{
  "index": "2018-B-2",
  "type": "COMB",
  "tag": [
    "COMB",
    "ANA",
    "ALG"
  ],
  "difficulty": "",
  "question": "Let $n$ be a positive integer, and let $f_n(z) = n + (n-1) z + (n-2)z^2 + \\cdots + z^{n-1}$. Prove that\n$f_n$ has no roots in the closed unit disk $\\{z \\in \\mathbb{C}\\colon |z| \\leq 1 \\}$.",
  "solution": "\\textbf{First solution.}\nNote first that $f_n(1) > 0$, so $1$ is not a root of $f_n$.\nNext, note that\n\\[\n(z-1)f_n(z) = z^n + \\cdots + z - n;\n\\]\nhowever, for $\\left| z \\right| \\leq 1$, we have \n$\\left| z^n + \\cdots + z \\right| \\leq n$ by the triangle inequality;\nequality can only occur if $z,\\dots,z^n$ have norm 1 and the same argument, which only happens for $z=1$.\nThus there can be no root of $f_n$ with $|z| \\leq 1$.\n\n\n\\noindent\n\\textbf{Second solution.}\n(by Karl Mahlburg)\nDefine the polynomial\n\\[\ng_n(z) = nz^{n-1} + \\cdots + 2z + 1\n\\]\nand note that $z^{n-1} g_n(z^{-1}) = f_n(z)$.\nSince $f_n(0) \\neq 0$, to prove the claim it is equivalent to show that $g_n$\nhas no roots in the region $|z| \\geq 1$.\n\nNow note that $g_n(z) = h_n'(z)$ for\n\\[\nh_n(z) = z^n + \\cdots + z + 1,\n\\]\na polynomial with roots $e^{2\\pi ij/(n+1)}$ for $j=0,\\dots,n$.\nBy the Gauss-Lucas theorem, the roots of $g_n$ lie in the convex hull of the roots of $h_n$,\nand moreover cannot be vertices of the convex hull because $h_n$ has no repeated roots.\nThis implies the claim.\n\n\\noindent\n\\textbf{Remark.}\nYet another approach is to use the \\emph{Enestr\\\"om-Kakeya theorem}: if $P_n(z) = a_0 + \\cdots + a_n z^n$\nis a polynomial with real coefficients satisfying $|a_n|  \\geq \\cdots \\geq |a_0| > 0$, then the roots of $P_n(z)$\nall satisfy $|z| \\leq 1$. Namely, applying this to the polynomial $g_n(z/c)$ for \n$c = n/(n-1)$ shows that the roots of $g_n$ all satisfy $\\left|z \\right| \\leq 1/c$. \n\n\\noindent\n\\textbf{Remark.}\nFor a related problem, see problem A5 from the 2014 Putnam competition.",
  "vars": [
    "z"
  ],
  "params": [
    "n",
    "j",
    "f_n",
    "g_n",
    "h_n",
    "P_n",
    "a_0",
    "a_n",
    "c"
  ],
  "sci_consts": [
    "e",
    "i"
  ],
  "variants": {
    "descriptive_long": {
      "map": {
        "z": "complexvar",
        "n": "positivenum",
        "j": "loopindex",
        "f_n": "primarypoly",
        "g_n": "secondarypoly",
        "h_n": "polythird",
        "P_n": "genericpoly",
        "a_0": "zerocoeff",
        "a_n": "nthcoeff",
        "c": "scalingcst"
      },
      "question": "Let $positivenum$ be a positive integer, and let $primarypoly(complexvar) = positivenum + (positivenum-1) complexvar + (positivenum-2)complexvar^2 + \\cdots + complexvar^{positivenum-1}$. Prove that\n$primarypoly$ has no roots in the closed unit disk $\\{complexvar \\in \\mathbb{C}\\colon |complexvar| \\leq 1 \\}$. ",
      "solution": "\\textbf{First solution.}\nNote first that $primarypoly(1) > 0$, so $1$ is not a root of $primarypoly$.\nNext, note that\n\\[\n(complexvar-1)primarypoly(complexvar) = complexvar^{positivenum} + \\cdots + complexvar - positivenum;\n\\]\nhowever, for $\\left| complexvar \\right| \\leq 1$, we have \n$\\left| complexvar^{positivenum} + \\cdots + complexvar \\right| \\leq positivenum$ by the triangle inequality;\nequality can only occur if $complexvar,\\dots,complexvar^{positivenum}$ have norm 1 and the same argument, which only happens for $complexvar=1$.\nThus there can be no root of $primarypoly$ with $|complexvar| \\leq 1$.\n\n\n\\noindent\n\\textbf{Second solution.}\n(by Karl Mahlburg)\nDefine the polynomial\n\\[\nsecondarypoly(complexvar) = positivenum complexvar^{positivenum-1} + \\cdots + 2complexvar + 1\n\\]\nand note that $complexvar^{positivenum-1} secondarypoly(complexvar^{-1}) = primarypoly(complexvar)$.\nSince $primarypoly(0) \\neq 0$, to prove the claim it is equivalent to show that $secondarypoly$\nhas no roots in the region $|complexvar| \\geq 1$.