution": "Throughout write\n\\[\nr_j:=|w_j|,\\qquad\nG:=(r_1r_2\\cdots r_N)^{1/N}.\n\\]\n\n\\textbf{1.  Geometric mean of the root moduli}  \nFactoring $Q$ yields\n\\[\nQ(z)=c_N\\prod_{j=1}^{N}(z-w_j),\\qquad \n(-1)^N \\frac{c_0}{c_N}=w_1\\cdots w_N.\n\\]\nHence\n\\[\nG=\\Bigl|\\frac{c_0}{c_N}\\Bigr|^{1/N}.\n\\tag{1}\n\\]\n\n\\textbf{2.  Power–mean inequality}  \nFor every $\\beta>0$ and every positive real numbers\n$r_1,\\dots ,r_N$,\n\\[\n\\mu_\\beta\\ge G\n\\tag{2}\n\\]\n(the classical power–mean hierarchy).  Consequently the universal\nconstant $M_{N,\\beta}$ is determined by the \\emph{minimum} of the ratio\n$c_0/c_N$ under (I)–(III).\n\n\\textbf{3.  Logarithmic variables}  \nPut $x_k:=\\ln c_k\\;(0\\le k\\le N)$ and denote forward differences by\n$d_k:=x_{k+1}-x_k$.  Then  \n\n(a) $\\ln 3\\le x_0\\le x_1\\le\\cdots\\le x_N\\le\\ln(3N)$;\n\n(b) $0\\le d_0\\le d_1\\le\\cdots\\le d_{N-1}$ (discrete convexity);\n\n(c) $\\tfrac{1}{N+1}\\sum_{k=0}^{N}x_k=\\Lambda$.\n\nWe must minimise\n\\[\n\\Phi(x_0,\\dots ,x_N):=x_0-x_N\n\\tag{3}\n\\]\nsubject to (a)–(c).\n\n\\textbf{4.  A discrete convexity lemma}\n\nLemma 1.  \nFor every discrete convex sequence\n$(x_k)_{k=0}^{N}$ one has\n\\[\nx_0\\;\\le\\;\\frac1{N+1}\\sum_{k=0}^{N}x_k\n\\;\\le\\;\\frac{x_0+x_N}{2}.\n\\tag{4}\n\\]\n\n\\emph{Proof.}  \nSet $s_k:=x_k-x_0\\;(0\\le k\\le N)$.  Then $s_0=0$, $s_k\\ge 0$ and the\ndifferences $s_{k+1}-s_k=d_k$ form a non–decreasing sequence.  Hence\n$s_k\\le\\frac{k}{N}\\,s_N$ for all $k$, so\n\\[\n\\sum_{k=0}^{N}s_k\\le\\sum_{k=0}^{N}\\frac{k}{N}s_N\n=\\frac{N}{2}s_N.\n\\]\nDividing by $N+1$ and restoring the $x$–notation gives the right–hand\nside of (4).  The left–hand side is obvious.  $\\square$\n\n\\textbf{5.  Extremal sequences are affine}\n\nLemma 2 (rigidity).  \nIf an admissible sequence attains equality\n$\\tfrac{1}{N+1}\\sum_{k=0}^{N}x_k=\\tfrac{1}{2}(x_0+x_N)$ in (4), then\n\\[\nx_k=x_0+\\frac{k}{N}(x_N-x_0),\\qquad 0\\le k\\le N.\n\\tag{5}\n\\]\n\n\\emph{Proof.}  \nDefine $y_k:=x_0+\\frac{k}{N}(x_N-x_0)$.  The discrete convexity of\n$(x_k)$ implies $x_k\\le y_k$ for $0\\le k\\le N$, while the two sequences\nshare the same endpoints and the same arithmetic mean.  Hence their\ntermwise difference is a non–negative sequence with vanishing sum,\nforcing $x_k=y_k$ for every $k$.  $\\square$\n\n\\textbf{6.  Optimisation of the endpoints}  \nLemma 1 implies\n\\[\nx_0+x_N\\ge 2\\Lambda\\ge 2\\ln(3\\sqrt N)=\\ln(9N).\n\\tag{6}\n\\]\nPut $S:=x_0+x_N$ and $D:=x_N-x_0$.  Conditions (a) are rewritten as\n\\[\n2\\ln 3\\le S-D,\\quad S+D\\le 2\\ln(3N).\n\\tag{7}\n\\]\nFor fixed $S$ the maximal admissible $D$ equals\n\\[\nD_{\\max}(S)=\\min\\bigl\\{\\,S-2\\ln 3,\\;2\\ln(3N)-S\\,\\bigr\\}.\n\\]\nThe two arguments of the minimum are increasing and decreasing in $S$,\nrespectively, and they are equal at\n\\[\nS_0:=\\ln(9N).