{
  "index": "1971-A-4",
  "type": "ANA",
  "tag": [
    "ANA",
    "ALG"
  ],
  "difficulty": "",
  "question": "A-4. Show that for \\( 0<\\varepsilon<1 \\) the expression \\( (x+y)^{n}\\left(x^{2}-(2-\\varepsilon) x y+y^{2}\\right) \\) is a polynomial with positive coefficients for \\( n \\) sufficiently large and integral. For \\( \\varepsilon=.002 \\) find the smallest admissible value of \\( n \\).",
  "solution": "A-4 In the expansion of \\( (x+y)^{n}\\left(x^{2}-(2-\\varepsilon) x y+y^{2}\\right) \\) the coefficient of \\( x^{k+1} y^{n+1-k} \\) is\n\\[\n\\begin{array}{l}\n\\binom{n}{k-1}-(2-\\varepsilon)\\binom{n}{k}+\\binom{n}{k+1} \\\\\n\\quad=\\binom{n}{k}\\left\\{\\frac{k}{n-k+1}+\\frac{n-k}{k+1}-(2-\\varepsilon)\\right\\} .\n\\end{array}\n\\]\n\nNow for fixed \\( n \\) consider the expression\n\\[\n\\phi(k)=\\frac{k}{n-k+1}+\\frac{n-k}{k+1}-(2-\\varepsilon) .\n\\]\n\nIf \\( k \\) is taken to be a continuous positive variable\n\\[\n\\phi^{\\prime}(k)=\\frac{(n+1)\\left\\{(k+1)^{2}-(n-k+1)^{2}\\right\\}}{(n-k+1)^{2}(k+1)^{2}}\n\\]\n\nHence \\( \\phi^{\\prime}(k)=0 \\) at \\( k=n / 2 \\) and it follows easily that \\( \\phi(k) \\) is minimum at \\( k=n / 2 \\).\nWe needn't consider end point minima since it easily follows that for \\( n>2 \\) the polynomial has its first two and last two coefficients positive. We may also note that if the two mid-terms in the expansion are non-positive for a given odd value of \\( n \\) then for the next larger value of \\( n \\) the mid-term remains non-positive. Hence if the midcoefficients become positive, the first value of \\( n \\) for which this occurs is odd. Now if \\( n \\) is odd and \\( k=\\frac{1}{2}(n+1) \\) then \\( \\phi(k)=\\frac{n-1}{n+3}-1+\\varepsilon \\), and \\( \\phi(k)>0 \\) for \\( n>\\frac{4}{\\varepsilon}-3 \\). If \\( \\varepsilon=.002, n>1997 \\) and \\( n \\) is odd. Hence the minimum \\( n \\) for which all terms are positive is 1999 .",
  "vars": [
    "x",
    "y",
    "k",
    "n",
    "\\\\phi"
  ],
  "params": [
    "\\\\varepsilon"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "xvariable",
        "y": "yvariable",
        "k": "indexer",
        "n": "exponent",
        "\\phi": "coeffunction",
        "\\varepsilon": "epsilonconst"
      },
      "question": "A-4. Show that for \\( 0<epsilonconst<1 \\) the expression \\( (xvariable+yvariable)^{exponent}\\left(xvariable^{2}-(2-epsilonconst) xvariable yvariable+yvariable^{2}\\right) \\) is a polynomial with positive coefficients for \\( exponent \\) sufficiently large and integral. For \\( epsilonconst=.002 \\) find the smallest admissible value of \\( exponent \\).",
      "solution": "A-4 In the expansion of \\( (xvariable+yvariable)^{exponent}\\left(xvariable^{2}-(2-epsilonconst) xvariable yvariable+yvariable^{2}\\right) \\) the coefficient of \\( xvariable^{indexer+1} yvariable^{exponent+1-indexer} \\) is\n\\[\n\\begin{array}{l}\n\\binom{exponent}{indexer-1}-(2-epsilonconst)\\binom{exponent}{indexer}+\\binom{exponent}{indexer+1} \\\\\n\\quad=\\binom{exponent}{indexer}\\left\\{\\frac{indexer}{exponent-indexer+1}+\\frac{exponent-indexer}{indexer+1}-(2-epsilonconst)\\right\\} .\n\\end{array}\n\\]\n\nNow for fixed \\( exponent \\) consider the expression\n\\[\ncoeffunction(indexer)=\\frac{indexer}{exponent-indexer+1}+\\frac{exponent-indexer}{indexer+1}-(2-epsilonconst) .\n\\]\n\nIf \\( indexer \\) is taken to be a continuous positive variable\n\\[\ncoeffunction^{\\prime}(indexer)=\\frac{(exponent+1)\\left\\{(indexer+1)^{2}-(exponent-indexer+1)^{2}\\right\\}}{(exponent-indexer+1)^{2}(indexer+1)^{2}}\n\\]\n\nHence \\( coeffunction^{\\prime}(indexer)=0 \\) at \\( indexer=exponent / 2 \\) and it follows easily that \\( coeffunction(indexer) \\) is minimum at \\( indexer=exponent / 2 \\).