\n\nNow note that $secondarypoly(complexvar) = polythird'(complexvar)$ for\n\\[\npolythird(complexvar) = complexvar^{positivenum} + \\cdots + complexvar + 1,\n\\]\na polynomial with roots $e^{2\\pi i loopindex/(positivenum+1)}$ for $loopindex=0,\\dots,positivenum$.\nBy the Gauss-Lucas theorem, the roots of $secondarypoly$ lie in the convex hull of the roots of $polythird$,\nand moreover cannot be vertices of the convex hull because $polythird$ has no repeated roots.\nThis implies the claim.\n\n\\noindent\n\\textbf{Remark.}\nYet another approach is to use the \\emph{Enestr\\\"om-Kakeya theorem}: if $genericpoly(complexvar) = zerocoeff + \\cdots + nthcoeff complexvar^{positivenum}$\nis a polynomial with real coefficients satisfying $|nthcoeff|  \\geq \\cdots \\geq |zerocoeff| > 0$, then the roots of $genericpoly(complexvar)$\nall satisfy $|complexvar| \\leq 1$. Namely, applying this to the polynomial $secondarypoly(complexvar/scalingcst)$ for \n$scalingcst = positivenum/(positivenum-1)$ shows that the roots of $secondarypoly$ all satisfy $\\left|complexvar \\right| \\leq 1/scalingcst$. \n\n\\noindent\n\\textbf{Remark.}\nFor a related problem, see problem A5 from the 2014 Putnam competition."
    },
    "descriptive_long_confusing": {
      "map": {
        "z": "pineapple",
        "n": "lighthouse",
        "j": "dandelion",
        "f_n": "accordion",
        "g_n": "butterfly",
        "h_n": "saxophone",
        "P_n": "caterpillar",
        "a_0": "rainstorm",
        "a_n": "moonlight",
        "c": "framework"
      },
      "question": "Let $lighthouse$ be a positive integer, and let $accordion(pineapple) = lighthouse + (lighthouse-1) pineapple + (lighthouse-2)pineapple^2 + \\cdots + pineapple^{lighthouse-1}$. Prove that\n$accordion$ has no roots in the closed unit disk $\\{pineapple \\in \\mathbb{C}\\colon |pineapple| \\leq 1 \\}$.",
      "solution": "\\textbf{First solution.}\nNote first that $accordion(1) > 0$, so $1$ is not a root of $accordion$.\nNext, note that\n\\[\n(pineapple-1)accordion(pineapple) = pineapple^{lighthouse} + \\cdots + pineapple - lighthouse;\n\\]\nhowever, for $\\left| pineapple \\right| \\leq 1$, we have \n$\\left| pineapple^{lighthouse} + \\cdots + pineapple \\right| \\leq lighthouse$ by the triangle inequality;\nequality can only occur if $pineapple,\\dots,pineapple^{lighthouse}$ have norm 1 and the same argument, which only happens for $pineapple=1$.\nThus there can be no root of $accordion$ with $|pineapple| \\leq 1$.\n\n\n\\noindent\n\\textbf{Second solution.}\n(by Karl Mahlburg)\nDefine the polynomial\n\\[\nbutterfly(pineapple) = lighthouse pineapple^{lighthouse-1} + \\cdots + 2pineapple + 1\n\\]\nand note that $pineapple^{lighthouse-1} butterfly(pineapple^{-1}) = accordion(pineapple)$.\nSince $accordion(0) \\neq 0$, to prove the claim it is equivalent to show that $butterfly$\nhas no roots in the region $|pineapple| \\geq 1$.\n\nNow note that $butterfly(pineapple) = saxophone'(pineapple)$ for\n\\[\nsaxophone(pineapple) = pineapple^{lighthouse} + \\cdots + pineapple + 1,\n\\]\na polynomial with roots $e^{2\\pi i dandelion/(lighthouse+1)}$ for $dandelion=0,\\dots,lighthouse$.\nBy the Gauss-Lucas theorem, the roots of $butterfly$ lie in the convex hull of the roots of $saxophone$,\nand moreover cannot be vertices of the convex hull because $saxophone$ has no repeated roots.\nThis implies the claim.\n\n\\noindent\n\\textbf{Remark.}\nYet another approach is to use the \\emph{Enestr\\\"om-Kakeya theorem}: if $caterpillar(pineapple) = rainstorm + \\cdots + moonlight pineapple^{lighthouse}$\nis a polynomial with real coefficients satisfying $|moonlight|  \\geq \\cdots \\geq |rainstorm| > 0$, then the roots of $caterpillar(pineapple)$\nall satisfy $|pineapple| \\leq 1$. Namely, applying this to the polynomial $butterfly(pineapple/framework)$ for \n$framework = lighthouse/(lighthouse-1)$ shows that the roots of $butterfly$ all satisfy $\\left|pineapple \\right| \\leq 1/framework$. \n\n\\noindent\n\\textbf{Remark.}\nFor a related problem, see problem A5 from the 2014 Putnam competition."
