{
  "index": "1966-A-4",
  "type": "NT",
  "tag": [
    "NT",
    "ALG"
  ],
  "difficulty": "",
  "question": "A-4. Prove that after deleting the perfect squares from the list of positive integers the number we find in the \\( n \\)th position is equal to \\( n+\\{\\sqrt{ } n\\} \\), where \\( \\{\\sqrt{ } n\\} \\) denotes the integer closest to \\( \\sqrt{ } n \\).",
  "solution": "A-4 To prove the formula by induction, it suffices to show that the difference \\( \\Delta=n+\\{\\sqrt{ } n\\}-(n-1+\\{\\sqrt{n-1}\\})=1 \\) or 2 , with the value 2 occurring if and only if the number \\( n+\\{\\sqrt{n-1}\\} \\) is a perfect square. For convenience, let \\( \\{\\sqrt{n-1}\\}=q \\). Then of course \\( q-\\frac{1}{2}<\\sqrt{n-1}<q+\\frac{1}{2} \\) or better \\( q^{2}-q+\\frac{1}{4}<n-1 \\) \\( <q^{2}+q+\\frac{1}{4} \\). This gives \\( q^{2}+5 / 4<n+\\{\\sqrt{n-1}\\}<(q+1)^{2}+\\frac{1}{4} \\). Therefore the number \\( n+\\{\\sqrt{n-1}\\} \\) is a perfect square if and only if \\( n=(q+1)^{2}-q \\). However, then and only then \\( \\sqrt{ } n>q+\\frac{1}{2}>\\sqrt{n-1} \\). In other words then and only then \\( \\{\\sqrt{ } n\\}-\\{\\sqrt{n-1}\\}=1 \\), because this difference is never greater than 1.",
  "vars": [
    "n",
    "\\\\Delta",
    "q"
  ],
  "params": [],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "n": "indexval",
        "\\Delta": "gapvalue",
        "q": "nearint"
      },
      "question": "A-4. Prove that after deleting the perfect squares from the list of positive integers the number we find in the \\( indexval \\)th position is equal to \\( indexval+\\{\\sqrt{ } indexval\\} \\), where \\( \\{\\sqrt{ } indexval\\} \\) denotes the integer closest to \\( \\sqrt{ } indexval \\).",
      "solution": "A-4 To prove the formula by induction, it suffices to show that the difference \\( gapvalue=indexval+\\{\\sqrt{ } indexval\\}-(indexval-1+\\{\\sqrt{indexval-1}\\})=1 \\) or 2, with the value 2 occurring if and only if the number \\( indexval+\\{\\sqrt{indexval-1}\\} \\) is a perfect square. For convenience, let \\( \\{\\sqrt{indexval-1}\\}=nearint \\). Then of course \\( nearint-\\frac{1}{2}<\\sqrt{indexval-1}<nearint+\\frac{1}{2} \\) or better \\( nearint^{2}-nearint+\\frac{1}{4}<indexval-1 \\) \\( <nearint^{2}+nearint+\\frac{1}{4} \\). This gives \\( nearint^{2}+5 / 4<indexval+\\{\\sqrt{indexval-1}\\}<(nearint+1)^{2}+\\frac{1}{4} \\). Therefore the number \\( indexval+\\{\\sqrt{indexval-1}\\} \\) is a perfect square if and only if \\( indexval=(nearint+1)^{2}-nearint \\). However, then and only then \\( \\sqrt{ } indexval>nearint+\\frac{1}{2}>\\sqrt{indexval-1} \\). In other words then and only then \\( \\{\\sqrt{ } indexval\\}-\\{\\sqrt{indexval-1}\\}=1 \\), because this difference is never greater than 1."