\nWe needn't consider end point minima since it easily follows that for \\( exponent>2 \\) the polynomial has its first two and last two coefficients positive. We may also note that if the two mid-terms in the expansion are non-positive for a given odd value of \\( exponent \\) then for the next larger value of \\( exponent \\) the mid-term remains non-positive. Hence if the midcoefficients become positive, the first value of \\( exponent \\) for which this occurs is odd. Now if \\( exponent \\) is odd and \\( indexer=\\frac{1}{2}(exponent+1) \\) then \\( coeffunction(indexer)=\\frac{exponent-1}{exponent+3}-1+epsilonconst \\), and \\( coeffunction(indexer)>0 \\) for \\( exponent>\\frac{4}{epsilonconst}-3 \\). If \\( epsilonconst=.002, exponent>1997 \\) and \\( exponent \\) is odd. Hence the minimum \\( exponent \\) for which all terms are positive is 1999 ."
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "seashell",
        "y": "marigold",
        "k": "driftwood",
        "n": "peregrine",
        "\\phi": "\\labyrinth",
        "\\varepsilon": "\\hazelnut"
      },
      "question": "A-4. Show that for \\( 0<\\hazelnut<1 \\) the expression \\( (seashell+marigold)^{peregrine}\\left(seashell^{2}-(2-\\hazelnut) seashell marigold+marigold^{2}\\right) \\) is a polynomial with positive coefficients for \\( peregrine \\) sufficiently large and integral. For \\( \\hazelnut=.002 \\) find the smallest admissible value of \\( peregrine \\).",
      "solution": "A-4 In the expansion of \\( (seashell+marigold)^{peregrine}\\left(seashell^{2}-(2-\\hazelnut) seashell marigold+marigold^{2}\\right) \\) the coefficient of \\( seashell^{driftwood+1} marigold^{peregrine+1-driftwood} \\) is\\n\\[\\n\\begin{array}{l}\\n\\binom{peregrine}{driftwood-1}-(2-\\hazelnut)\\binom{peregrine}{driftwood}+\\binom{peregrine}{driftwood+1} \\\\n\\quad=\\binom{peregrine}{driftwood}\\left\\{\\frac{driftwood}{peregrine-driftwood+1}+\\frac{peregrine-driftwood}{driftwood+1}-(2-\\hazelnut)\\right\\} .\\n\\end{array}\\n\\]\\n\\nNow for fixed \\( peregrine \\) consider the expression\\n\\[\\n\\labyrinth(driftwood)=\\frac{driftwood}{peregrine-driftwood+1}+\\frac{peregrine-driftwood}{driftwood+1}-(2-\\hazelnut) .\\n\\]\\n\\nIf \\( driftwood \\) is taken to be a continuous positive variable\\n\\[\\n\\labyrinth^{\\prime}(driftwood)=\\frac{(peregrine+1)\\left\\{(driftwood+1)^{2}-(peregrine-driftwood+1)^{2}\\right\\}}{(peregrine-driftwood+1)^{2}(driftwood+1)^{2}}\\n\\]\\n\\nHence \\( \\labyrinth^{\\prime}(driftwood)=0 \\) at \\( driftwood=peregrine / 2 \\) and it follows easily that \\( \\labyrinth(driftwood) \\) is minimum at \\( driftwood=peregrine / 2 \\).\\nWe needn't consider end point minima since it easily follows that for \\( peregrine>2 \\) the polynomial has its first two and last two coefficients positive. We may also note that if the two mid-terms in the expansion are non-positive for a given odd value of \\( peregrine \\) then for the next larger value of \\( peregrine \\) the mid-term remains non-positive. Hence if the midcoefficients become positive, the first value of \\( peregrine \\) for which this occurs is odd. Now if \\( peregrine \\) is odd and \\( driftwood=\\frac{1}{2}(peregrine+1) \\) then \\( \\labyrinth(driftwood)=\\frac{peregrine-1}{peregrine+3}-1+\\hazelnut \\), and \\( \\labyrinth(driftwood)>0 \\) for \\( peregrine>\\frac{4}{\\hazelnut}-3 \\). If \\( \\hazelnut=.002, peregrine>1997 \\) and \\( peregrine \\) is odd. Hence the minimum \\( peregrine \\) for which all terms are positive is 1999 ."