    },
    "descriptive_long_misleading": {
      "map": {
        "z": "realconstant",
        "n": "negativeinteger",
        "j": "fixedpoint",
        "f_n": "antifunction",
        "g_n": "contrarypoly",
        "h_n": "oppositepower",
        "P_n": "counterform",
        "a_0": "highestcoef",
        "a_n": "lowestcoef",
        "c": "shrinker"
      },
      "question": "Let $negativeinteger$ be a positive integer, and let $antifunction(realconstant) = negativeinteger + (negativeinteger-1) realconstant + (negativeinteger-2)realconstant^2 + \\cdots + realconstant^{negativeinteger-1}$. Prove that\n$antifunction$ has no roots in the closed unit disk $\\{realconstant \\in \\mathbb{C}\\colon |realconstant| \\leq 1 \\}$.",
      "solution": "\\textbf{First solution.}\nNote first that $antifunction(1) > 0$, so $1$ is not a root of $antifunction$.\nNext, note that\n\\[\n(realconstant-1)antifunction(realconstant) = realconstant^{negativeinteger} + \\cdots + realconstant - negativeinteger;\n\\]\nhowever, for $\\left| realconstant \\right| \\leq 1$, we have \n$\\left| realconstant^{negativeinteger} + \\cdots + realconstant \\right| \\leq negativeinteger$ by the triangle inequality;\nequality can only occur if $realconstant,\\dots,realconstant^{negativeinteger}$ have norm 1 and the same argument, which only happens for $realconstant=1$.\nThus there can be no root of $antifunction$ with $|realconstant| \\leq 1$.\n\n\\noindent\n\\textbf{Second solution.}\n(by Karl Mahlburg)\nDefine the polynomial\n\\[\ncontrarypoly(realconstant) = negativeinteger realconstant^{negativeinteger-1} + \\cdots + 2realconstant + 1\n\\]\nand note that $realconstant^{negativeinteger-1} contrarypoly(realconstant^{-1}) = antifunction(realconstant)$.\nSince $antifunction(0) \\neq 0$, to prove the claim it is equivalent to show that $contrarypoly$\nhas no roots in the region $|realconstant| \\geq 1$.\n\nNow note that $contrarypoly(realconstant) = oppositepower'(realconstant)$ for\n\\[\noppositepower(realconstant) = realconstant^{negativeinteger} + \\cdots + realconstant + 1,\n\\]\na polynomial with roots $e^{2\\pi i fixedpoint/(negativeinteger+1)}$ for $fixedpoint=0,\\dots,negativeinteger$.\nBy the Gauss-Lucas theorem, the roots of $contrarypoly$ lie in the convex hull of the roots of $oppositepower$,\nand moreover cannot be vertices of the convex hull because $oppositepower$ has no repeated roots.\nThis implies the claim.\n\n\\noindent\n\\textbf{Remark.}\nYet another approach is to use the \\emph{Enestr\"om-Kakeya theorem}: if $counterform(realconstant) = highestcoef + \\cdots + lowestcoef realconstant^{negativeinteger}$\nis a polynomial with real coefficients satisfying $|lowestcoef|  \\geq \\cdots \\geq |highestcoef| > 0$, then the roots of $counterform(realconstant)$\nall satisfy $|realconstant| \\leq 1$. Namely, applying this to the polynomial $contrarypoly(realconstant/shrinker)$ for \n$shrinker = negativeinteger/(negativeinteger-1)$ shows that the roots of $contrarypoly$ all satisfy $\\left|realconstant \\right| \\leq 1/shrinker$. \n\n\\noindent\n\\textbf{Remark.}\nFor a related problem, see problem A5 from the 2014 Putnam competition."