\n\\]\nBecause (6) shows $S\\ge 2\\Lambda\\ge S_0$, the feasible interval for $S$\nis $[\\,2\\Lambda,\\,2\\ln(3N)\\,]$, entirely to the \\emph{right} of $S_0$.\nHence $D_{\\max}(S)$ is decreasing in that interval and attains its\nmaximum at $S=2\\Lambda$.  Therefore\n\\[\nD_{\\max}=2\\bigl(\\ln(3N)-\\Lambda\\bigr).\n\\tag{8}\n\\]\n\n\\textbf{7.  The minimal coefficient ratio}  \nSince $\\Phi=-D$, we obtain from (8)\n\\[\n\\Bigl(\\frac{c_0}{c_N}\\Bigr)_{\\min}=\n\\exp\\!\\bigl(-D_{\\max}\\bigr)=\n\\exp\\!\\bigl(2\\Lambda-2\\ln(3N)\\bigr).\n\\tag{9}\n\\]\nSubstituting (9) in (2) and (1) gives the desired bound\n\\[\n\\boxed{%\nM_{N,\\beta}(3,3N,\\Lambda)=\n\\exp\\!\\Bigl(\\tfrac{2}{N}\\bigl(\\Lambda-\\ln(3N)\\bigr)\\Bigr)}.\n\\tag{10}\n\\]\n(The value is independent of $\\beta$, although $\\beta>0$ is required to\ndefine $\\mu_\\beta$.)\n\n\\textbf{8.  Sharpness and equality}  \n\n\\emph{Equal root moduli.}  \nEquality in (2) forces\n$r_1=\\dots=r_N=G$, i.e.\\ all roots have the same modulus.\n\n\\emph{Affine logarithmic sequence.}  \nEquality in (4) entails, by Lemma 2, that\n$x_k$ is affine as in (5).  Consequently\n\\[\n\\frac{c_{k+1}}{c_k}=\\rho\\quad(0\\le k\\le N-1),\n\\qquad \n\\rho:=\\exp\\!\\Bigl(\\tfrac{2}{N}\\bigl(\\ln(3N)-\\Lambda\\bigr)\\Bigr)\n\\ge 1.\n\\tag{11}\n\\]\n(The boundary value $\\rho=1$ occurs exactly when\n$\\Lambda=\\ln(3N)$.)\n\n\\emph{Determination of the scale factor.}  \nWrite $c_k=\\gamma\\,\\rho^{\\,k}$ $(0\\le k\\le N)$.  Condition (III) forces\n\\[\n\\Lambda=\\frac{1}{N+1}\\sum_{k=0}^{N}\\ln c_k\n        =\\ln\\gamma+\\frac{N}{2}\\ln\\rho\n\\quad\\Longrightarrow\\quad\n\\boxed{\\;\n\\gamma=\\exp\\!\\bigl(2\\Lambda-\\ln(3N)\\bigr)\n\\;}\n\\tag{12}\n\\]\nso the scale factor is \\emph{unique}.  Note that $\\gamma$ automatically\nsatisfies $3\\le \\gamma\\le 3N$ thanks to the bounds on $\\Lambda$.\n\n\\emph{Root modulus.}  \nEquations (1), (9) and (11) give\n\\[\nr_1=\\dots=r_N=\\rho^{-1}=M_{N,\\beta}.\n\\tag{13}\n\\]\n\n\\emph{Explicit equality polynomials.}  \nBecause all roots have modulus $\\rho^{-1}$ and their product equals\n$(-1)^N c_0/c_N=-\\rho^{-N}$, they must be the $N$ points\n$\\rho^{-1}\\zeta$ where $\\zeta$ ranges over the $(N+1)$st roots of unity\ndifferent from $1$.  Hence\n\\[\nQ(z)=\\gamma\\sum_{k=0}^{N}\\rho^{\\,k}z^{k}\n     =\\gamma\\,\\frac{1-(\\rho z)^{N+1}}{1-\\rho z}.\n\\tag{14}\n\\]\nFor $\\Lambda=\\ln(3N)$ one has $\\rho=1$, $\\gamma=3N$ and (14) reduces to\n$Q(z)=3N\\bigl(1+z+\\dots +z^{N}\\bigr)$, whose roots indeed lie on the\nunit circle; the bound (10) becomes $M_{N,\\beta}=1$.\n\n\\emph{Uniqueness.}  \nConversely, suppose $\\mu_\\beta=M_{N,\\beta}$.  Then equalities occur in\nboth (2) and Lemma&nbsp;1, whence $(x_k)$ is affine and the coefficients\nmust be the geometric progression (11) with \\emph{the unique} initial\nterm (12).  