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "verticalaxis",
        "y": "horizontalaxis",
        "k": "continuumvalue",
        "n": "fractionalmagnitude",
        "\\phi": "constantvalue",
        "\\varepsilon": "giganticdelta"
      },
      "question": "A-4. Show that for \\( 0<giganticdelta<1 \\) the expression \\( (verticalaxis+horizontalaxis)^{fractionalmagnitude}\\left(verticalaxis^{2}-(2-giganticdelta) verticalaxis horizontalaxis+horizontalaxis^{2}\\right) \\) is a polynomial with positive coefficients for \\( fractionalmagnitude \\) sufficiently large and integral. For \\( giganticdelta=.002 \\) find the smallest admissible value of \\( fractionalmagnitude \\).",
      "solution": "A-4 In the expansion of \\( (verticalaxis+horizontalaxis)^{fractionalmagnitude}\\left(verticalaxis^{2}-(2-giganticdelta) verticalaxis horizontalaxis+horizontalaxis^{2}\\right) \\) the coefficient of \\( verticalaxis^{continuumvalue+1} horizontalaxis^{fractionalmagnitude+1-continuumvalue} \\) is\n\\[\n\\begin{array}{l}\n\\binom{fractionalmagnitude}{continuumvalue-1}-(2-giganticdelta)\\binom{fractionalmagnitude}{continuumvalue}+\\binom{fractionalmagnitude}{continuumvalue+1} \\\\\n\\quad=\\binom{fractionalmagnitude}{continuumvalue}\\left\\{\\frac{continuumvalue}{fractionalmagnitude-continuumvalue+1}+\\frac{fractionalmagnitude-continuumvalue}{continuumvalue+1}-(2-giganticdelta)\\right\\} .\n\\end{array}\n\\]\n\nNow for fixed \\( fractionalmagnitude \\) consider the expression\n\\[\nconstantvalue(continuumvalue)=\\frac{continuumvalue}{fractionalmagnitude-continuumvalue+1}+\\frac{fractionalmagnitude-continuumvalue}{continuumvalue+1}-(2-giganticdelta) .\n\\]\n\nIf \\( continuumvalue \\) is taken to be a continuous positive variable\n\\[\nconstantvalue^{\\prime}(continuumvalue)=\\frac{(fractionalmagnitude+1)\\left\\{(continuumvalue+1)^{2}-(fractionalmagnitude-continuumvalue+1)^{2}\\right\\}}{(fractionalmagnitude-continuumvalue+1)^{2}(continuumvalue+1)^{2}}\n\\]\n\nHence \\( constantvalue^{\\prime}(continuumvalue)=0 \\) at \\( continuumvalue=fractionalmagnitude / 2 \\) and it follows easily that \\( constantvalue(continuumvalue) \\) is minimum at \\( continuumvalue=fractionalmagnitude / 2 \\).\nWe needn't consider end point minima since it easily follows that for \\( fractionalmagnitude>2 \\) the polynomial has its first two and last two coefficients positive. We may also note that if the two mid-terms in the expansion are non-positive for a given odd value of \\( fractionalmagnitude \\) then for the next larger value of \\( fractionalmagnitude \\) the mid-term remains non-positive. Hence if the midcoefficients become positive, the first value of \\( fractionalmagnitude \\) for which this occurs is odd. Now if \\( fractionalmagnitude \\) is odd and \\( continuumvalue=\\frac{1}{2}(fractionalmagnitude+1) \\) then \\( constantvalue(continuumvalue)=\\frac{fractionalmagnitude-1}{fractionalmagnitude+3}-1+giganticdelta \\), and \\( constantvalue(continuumvalue)>0 \\) for \\( fractionalmagnitude>\\frac{4}{giganticdelta}-3 \\). If \\( giganticdelta=.002, fractionalmagnitude>1997 \\) and \\( fractionalmagnitude \\) is odd. Hence the minimum \\( fractionalmagnitude \\) for which all terms are positive is 1999 ."