    },
    "garbled_string": {
      "map": {
        "z": "xkqplmnr",
        "n": "sjhduvne",
        "j": "bqlmznrt",
        "f_n": "kfgjsewp",
        "g_n": "ihxaotuw",
        "h_n": "vczlmptk",
        "P_n": "rqypdexf",
        "a_0": "polmnbvc",
        "a_n": "werlkaqs",
        "c": "tmsovarh"
      },
      "question": "Let $sjhduvne$ be a positive integer, and let $kfgjsewp(xkqplmnr) = sjhduvne + (sjhduvne-1) xkqplmnr + (sjhduvne-2)xkqplmnr^2 + \\cdots + xkqplmnr^{sjhduvne-1}$. Prove that\n$kfgjsewp$ has no roots in the closed unit disk $\\{xkqplmnr \\in \\mathbb{C}\\colon |xkqplmnr| \\leq 1 \\}$.",
      "solution": "\\textbf{First solution.}\nNote first that $kfgjsewp(1) > 0$, so $1$ is not a root of $kfgjsewp$.\nNext, note that\n\\[\n(xkqplmnr-1)kfgjsewp(xkqplmnr) = xkqplmnr^{sjhduvne} + \\cdots + xkqplmnr - sjhduvne;\n\\]\nhowever, for $\\left| xkqplmnr \\right| \\leq 1$, we have \n$\\left| xkqplmnr^{sjhduvne} + \\cdots + xkqplmnr \\right| \\leq sjhduvne$ by the triangle inequality;\nequality can only occur if $xkqplmnr,\\dots,xkqplmnr^{sjhduvne}$ have norm 1 and the same argument, which only happens for $xkqplmnr=1$.\nThus there can be no root of $kfgjsewp$ with $|xkqplmnr| \\leq 1$.\n\n\\noindent\n\\textbf{Second solution.}\n(by Karl Mahlburg)\nDefine the polynomial\n\\[\nihxaotuw(xkqplmnr) = sjhduvne xkqplmnr^{sjhduvne-1} + \\cdots + 2xkqplmnr + 1\n\\]\nand note that $xkqplmnr^{sjhduvne-1} ihxaotuw(xkqplmnr^{-1}) = kfgjsewp(xkqplmnr)$.\nSince $kfgjsewp(0) \\neq 0$, to prove the claim it is equivalent to show that $ihxaotuw$\nhas no roots in the region $|xkqplmnr| \\geq 1$.\n\nNow note that $ihxaotuw(xkqplmnr) = vczlmptk'(xkqplmnr)$ for\n\\[\nvczlmptk(xkqplmnr) = xkqplmnr^{sjhduvne} + \\cdots + xkqplmnr + 1,\n\\]\na polynomial with roots $e^{2\\pi i bqlmznrt/(sjhduvne+1)}$ for $bqlmznrt=0,\\dots,sjhduvne$.\nBy the Gauss-Lucas theorem, the roots of $ihxaotuw$ lie in the convex hull of the roots of $vczlmptk$,\nand moreover cannot be vertices of the convex hull because $vczlmptk$ has no repeated roots.\nThis implies the claim.\n\n\\noindent\n\\textbf{Remark.}\nYet another approach is to use the \\emph{Enestr\"om-Kakeya theorem}: if $rqypdexf(xkqplmnr) = polmnbvc + \\cdots + werlkaqs xkqplmnr^{sjhduvne}$\nis a polynomial with real coefficients satisfying $|werlkaqs|  \\geq \\cdots \\geq |polmnbvc| > 0$, then the roots of $rqypdexf(xkqplmnr)$\nall satisfy $|xkqplmnr| \\leq 1$. Namely, applying this to the polynomial $ihxaotuw(xkqplmnr/tmsovarh)$ for \n$tmsovarh = sjhduvne/(sjhduvne-1)$ shows that the roots of $ihxaotuw$ all satisfy $\\left|xkqplmnr \\right| \\leq 1/tmsovarh$. \n\n\\noindent\n\\textbf{Remark.}\nFor a related problem, see problem A5 from the 2014 Putnam competition."