Thus the only admissible coefficient family (up to\npermutation of the roots, which does not change the polynomial) is\n(14).\n\n\\textbf{9.  Answer to (a)–(b)}  \n\n(a)  The largest constant is given by (10).\n\n(b)  Equality holds precisely for the polynomial (14), whose coefficients\nform the geometric progression $c_k=\\gamma\\rho^{\\,k}$ with\n$\\rho$ from (11) and $\\gamma$ from (12); the $N$ roots are\n$\\rho^{-1}\\zeta$, where $\\zeta$ runs through the $(N+1)$st roots of\nunity distinct from $1$.\nNo other admissible polynomial attains the bound.",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.859343",
        "was_fixed": false,
        "difficulty_analysis": "Compared with the original setting, the enhanced variant is harder in\nseveral independent directions.\n\n1.  Additional structural constraints  \n    • The coefficients are required to be **log-convex** (condition II),\n      introducing majorisation/convexity considerations absent from the\n      original problem.  \n    • A **logarithmic mean** of the coefficients is prescribed\n      (condition III), so the optimisation problem is now\n      constrained by a *global* nonlinear condition instead of mere\n      endpoint bounds.\n\n2.  Variable parameter  \n    The root‐mean to be bounded is the\n    $\\beta$–power mean, so the solver has to work simultaneously for a\n    continuum of exponents.  Establishing a bound that is uniform in\n    $\\beta$ forces the use of power–mean theory rather than the single\n    $\\beta=1$ case of the kernel problem.\n\n3.  Deeper optimisation theory  \n    To identify the extremal sequence of coefficients one must invoke\n    Karamata’s theorem, the Hardy–Littlewood–Pólya majorisation\n    principle, or a full Lagrange–multiplier analysis on an\n    $(N+1)$–dimensional simplex subject to convex ordering (II) and the\n    nonlinear moment constraint (III).  None of these tools is touched\n    in the original problem, which relies only on a single application\n    of $\\mathrm{AM}\\ge\\mathrm{GM}$.\n\n4.  Verifying attainability  \n    Showing that the geometric progression really satisfies all three\n    coefficient conditions and yields equality for every $\\beta$\n    requires a careful scaling argument, followed by an explicit\n    localisation of the roots via a Möbius transformation—again a layer\n    of analysis that is unnecessary for the kernel exercise.\n\nAll these ingredients oblige the solver to weave together root‐product\nformulas, power–mean inequalities, majorisation theory and explicit\nfactorisation—making the enhanced variant substantially more technical\nand conceptually deeper than both the original problem and the current\nkernel variant."