    },
    "descriptive_long_confusing": {
      "map": {
        "n": "candlestick",
        "\\Delta": "heliograph",
        "q": "tangerine"
      },
      "question": "A-4. Prove that after deleting the perfect squares from the list of positive integers the number we find in the \\( candlestick \\)th position is equal to \\( candlestick+\\{\\sqrt{ } candlestick\\} \\), where \\( \\{\\sqrt{ } candlestick\\} \\) denotes the integer closest to \\( \\sqrt{ } candlestick \\).",
      "solution": "A-4 To prove the formula by induction, it suffices to show that the difference \\( heliograph=candlestick+\\{\\sqrt{ } candlestick\\}-(candlestick-1+\\{\\sqrt{candlestick-1}\\})=1 \\) or 2 , with the value 2 occurring if and only if the number \\( candlestick+\\{\\sqrt{candlestick-1}\\} \\) is a perfect square. For convenience, let \\( \\{\\sqrt{candlestick-1}\\}=tangerine \\). Then of course \\( tangerine-\\frac{1}{2}<\\sqrt{candlestick-1}<tangerine+\\frac{1}{2} \\) or better \\( tangerine^{2}-tangerine+\\frac{1}{4}<candlestick-1 \\) \\( <tangerine^{2}+tangerine+\\frac{1}{4} \\). This gives \\( tangerine^{2}+5 / 4<candlestick+\\{\\sqrt{candlestick-1}\\}<(tangerine+1)^{2}+\\frac{1}{4} \\). Therefore the number \\( candlestick+\\{\\sqrt{candlestick-1}\\} \\) is a perfect square if and only if \\( candlestick=(tangerine+1)^{2}-tangerine \\). However, then and only then \\( \\sqrt{ } candlestick>tangerine+\\frac{1}{2}>\\sqrt{candlestick-1} \\). In other words then and only then \\( \\{\\sqrt{ } candlestick\\}-\\{\\sqrt{candlestick-1}\\}=1 \\), because this difference is never greater than 1."
    },
    "descriptive_long_misleading": {
      "map": {
        "n": "continuousvalue",
        "\\\\Delta": "summation",
        "q": "irrationalvalue"
      },
      "question": "A-4. Prove that after deleting the perfect squares from the list of positive integers the number we find in the \\( continuousvalue \\)th position is equal to \\( continuousvalue+\\{\\sqrt{ } continuousvalue\\} \\), where \\( \\{\\sqrt{ } continuousvalue\\} \\) denotes the integer closest to \\( \\sqrt{ } continuousvalue \\).",
      "solution": "A-4 To prove the formula by induction, it suffices to show that the difference \\( summation=continuousvalue+\\{\\sqrt{ } continuousvalue\\}-(continuousvalue-1+\\{\\sqrt{continuousvalue-1}\\})=1 \\) or 2 , with the value 2 occurring if and only if the number \\( continuousvalue+\\{\\sqrt{continuousvalue-1}\\} \\) is a perfect square. For convenience, let \\( \\{\\sqrt{continuousvalue-1}\\}=irrationalvalue \\). Then of course \\( irrationalvalue-\\frac{1}{2}<\\sqrt{continuousvalue-1}<irrationalvalue+\\frac{1}{2} \\) or better \\( irrationalvalue^{2}-irrationalvalue+\\frac{1}{4}<continuousvalue-1 \\) \\( <irrationalvalue^{2}+irrationalvalue+\\frac{1}{4} \\). This gives \\( irrationalvalue^{2}+5 / 4<continuousvalue+\\{\\sqrt{continuousvalue-1}\\}<(irrationalvalue+1)^{2}+\\frac{1}{4} \\). Therefore the number \\( continuousvalue+\\{\\sqrt{continuousvalue-1}\\} \\) is a perfect square if and only if \\( continuousvalue=(irrationalvalue+1)^{2}-irrationalvalue \\). However, then and only then \\( \\sqrt{ } continuousvalue>irrationalvalue+\\frac{1}{2}>\\sqrt{continuousvalue-1} \\). In other words then and only then \\( \\{\\sqrt{ } continuousvalue\\}-\\{\\sqrt{continuousvalue-1}\\}=1 \\), because this difference is never greater than 1."