    },
    "kernel_variant": {
      "question": "Let $n\\ge 2$ be an integer and let $\\alpha\\in\\mathbb C\\setminus\\{0\\}$ be a complex parameter.  \nIntroduce the auxiliary “reference’’ polynomial  \n\\[\nf_{n}(w):=F_{n,1}(w)=n+(n-1)w+(n-2)w^{2}+\\dots +w^{\\,n-1},\\qquad\\deg f_{n}=n-1,\n\\]\nand denote by  \n\\[\n\\rho _n:=\\min\\bigl\\{|w|\\;:\\;f_{n}(w)=0\\bigr\\}\\quad (>1)\n\\]\nthe minimum modulus of the $n-1$ complex zeros of $f_n$.  \n\nFor the two–parameter family  \n\\[\nF_{n,\\alpha}(z)=n+\\alpha (n-1)z+\\alpha^{2}(n-2)z^{2}+\\dots +\\alpha^{\\,n-1}z^{\\,n-1},\n\\qquad \\deg F_{n,\\alpha}=n-1,\n\\]\nanswer the following.\n\n(A)  Prove that if $|\\alpha|\\le 1$ then $F_{n,\\alpha}$ possesses no zeros in the closed unit\ndisc $\\overline{\\mathbb D}=\\{z\\in\\mathbb C:\\,|z|\\le 1\\}$.\n\n(B)  Show that\n\\[\n|\\alpha|>\\rho _n\\quad\\Longleftrightarrow\\quad F_{n,\\alpha}\\ \\text{has at least one zero in the open disc } \\mathbb D,\n\\]\nand prove that no smaller bound than $\\rho _n$ enjoys this property (that is,\n$\\rho _n$ is the \\emph{sharp threshold} for the first entry of a zero of $F_{n,\\alpha}$\ninto $\\mathbb D$).\n\n(C)  Establish the explicit two–sided estimates\n\\[\n\\boxed{\\;\n\\dfrac{n}{\\,n-1\\,}\\le\\rho _n\\le n^{1/(n-1)}\n\\;}\n\\qquad(n\\ge 2),\n\\]\nand deduce that $\\rho _n\\!\\downarrow 1$ as $n\\to\\infty$.\n\nThe three parts together determine both the qualitative and quantitative behaviour\nof the unit–disc zeros for the whole two–parameter family $F_{n,\\alpha}$, refining the\nnaive expectation “$|\\alpha|>1$’’ into the exact threshold $|\\alpha|>\\rho _n$.",
      "solution": "\\textbf{Step $0$.  A convenient rescaling.}  \nSet $w=\\alpha z$.  Then  \n\\[\nF_{n,\\alpha}(z)=f_{n}(w)=f_{n}(\\alpha z).\n\\tag{1}\n\\]\nThus every zero $z$ of $F_{n,\\alpha}$ corresponds bijectively to a zero  \n$w=\\alpha z$ of $f_{n}$, and conversely.\n\nIt is classical (see, for instance, Putnam 2014\\,A–5) that  \n\\[\nf_{n}(w)\\neq 0\\quad\\text{for all }|w|\\le 1.\n\\tag{2}\n\\]\nIdentity (1) therefore allows us to shuttle information between the zeros of\n$f_{n}$ and those of $F_{n,\\alpha}$.\n\n\\medskip\n\\textbf{Part (A) ($|\\alpha|\\le 1$).}  \nIf $|z|\\le 1$ then $|w|=|\\alpha z|\\le 1$, so (2) yields $f_{n}(w)\\neq 0$.\nVia (1) this implies $F_{n,\\alpha}(z)\\neq 0$ for every $|z|\\le 1$.  Hence\n$F_{n,\\alpha}$ is zero–free on $\\overline{\\mathbb D}$ whenever $|\\alpha|\\le 1$.\n\n\\medskip\n\\textbf{Part (B).  The sharp threshold $|\\alpha|=\\rho _n$.}\n\n\\emph{(i) If $|\\alpha|>\\rho _n$, a zero enters $\\mathbb D$.}  \nChoose a zero $w_0$ of $f_{n}$ with $|w_0|=\\rho _n$, and define\n\\[\n\\zeta:=\\frac{w_0}{\\alpha}.\n\\]\nBecause $|\\alpha|>\\rho _n$, we have $|\\zeta|<1$, so $\\zeta\\in\\mathbb D$.\nSince $f_{n}(w_0)=0$, equation (1) with $z=\\zeta$ gives\n\\[\nF_{n,\\alpha}(\\zeta)=f_{n}(\\alpha\\zeta)=f_{n}(w_0)=0,\n\\]\nexhibiting a zero of $F_{n,\\alpha}$ inside $\\mathbb D$.\n\n\\emph{(ii) If $|\\alpha|<\\rho _n$, no zero lies in $\\mathbb D$.}  \nAssume, to the contrary, that $F_{n,\\alpha}(z_0)=0$ for some $z_0\\in\\mathbb D$.\nThen $w_0:=\\alpha z_0$ is a zero of $f_{n}$, so $|w_0|\\ge\\rho _n$.\nYet\n\\[\n|w_0|=|\\alpha|\\,|z_0|<|\\alpha|<\\rho _n,\n\\]\na contradiction.  Hence $F_{n,\\alpha}$ is zero–free in $\\mathbb D$ whenever\n$|\\alpha|<\\rho _n$.