    },
    "garbled_string": {
      "map": {
        "x": "qzxwvtnp",
        "y": "hjgrksla",
        "k": "asldkfjq",
        "n": "pwoeiruty",
        "\\phi": "\\haldjskq",
        "\\varepsilon": "\\bcsrtmxz"
      },
      "question": "A-4. Show that for \\( 0<\\bcsrtmxz<1 \\) the expression \\( (qzxwvtnp+hjgrksla)^{pwoeiruty}\\left(qzxwvtnp^{2}-(2-\\bcsrtmxz) qzxwvtnp hjgrksla+hjgrksla^{2}\\right) \\) is a polynomial with positive coefficients for \\( pwoeiruty \\) sufficiently large and integral. For \\( \\bcsrtmxz=.002 \\) find the smallest admissible value of \\( pwoeiruty \\).",
      "solution": "A-4 In the expansion of \\( (qzxwvtnp+hjgrksla)^{pwoeiruty}\\left(qzxwvtnp^{2}-(2-\\bcsrtmxz) qzxwvtnp hjgrksla+hjgrksla^{2}\\right) \\) the coefficient of \\( qzxwvtnp^{asldkfjq+1} hjgrksla^{pwoeiruty+1-asldkfjq} \\) is\n\\[\n\\begin{array}{l}\n\\binom{pwoeiruty}{asldkfjq-1}-(2-\\bcsrtmxz)\\binom{pwoeiruty}{asldkfjq}+\\binom{pwoeiruty}{asldkfjq+1} \\\\\n\\quad=\\binom{pwoeiruty}{asldkfjq}\\left\\{\\frac{asldkfjq}{pwoeiruty-asldkfjq+1}+\\frac{pwoeiruty-asldkfjq}{asldkfjq+1}-(2-\\bcsrtmxz)\\right\\} .\n\\end{array}\n\\]\n\nNow for fixed \\( pwoeiruty \\) consider the expression\n\\[\n\\haldjskq(asldkfjq)=\\frac{asldkfjq}{pwoeiruty-asldkfjq+1}+\\frac{pwoeiruty-asldkfjq}{asldkfjq+1}-(2-\\bcsrtmxz) .\n\\]\n\nIf \\( asldkfjq \\) is taken to be a continuous positive variable\n\\[\n\\haldjskq^{\\prime}(asldkfjq)=\\frac{(pwoeiruty+1)\\left\\{(asldkfjq+1)^{2}-(pwoeiruty-asldkfjq+1)^{2}\\right\\}}{(pwoeiruty-asldkfjq+1)^{2}(asldkfjq+1)^{2}}\n\\]\n\nHence \\( \\haldjskq^{\\prime}(asldkfjq)=0 \\) at \\( asldkfjq=pwoeiruty / 2 \\) and it follows easily that \\( \\haldjskq(asldkfjq) \\) is minimum at \\( asldkfjq=pwoeiruty / 2 \\).\nWe needn't consider end point minima since it easily follows that for \\( pwoeiruty>2 \\) the polynomial has its first two and last two coefficients positive. We may also note that if the two mid-terms in the expansion are non-positive for a given odd value of \\( pwoeiruty \\) then for the next larger value of \\( pwoeiruty \\) the mid-term remains non-positive. Hence if the midcoefficients become positive, the first value of \\( pwoeiruty \\) for which this occurs is odd. Now if \\( pwoeiruty \\) is odd and \\( asldkfjq=\\frac{1}{2}(pwoeiruty+1) \\) then \\( \\haldjskq(asldkfjq)=\\frac{pwoeiruty-1}{pwoeiruty+3}-1+\\bcsrtmxz \\), and \\( \\haldjskq(asldkfjq)>0 \\) for \\( pwoeiruty>\\frac{4}{\\bcsrtmxz}-3 \\). If \\( \\bcsrtmxz=.002, pwoeiruty>1997 \\) and \\( pwoeiruty \\) is odd. Hence the minimum \\( pwoeiruty \\) for which all terms are positive is 1999 ."