      }
    },
    "original_kernel_variant": {
      "question": "Fix an integer $N\\ge 2$ and a real parameter $\\beta>0$.  \nLet  \n\\[\nQ(z)=\\sum_{k=0}^{N}c_k z^{k},\\qquad c_N\\ne 0 ,\n\\]\nbe a complex polynomial whose \\emph{real, positive} coefficients satisfy  \n\n(I) (upper–lower strip) $3\\le c_0\\le c_1\\le\\cdots\\le c_N\\le 3N$;  \n\n(II) (log–convexity)  \n\\[\n\\frac{c_{k+1}}{c_k}\\le\\frac{c_{k+2}}{c_{k+1}},\\qquad 0\\le k\\le N-2;\n\\]\n\n(III) (prescribed logarithmic mean)  \n\\[\n\\frac{1}{N+1}\\sum_{k=0}^{N}\\ln c_k=\\Lambda,\\qquad \n\\ln(3\\sqrt N)\\le \\Lambda\\le \\ln(3N).\n\\]\n\nDenote the (complex) roots of $Q$ by $w_1,\\dots ,w_N$ and introduce the\norder–$\\beta$ power mean of their moduli\n\\[\n\\mu_\\beta=\\Bigl(\\tfrac{|w_1|^\\beta+\\dots+|w_N|^\\beta}{N}\\Bigr)^{1/\\beta},\n\\qquad \\beta>0.\n\\]\n\n(a)  Determine, in closed form, the largest constant\n\\[\nM_{N,\\beta}=M_{N,\\beta}(3,3N,\\Lambda)\n\\]\nsuch that\n\\[\n\\mu_\\beta\\ge M_{N,\\beta}\n\\]\nfor \\emph{every} polynomial whose coefficients obey\n\\textup{(I)}, \\textup{(II)} and \\textup{(III)}.\n\n(b)  For all admissible triples $(N,\\beta,\\Lambda)$ describe, \\emph{up\nto permutations of the roots}, all coefficient families for which the\nequality $\\mu_\\beta=M_{N,\\beta}$ is attained.",
      "solution": "Throughout write\n\\[\nr_j:=|w_j|,\\qquad\nG:=(r_1r_2\\cdots r_N)^{1/N}.\n\\]\n\n\\textbf{1.  Geometric mean of the root moduli}  \nFactoring $Q$ yields\n\\[\nQ(z)=c_N\\prod_{j=1}^{N}(z-w_j),\\qquad \n(-1)^N \\frac{c_0}{c_N}=w_1\\cdots w_N.\n\\]\nHence\n\\[\nG=\\Bigl|\\frac{c_0}{c_N}\\Bigr|^{1/N}.\n\\tag{1}\n\\]\n\n\\textbf{2.  Power–mean inequality}  \nFor every $\\beta>0$ and every positive real numbers\n$r_1,\\dots ,r_N$,\n\\[\n\\mu_\\beta\\ge G\n\\tag{2}\n\\]\n(the classical power–mean hierarchy).  Consequently the universal\nconstant $M_{N,\\beta}$ is determined by the \\emph{minimum} of the ratio\n$c_0/c_N$ under (I)–(III).\n\n\\textbf{3.  Logarithmic variables}  \nPut $x_k:=\\ln c_k\\;(0\\le k\\le N)$ and denote forward differences by\n$d_k:=x_{k+1}-x_k$.  Then  \n\n(a) $\\ln 3\\le x_0\\le x_1\\le\\cdots\\le x_N\\le\\ln(3N)$;\n\n(b) $0\\le d_0\\le d_1\\le\\cdots\\le d_{N-1}$ (discrete convexity);\n\n(c) $\\tfrac{1}{N+1}\\sum_{k=0}^{N}x_k=\\Lambda$.\n\nWe must minimise\n\\[\n\\Phi(x_0,\\dots ,x_N):=x_0-x_N\n\\tag{3}\n\\]\nsubject to (a)–(c).\n\n\\textbf{4.  A discrete convexity lemma}\n\nLemma 1.  \nFor every discrete convex sequence\n$(x_k)_{k=0}^{N}$ one has\n\\[\nx_0\\;\\le\\;\\frac1{N+1}\\sum_{k=0}^{N}x_k\n\\;\\le\\;\\frac{x_0+x_N}{2}.\n\\tag{4}\n\\]\n\n\\emph{Proof.}  \nSet $s_k:=x_k-x_0\\;(0\\le k\\le N)$.  Then $s_0=0$, $s_k\\ge 0$ and the\ndifferences $s_{k+1}-s_k=d_k$ form a non–decreasing sequence.  Hence\n$s_k\\le\\frac{k}{N}\\,s_N$ for all $k$, so\n\\[\n\\sum_{k=0}^{N}s_k\\le\\sum_{k=0}^{N}\\frac{k}{N}s_N\n=\\frac{N}{2}s_N.\n\\]\nDividing by $N+1$ and restoring the $x$–notation gives the right–hand\nside of (4).  The left–hand side is obvious.  $\\square$\n\n\\textbf{5.  Extremal sequences are affine}\n\nLemma 2 (rigidity).  \nIf an admissible sequence attains equality\n$\\tfrac{1}{N+1}\\sum_{k=0}^{N}x_k=\\tfrac{1}{2}(x_0+x_N)$ in (4), then\n\\[\nx_k=x_0+\\frac{k}{N}(x_N-x_0),\\qquad 0\\le k\\le N.