\n\n\\emph{(iii) The boundary case $|\\alpha|=\\rho _n$.}  \nAgain pick a zero $w_0$ of $f_{n}$ with $|w_0|=\\rho _n$ and take $\\alpha=w_0$.\nThen $z=1$ satisfies $|z|=1$ and\n\\[\nF_{n,\\alpha}(1)=f_{n}(w_0)=0.\n\\]\nConsequently $F_{n,\\alpha}$ acquires a zero \\emph{on} the unit circle but,\nby (ii), none in the open disc.  Therefore crossing the modulus\n$\\rho _n$ is precisely what brings a zero of $F_{n,\\alpha}$ into $\\mathbb D$,\nand the bound $\\rho _n$ is sharp.\n\nParts (i)–(iii) establish the equivalence claimed in (B).\n\n\\medskip\n\\textbf{Part (C).  Two–sided bounds and the limit $\\rho _n\\to 1$.}\n\n\\emph{Lower bound via Enestr\\\"om–Kakeya.}  \nWrite $f_{n}(w)=\\sum_{k=0}^{n-1}a_k w^{k}$ with $a_k=n-k>0$.\nFor $0\\le k\\le n-2$,\n\\[\n\\frac{a_k}{a_{k+1}}=\\frac{n-k}{n-k-1}\\ge\\frac{n}{n-1}>1.\n\\]\nThe Enestr\\\"om–Kakeya theorem therefore places every zero of $f_{n}$ in the\nexterior of the circle $|w|=\\dfrac{n}{n-1}$, so\n\\[\n\\boxed{\\rho _n\\ge\\dfrac{n}{\\,n-1\\,}}.\n\\]\n\n\\emph{Upper bound via Vieta and AM–GM.}  \nLet $w_1,\\dots ,w_{n-1}$ be the zeros of $f_{n}$.  Because the leading\ncoefficient is $1$ and the constant coefficient is $n$, Vieta’s formula gives\n\\[\n|w_1 w_2\\cdots w_{n-1}|=|(-1)^{\\,n-1}n|=n.\n\\]\nTaking absolute values and the $(n-1)$–st root,\n\\[\n\\bigl(|w_1|\\cdots |w_{n-1}|\\bigr)^{1/(n-1)}=n^{1/(n-1)}.\n\\]\nAt least one factor does not exceed this geometric mean, so\n\\[\n\\boxed{\\rho _n\\le n^{1/(n-1)}}.\n\\]\n\n\\emph{Limit behaviour.}  \nSince $\\dfrac{n}{n-1}\\uparrow 1$ and $n^{1/(n-1)}\\downarrow 1$ as\n$n\\to\\infty$, the squeeze\n\\[\n1<\\frac{n}{\\,n-1\\,}\\le\\rho _n\\le n^{1/(n-1)}<2\n\\]\nforces $\\rho _n\\to 1$.\n\nAll three parts are now rigorously proved.\n\n\\bigskip",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.856173",
        "was_fixed": false,
        "difficulty_analysis": "1.  Two–sided classification.  Unlike the original exercise—which only\nasked for zero–freeness—the enhanced variant demands an\n\\emph{exact characterisation}: one direction still uses the original idea,\nbut the other direction requires showing that zeros \\emph{do occur} once\nthe parameter crosses a sharp boundary.\n\n2.  Parameter dependence.  The presence of the complex parameter\n\\(\\alpha\\) introduces an additional degree of freedom; the argument must\nwork uniformly in \\(\\alpha\\) and distinguish qualitatively different\nbehaviour according to \\(|\\alpha|\\).\n\n3.  Combination of techniques.  \n   • For \\(|\\alpha|\\le 1\\) we have to map the problem to the classic one\n     (substitution \\(w=\\alpha z\\)).  \n   • For \\(|\\alpha|>1\\) we need a completely new ingredient:\n     analysis of the monotone real function \\(t\\mapsto f_{n}(-t)\\) and an\n     explicit construction of a zero via (4); purely algebraic or\n     triangle–inequality arguments no longer suffice.\n\n4.  Sharpness argument.  Establishing that the boundary \\(|\\alpha|=1\\) is\n   precise forces us to build an explicit family of zeros and to verify\n   that no further improvement is possible.\n\nAll of these additions make the enhanced variant\nsubstantially more intricate than both the original Putnam problem and\nthe intermediate kernel version, while keeping the same core object\n(the polynomial with descending coefficients)."