    },
    "kernel_variant": {
      "question": "Let 0<\\varepsilon<1 and set\n\nc:=\\frac{7}{4}-\\varepsilon\\qquad\\bigl(\\tfrac34<c<\\tfrac74\\bigr).\n\n(a)  Show that for every sufficiently large positive integer n all coefficients of the homogeneous polynomial\n\n           (x+y)^{n}\\bigl(x^{2}-cxy+y^{2}\\bigr)\n\nare positive.\n\n(b)  When \\varepsilon=0.05 (so c=1.7) determine the smallest positive integer n for which every coefficient is positive.",
      "solution": "Throughout write\n\n         P_{n}(x,y):=(x+y)^{n}\\bigl(x^{2}-cxy+y^{2}\\bigr),\\qquad c=\\frac74-\\varepsilon\\;(0<\\varepsilon<1).\n\nThe expansion of P_{n} is a homogeneous form of degree n+2.  We examine its coefficients.\n\n1.  An explicit expression for the coefficients\n------------------------------------------------\nFix k with 0\\le k\\le n+2.  The monomial x^{k}y^{n+2-k} can appear in three ways:\n  •  from x^{2}(x+y)^{n} by taking x^{k-2}y^{n-k},\n  •  from -cxy(x+y)^{n} by taking x^{k-1}y^{n-k+1},\n  •  from y^{2}(x+y)^{n} by taking x^{k}y^{n-k}.\nHence\n\n      a_{k}:=[x^{k}y^{n+2-k}]P_{n}=\binom{n}{k-2}-c\\binom{n}{k-1}+\\binom{n}{k},   (1)\n\nwhere, by convention, \\binom{n}{j}=0 for j<0 or j>n.\n\nClearly a_{0}=a_{n+2}=1>0.  To study the remaining coefficients set\n\n          r:=k-1\\qquad(0\\le r\\le n),                                     (2)\n\na_{k}=b_{r} with\n\n      b_{r}:=\\binom{n}{r-1}-c\\binom{n}{r}+\\binom{n}{r+1}.                 (3)\n\nBecause \\binom{n}{r}>0, divide (3) by it:\n\n \\frac{\\binom{n}{r-1}}{\\binom{n}{r}}=\\frac{r}{n-r+1},\\qquad\n \\frac{\\binom{n}{r+1}}{\\binom{n}{r}}=\\frac{n-r}{r+1},\n\nso that\n\n      b_{r}=\\binom{n}{r}\\,\\varphi_{n}(r),                                 (4)\n\n      \\varphi_{n}(r):=\\frac{r}{n-r+1}+\\frac{n-r}{r+1}-c.                  (5)\n\nBecause \\binom{n}{r}>0, the sign of b_{r} is the sign of \\varphi_{n}(r).\n\n2.  The minimum value of \\varphi_{n}\n--------------------------------------\nTreat r as a real variable on [0,n].  Differentiating (5),\n\n  \\varphi'_{n}(r)=\\frac{(n+1)\\bigl((r+1)^{2}-(n-r+1)^{2}\\bigr)}{(r+1)^{2}(n-r+1)^{2}}.\n\nThus \\varphi'_{n}(r)=0 when r=n/2, and \\varphi''_{n}(n/2)>0, so the unique minimum occurs at r=n/2.  Hence the smallest value of \\varphi_{n}(r) for integral r is attained at\n\n   r=\\frac{n}{2}\\;(n\\text{ even}),\\qquad r=\\frac{n-1}{2}\\text{ or }\\frac{n+1}{2}\\;(n\\text{ odd}).\n\n3.  The two parities\n---------------------\n(a)  n even, n=2m.  Take r=m in (5):\n\n      \\varphi_{n,\\min}=\\frac{m}{m+1}+\\frac{m}{m+1}-c=\\frac{2m}{m+1}-c=\n      \\frac{2n}{n+2}-c.                                             (6)\n\nAll coefficients are positive iff \\varphi_{n,\\min}>0, i.e.\n\n      \\frac{2n}{n+2}>c\\quad\\Longleftrightarrow\\quad n>\\alpha,\n\n      \\alpha:=\\frac{2c}{2-c}.                                      (7)\n\n(b)  n odd, n=2m+1.  Taking r=m (or m+1) gives\n\n      \\varphi_{n,\\min}=1+\\frac{m}{m+2}-c=1+\\frac{n-1}{n+3}-c.       (8)\n\nRequiring \\varphi_{n,\\min}>0 yields\n\n      1+\\frac{n-1}{n+3}>c\\quad\\Longleftrightarrow\\quad n>\\beta,\n\n      \\beta:=\\frac{3c-2}{2-c}.                                     (9)\n\nBecause \\alpha-\\beta=1, we have \\alpha=\\beta+1.  Thus the even bound is always exactly one more than the odd bound.\n\n4.  