\n\\tag{5}\n\\]\n\n\\emph{Proof.}  \nDefine $y_k:=x_0+\\frac{k}{N}(x_N-x_0)$.  The discrete convexity of\n$(x_k)$ implies $x_k\\le y_k$ for $0\\le k\\le N$, while the two sequences\nshare the same endpoints and the same arithmetic mean.  Hence their\ntermwise difference is a non–negative sequence with vanishing sum,\nforcing $x_k=y_k$ for every $k$.  $\\square$\n\n\\textbf{6.  Optimisation of the endpoints}  \nLemma 1 implies\n\\[\nx_0+x_N\\ge 2\\Lambda\\ge 2\\ln(3\\sqrt N)=\\ln(9N).\n\\tag{6}\n\\]\nPut $S:=x_0+x_N$ and $D:=x_N-x_0$.  Conditions (a) are rewritten as\n\\[\n2\\ln 3\\le S-D,\\quad S+D\\le 2\\ln(3N).\n\\tag{7}\n\\]\nFor fixed $S$ the maximal admissible $D$ equals\n\\[\nD_{\\max}(S)=\\min\\bigl\\{\\,S-2\\ln 3,\\;2\\ln(3N)-S\\,\\bigr\\}.\n\\]\nThe two arguments of the minimum are increasing and decreasing in $S$,\nrespectively, and they are equal at\n\\[\nS_0:=\\ln(9N).\n\\]\nBecause (6) shows $S\\ge 2\\Lambda\\ge S_0$, the feasible interval for $S$\nis $[\\,2\\Lambda,\\,2\\ln(3N)\\,]$, entirely to the \\emph{right} of $S_0$.\nHence $D_{\\max}(S)$ is decreasing in that interval and attains its\nmaximum at $S=2\\Lambda$.  Therefore\n\\[\nD_{\\max}=2\\bigl(\\ln(3N)-\\Lambda\\bigr).\n\\tag{8}\n\\]\n\n\\textbf{7.  The minimal coefficient ratio}  \nSince $\\Phi=-D$, we obtain from (8)\n\\[\n\\Bigl(\\frac{c_0}{c_N}\\Bigr)_{\\min}=\n\\exp\\!\\bigl(-D_{\\max}\\bigr)=\n\\exp\\!\\bigl(2\\Lambda-2\\ln(3N)\\bigr).\n\\tag{9}\n\\]\nSubstituting (9) in (2) and (1) gives the desired bound\n\\[\n\\boxed{%\nM_{N,\\beta}(3,3N,\\Lambda)=\n\\exp\\!\\Bigl(\\tfrac{2}{N}\\bigl(\\Lambda-\\ln(3N)\\bigr)\\Bigr)}.\n\\tag{10}\n\\]\n(The value is independent of $\\beta$, although $\\beta>0$ is required to\ndefine $\\mu_\\beta$.)\n\n\\textbf{8.  Sharpness and equality}  \n\n\\emph{Equal root moduli.}  \nEquality in (2) forces\n$r_1=\\dots=r_N=G$, i.e.\\ all roots have the same modulus.\n\n\\emph{Affine logarithmic sequence.}  \nEquality in (4) entails, by Lemma 2, that\n$x_k$ is affine as in (5).  Consequently\n\\[\n\\frac{c_{k+1}}{c_k}=\\rho\\quad(0\\le k\\le N-1),\n\\qquad \n\\rho:=\\exp\\!\\Bigl(\\tfrac{2}{N}\\bigl(\\ln(3N)-\\Lambda\\bigr)\\Bigr)\n\\ge 1.\n\\tag{11}\n\\]\n(The boundary value $\\rho=1$ occurs exactly when\n$\\Lambda=\\ln(3N)$.)\n\n\\emph{Determination of the scale factor.}  \nWrite $c_k=\\gamma\\,\\rho^{\\,k}$ $(0\\le k\\le N)$.  Condition (III) forces\n\\[\n\\Lambda=\\frac{1}{N+1}\\sum_{k=0}^{N}\\ln c_k\n        =\\ln\\gamma+\\frac{N}{2}\\ln\\rho\n\\quad\\Longrightarrow\\quad\n\\boxed{\\;\n\\gamma=\\exp\\!\\bigl(2\\Lambda-\\ln(3N)\\bigr)\n\\;}\n\\tag{12}\n\\]\nso the scale factor is \\emph{unique}.  Note that $\\gamma$ automatically\nsatisfies $3\\le \\gamma\\le 3N$ thanks to the bounds on $\\Lambda$.\n\n\\emph{Root modulus.}  \nEquations (1), (9) and (11) give\n\\[\nr_1=\\dots=r_N=\\rho^{-1}=M_{N,\\beta}.\n\\tag{13}\n\\]\n\n\\emph{Explicit equality polynomials.}  \nBecause all roots have modulus $\\rho^{-1}$ and their product equals\n$(-1)^N c_0/c_N=-\\rho^{-N}$, they must be the $N$ points\n$\\rho^{-1}\\zeta$ where $\\zeta$ ranges over the $(N+1)$st roots of unity\ndifferent from $1$.  Hence\n\\[\nQ(z)=\\gamma\\sum_{k=0}^{N}\\rho^{\\,k}z^{k}\n     =\\gamma\\,\\frac{1-(\\rho z)^{N+1}}{1-\\rho z}.