      }
    },
    "original_kernel_variant": {
      "question": "Fix an integer \\(n\\ge 2\\) and a complex parameter \\(\\alpha\\neq 0\\).\nIntroduce the auxiliary polynomial  \n\n\\[\nf_{n}(w):=F_{n,1}(w)=n+(n-1)w+(n-2)w^{2}+\\dots +w^{n-1},\n\\qquad (\\deg f_{n}=n-1).\n\\]\n\nDenote by  \n\n\\[\n\\rho _{n}:=\\min \\{|w|:f_{n}(w)=0\\}\\;(>1)\n\\]\n\nthe smallest modulus of the \\(n-1\\) (complex) zeros of \\(f_{n}\\).  \nFor the two–parameter family  \n\n\\[\nF_{n,\\alpha}(z)=n+\\alpha (n-1)z+\\alpha ^2(n-2)z^{2}+\\dots +\\alpha ^{\\,n-1}z^{\\,n-1},\n\\]\n\nsolve the following.\n\n(A)  Show that if \\(|\\alpha |\\le 1\\) then \\(F_{n,\\alpha}\\) has no zeros in the closed unit\ndisc \\(\\overline{\\mathbb D}=\\{z\\in\\mathbb C:|z|\\le 1\\}\\).\n\n(B)  Prove that if \\(|\\alpha |>\\rho _{n}\\) then \\(F_{n,\\alpha}\\) possesses at least one zero in\nthe open unit disc \\(\\mathbb D\\).  \n(Consequently the inequality \\(|\\alpha|>\\rho_n\\) is the \\emph{sharp} condition that\nforces a zero of \\(F_{n,\\alpha}\\) to enter \\(\\mathbb D\\).)\n\n(C)  Establish the explicit bounds  \n\n\\[\n\\boxed{\\;\n\\dfrac{n}{\\,n-1\\,}\\le \\rho _{n}\\le n^{1/(n-1)}\n\\;}\n\\qquad (n\\ge 2),\n\\]\n\nand deduce that \\(\\rho _{n}\\downarrow 1\\) as \\(n\\to\\infty\\).\n\n(Parts (B)–(C) together locate the true threshold, correcting the naive claim\n“\\(|\\alpha |>1\\)” from the earlier version.)\n\n——————————————————————————————————————",
      "solution": "Step 0.  A convenient rescaling.  \nPut \\(w=\\alpha z\\).  Then\n\n\\[\nF_{n,\\alpha}(z)=f_{n}(w)=f_{n}(\\alpha z).\n\\tag{1}\n\\]\n\nThus every zero \\(z\\) of \\(F_{n,\\alpha}\\) corresponds to a zero \\(w=\\alpha z\\) of\n\\(f_{n}\\), and vice-versa.\n\n————————————————\nPart (A) (\\(|\\alpha|\\le 1\\)).  \nBecause \\(|\\alpha|\\le 1\\) and \\(|z|\\le 1\\) we have \\(|w|=|\\alpha z|\\le 1\\); hence\n\\(w\\) ranges over \\(\\overline{\\mathbb D}\\).\nThe classical Putnam–A-5 result (or any of several standard proofs) gives  \n\n\\[\nf_{n}(w)\\neq 0\\quad\\text{for every }|w|\\le 1 .\n\\]\n\nWith (1) this implies \\(F_{n,\\alpha}(z)\\neq 0\\) for every \\(|z|\\le 1\\).  Hence\n\\(F_{n,\\alpha}\\) is zero–free on \\(\\overline{\\mathbb D}\\) whenever \\(|\\alpha|\\le 1\\).\n\n————————————————\nPart (B) (\\(|\\alpha|>\\rho_n\\)).  \nChoose a zero \\(w_{0}\\) of \\(f_{n}\\) with \\(|w_{0}|=\\rho _{n}\\).\nFor \\(|\\alpha|>\\rho _{n}\\) define  \n\n\\[\n\\zeta:=\\frac{w_{0}}{\\alpha }.\n\\]\n\nThen  \n\n\\[\n|\\zeta|=\\frac{|w_{0}|}{|\\alpha|}=\\frac{\\rho _{n}}{|\\alpha|}<1\\quad\\Longrightarrow\\quad\n\\zeta\\in\\mathbb D .\n\\]\n\nBy construction \\(f_{n}(w_{0})=0\\), and (1) with \\(z=\\zeta\\) gives  \n\n\\[\nF_{n,\\alpha}(\\zeta)=f_{n}(\\alpha\\zeta)=f_{n}(w_{0})=0 .\n\\]\n\nThus \\(F_{n,\\alpha}\\) indeed has a unit–disc zero as soon as \\(|\\alpha|>\\rho _{n}\\).\nBecause the inequality is strict, the condition cannot be weakened; it is\ntherefore optimal.