The least admissible n is odd\n----------------------------------\nLet\n     R:=\\beta=\\frac{3c-2}{2-c}.                                      (10)\n\nDefine\n     N_{\\text{odd}}:=\\text{the smallest odd integer strictly exceeding }R,  (11)\n     N_{\\text{even}}:=\\text{the smallest even integer strictly exceeding }R+1=\\alpha. (12)\n\nBecause R+1>R and N_{\\text{even}} is even, necessarily N_{\\text{even}}>N_{\\text{odd}}.  Moreover, if an even integer n satisfies n>\\alpha, then n-1 is odd and\n\n      n-1>\\alpha-1=\\beta=R,\n\nso n-1 already meets the odd inequality.  Therefore no even n can be minimal: the first n for which every coefficient is positive is precisely N_{\\text{odd}}.\n\nEquivalently\n\n     n_{\\min}(\\varepsilon)=\\begin{cases}\n        \\lceil R\\rceil      & \\text{if }\\lceil R\\rceil\\text{ is odd},\\\\[4pt]\n        \\lceil R\\rceil+1    & \\text{if }\\lceil R\\rceil\\text{ is even}.\n     \\end{cases}                                               (13)\n\n5.  Outer coefficients\n-----------------------\nFor completeness we note that (1) gives\n\n  a_{1}=n-c>n-\\tfrac74>0\\;(n\\ge2), \\qquad a_{n+1}=a_{1},\n\nso possible sign problems occur only among the central coefficients that have already been handled.\n\n6.  Numerics for \\boldsymbol{\\varepsilon=0.05}\n----------------------------------------------\nHere c=1.7, so\n\n      R=\\beta=\\frac{3\\cdot1.7-2}{2-1.7}=\\frac{5.1-2}{0.3}=\\frac{31}{3}=10.\\overline{3}.\n\nThe least odd integer exceeding 10.\\overline{3} is 11; that is N_{\\text{odd}}=11.  Checking (8) directly,\n\n      \\varphi_{11,\\min}=1+\\frac{10}{14}-1.7=1+\\frac57-1.7\\approx0.0143>0,\n\nso every coefficient of P_{11} is positive.  For n=10 (even) we have, by (6),\n\n      \\varphi_{10,\\min}=\\frac{20}{12}-1.7\\approx-0.033<0,\n\nso n=10 fails.  Thus\n\n                 n_{\\min}(\\varepsilon=0.05)=11.                (14)\n\n7.  Summary for general \\boldsymbol{\\varepsilon}\n------------------------------------------------\nWith c=\\frac74-\\varepsilon set R as in (10).  All coefficients of P_{n} are positive whenever\n\n      n\\ge n_{\\min}(\\varepsilon):=\\begin{cases}\n         \\lceil R\\rceil      & \\text{if }\\lceil R\\rceil\\text{ is odd},\\\\[4pt]\n         \\lceil R\\rceil+1    & \\text{if }\\lceil R\\rceil\\text{ is even}.\n      \\end{cases}\n\nThis bound is best possible: n_{\\min}(\\varepsilon)-1 fails because it is either \\le R (odd case) or \\le R+1 (even case), violating the relevant inequality.",
      "_meta": {
        "core_steps": [
          "Express each coefficient as  C(n,k)=binom(n,k-1)- (2-ε)·binom(n,k)+binom(n,k+1)=binom(n,k)·φ(k).",
          "View k as continuous; compute φ'(k) and find its unique minimum at k=n/2.",
          "Note endpoint coefficients are already positive when n>2, so overall positivity ⇔ φ(k) at its minimum is positive.",
          "Evaluate φ at k=n/2, obtain φ_min = (n-1)/(n+3) -1 + ε = ε - 4/(n+3) >0 ⇒ n > 4/ε - 3; take the least odd n satisfying this."
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Chosen numerical value of ε (small positive tolerance)",
            "original": "0.002"
          },
          "slot2": {
            "description": "Constant ‘2’ in the middle term coefficient of the quadratic x² − (2−ε)xy + y²",
            "original": "2"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "proof",
  "iteratively_fixed": true
}