\n\\tag{14}\n\\]\nFor $\\Lambda=\\ln(3N)$ one has $\\rho=1$, $\\gamma=3N$ and (14) reduces to\n$Q(z)=3N\\bigl(1+z+\\dots +z^{N}\\bigr)$, whose roots indeed lie on the\nunit circle; the bound (10) becomes $M_{N,\\beta}=1$.\n\n\\emph{Uniqueness.}  \nConversely, suppose $\\mu_\\beta=M_{N,\\beta}$.  Then equalities occur in\nboth (2) and Lemma&nbsp;1, whence $(x_k)$ is affine and the coefficients\nmust be the geometric progression (11) with \\emph{the unique} initial\nterm (12).  Thus the only admissible coefficient family (up to\npermutation of the roots, which does not change the polynomial) is\n(14).\n\n\\textbf{9.  Answer to (a)–(b)}  \n\n(a)  The largest constant is given by (10).\n\n(b)  Equality holds precisely for the polynomial (14), whose coefficients\nform the geometric progression $c_k=\\gamma\\rho^{\\,k}$ with\n$\\rho$ from (11) and $\\gamma$ from (12); the $N$ roots are\n$\\rho^{-1}\\zeta$, where $\\zeta$ runs through the $(N+1)$st roots of\nunity distinct from $1$.\nNo other admissible polynomial attains the bound.",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.654394",
        "was_fixed": false,
        "difficulty_analysis": "Compared with the original setting, the enhanced variant is harder in\nseveral independent directions.\n\n1.  Additional structural constraints  \n    • The coefficients are required to be **log-convex** (condition II),\n      introducing majorisation/convexity considerations absent from the\n      original problem.  \n    • A **logarithmic mean** of the coefficients is prescribed\n      (condition III), so the optimisation problem is now\n      constrained by a *global* nonlinear condition instead of mere\n      endpoint bounds.\n\n2.  Variable parameter  \n    The root‐mean to be bounded is the\n    $\\beta$–power mean, so the solver has to work simultaneously for a\n    continuum of exponents.  Establishing a bound that is uniform in\n    $\\beta$ forces the use of power–mean theory rather than the single\n    $\\beta=1$ case of the kernel problem.\n\n3.  Deeper optimisation theory  \n    To identify the extremal sequence of coefficients one must invoke\n    Karamata’s theorem, the Hardy–Littlewood–Pólya majorisation\n    principle, or a full Lagrange–multiplier analysis on an\n    $(N+1)$–dimensional simplex subject to convex ordering (II) and the\n    nonlinear moment constraint (III).  None of these tools is touched\n    in the original problem, which relies only on a single application\n    of $\\mathrm{AM}\\ge\\mathrm{GM}$.\n\n4.  Verifying attainability  \n    Showing that the geometric progression really satisfies all three\n    coefficient conditions and yields equality for every $\\beta$\n    requires a careful scaling argument, followed by an explicit\n    localisation of the roots via a Möbius transformation—again a layer\n    of analysis that is unnecessary for the kernel exercise.\n\nAll these ingredients oblige the solver to weave together root‐product\nformulas, power–mean inequalities, majorisation theory and explicit\nfactorisation—making the enhanced variant substantially more technical\nand conceptually deeper than both the original problem and the current\nkernel variant."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}