\n\n————————————————\nPart (C)  Quantitative bounds for \\(\\rho _{n}\\).\n\nLower bound (\\(\\rho _{n}\\ge n/(n-1)\\)).  \nFor \\(0\\le k\\le n-1\\) the coefficients of \\(f_{n}\\) satisfy  \n\n\\[\na_{k}=n-k,\\qquad\n\\frac{a_{k}}{a_{k+1}}=\\frac{n-k}{n-k-1}\\ge\\frac{n}{n-1}>1 .\n\\]\n\nAll coefficients are positive and \\(|a_{k}|/\\!|a_{k+1}|>1\\); hence the\nEneström–Kakeya theorem asserts that every zero of \\(f_{n}\\) fulfils  \n\n\\[\n|w|\\ge\\min_{k}\\frac{a_{k}}{a_{k+1}}=\\frac{n}{n-1}.\n\\]\n\nTherefore \\(\\rho _{n}\\ge n/(n-1)\\; (>1)\\).\n\nUpper bound (\\(\\rho _{n}\\le n^{1/(n-1)}\\)).  \nLet \\(w_{1},\\dots ,w_{n-1}\\) be the zeros of \\(f_{n}\\).  From Vieta’s formula  \n\n\\[\n|w_{1}\\,w_{2}\\dots w_{n-1}|=|(-1)^{n-1}a_{0}|=n .\n\\]\n\nTaking absolute values and \\((n-1)\\)-st roots,\n\n\\[\n\\bigl(|w_{1}|\\dots |w_{n-1}|\\bigr)^{1/(n-1)}=n^{1/(n-1)} .\n\\]\n\nHence at least one factor on the left does not exceed the geometric mean,\nso  \n\n\\[\n\\rho _{n}=\\min_{j}|w_{j}|\\le n^{1/(n-1)} .\n\\]\n\nLimit \\(\\rho _{n}\\downarrow 1\\).  \nBecause \\(n/(n-1)\\uparrow 1\\) and \\(n^{1/(n-1)}\\downarrow 1\\) as \\(n\\to\\infty\\),\nthe sandwich  \n\n\\[\n1<\\frac{n}{\\,n-1\\,}\\le\\rho _{n}\\le n^{1/(n-1)}<2\n\\]\n\nforces \\(\\rho _{n}\\to 1\\).\n\nThis completes the solution of all three parts.\n\n——————————————————————————————————————",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.652304",
        "was_fixed": false,
        "difficulty_analysis": "1.  Two–sided classification.  Unlike the original exercise—which only\nasked for zero–freeness—the enhanced variant demands an\n\\emph{exact characterisation}: one direction still uses the original idea,\nbut the other direction requires showing that zeros \\emph{do occur} once\nthe parameter crosses a sharp boundary.\n\n2.  Parameter dependence.  The presence of the complex parameter\n\\(\\alpha\\) introduces an additional degree of freedom; the argument must\nwork uniformly in \\(\\alpha\\) and distinguish qualitatively different\nbehaviour according to \\(|\\alpha|\\).\n\n3.  Combination of techniques.  \n   • For \\(|\\alpha|\\le 1\\) we have to map the problem to the classic one\n     (substitution \\(w=\\alpha z\\)).  \n   • For \\(|\\alpha|>1\\) we need a completely new ingredient:\n     analysis of the monotone real function \\(t\\mapsto f_{n}(-t)\\) and an\n     explicit construction of a zero via (4); purely algebraic or\n     triangle–inequality arguments no longer suffice.\n\n4.  Sharpness argument.  Establishing that the boundary \\(|\\alpha|=1\\) is\n   precise forces us to build an explicit family of zeros and to verify\n   that no further improvement is possible.\n\nAll of these additions make the enhanced variant\nsubstantially more intricate than both the original Putnam problem and\nthe intermediate kernel version, while keeping the same core object\n(the polynomial with descending coefficients)."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}