    },
    "garbled_string": {
      "map": {
        "n": "kaflgrum",
        "\\Delta": "zqplmndr",
        "q": "hsvqrtpo"
      },
      "question": "A-4. Prove that after deleting the perfect squares from the list of positive integers the number we find in the \\( kaflgrum \\)th position is equal to \\( kaflgrum+\\{\\sqrt{ } kaflgrum\\} \\), where \\( \\{\\sqrt{ } kaflgrum\\} \\) denotes the integer closest to \\( \\sqrt{ } kaflgrum \\).",
      "solution": "A-4 To prove the formula by induction, it suffices to show that the difference \\( zqplmndr=kaflgrum+\\{\\sqrt{ } kaflgrum\\}-(kaflgrum-1+\\{\\sqrt{kaflgrum-1}\\})=1 \\) or 2 , with the value 2 occurring if and only if the number \\( kaflgrum+\\{\\sqrt{kaflgrum-1}\\} \\) is a perfect square. For convenience, let \\( \\{\\sqrt{kaflgrum-1}\\}=hsvqrtpo \\). Then of course \\( hsvqrtpo-\\frac{1}{2}<\\sqrt{kaflgrum-1}<hsvqrtpo+\\frac{1}{2} \\) or better \\( hsvqrtpo^{2}-hsvqrtpo+\\frac{1}{4}<kaflgrum-1 \\) \\( <hsvqrtpo^{2}+hsvqrtpo+\\frac{1}{4} \\). This gives \\( hsvqrtpo^{2}+5 / 4<kaflgrum+\\{\\sqrt{kaflgrum-1}\\}<(hsvqrtpo+1)^{2}+\\frac{1}{4} \\). Therefore the number \\( kaflgrum+\\{\\sqrt{kaflgrum-1}\\} \\) is a perfect square if and only if \\( kaflgrum=(hsvqrtpo+1)^{2}-hsvqrtpo \\). However, then and only then \\( \\sqrt{ } kaflgrum>hsvqrtpo+\\frac{1}{2}>\\sqrt{kaflgrum-1} \\). In other words then and only then \\( \\{\\sqrt{ } kaflgrum\\}-\\{\\sqrt{kaflgrum-1}\\}=1 \\), because this difference is never greater than 1."
    },
    "kernel_variant": {
      "question": "Consider the ordinary list of positive integers  \n\\[\n1,2,3,4,5,6,\\dots\n\\]\nand erase every perfect square.  \nLet  \n\\[\na_1,a_2,a_3,\\dots\\qquad (a_1=2,\\;a_2=3,\\;a_3=5,\\dots)\n\\]\nbe the surviving integers, and introduce the auxiliary sequences  \n\n\\[\n\\Delta_n:=a_n-a_{\\,n-1}\\quad (n\\ge 2),\\qquad\nS_n:=\\sum_{k=1}^{n} a_k ,\\qquad\nP_n:=\\sum_{k=1}^{n}\\bigl(a_k-k\\bigr).\n\\]\n\n1.  (a)  Prove that $\\Delta_n\\in\\{1,2\\}$ for every $n\\ge 2$.  \n    (b)  Show that  \n    \\[\n        \\Delta_n=2\\;\\Longleftrightarrow\\;\n        n+\\lfloor\\sqrt n\\rfloor\\text{ is a perfect square}.\n    \\]\n\n2.  Establish the explicit formula  \n    \\[\n        \\boxed{\\;a_n=n+\\Bigl\\lfloor\\sqrt n+\\tfrac12\\Bigr\\rfloor\\;} .\\tag{$\\star$}\n    \\]\n\n3.  For the remainder of the problem write  \n    \\[\n        m:=\\Bigl\\lfloor\\sqrt n+\\tfrac12\\Bigr\\rfloor,\n        \\qquad\n        r:=n-m^{2}+m ,\n    \\]\n    so that $0\\le r\\le 2m$.  \n    Prove the identity  \n    \\[\n        \\boxed{\\;\n        S_n=\\frac{n(n+1)}{2}+\\frac{m(m-1)(2m-1)}{3}+m\\,r\n        \\;} .\\tag{$\\star\\star$}\n    \\]\n\n4.  Deduce an explicit expression for the “excess–sum’’\n    \\[\n        P_n\n        =S_n-\\frac{n(n+1)}{2}\n        =\\frac{m(m-1)(2m-1)}{3}+m\\,r\n        =mn-\\frac{m\\bigl(m^{2}-1\\bigr)}{3}.\n    \\]\n\n5.  Finally, prove the asymptotic expansion  \n    \\[\n        \\boxed{\\;\n        S_n=\\frac12\\,n^{2}+\\frac23\\,n^{3/2}+O(n)\n        \\;} .\\tag{$\\star\\star\\star$}\n    \\]\n    (and therefore $P_n=\\frac23\\,n^{3/2}+O(n)$).\n\nA complete proof of every item is required.",
      "solution": "Throughout we set  \n\\[\n\\rho(N):=N-\\lfloor\\sqrt N\\rfloor ,\n\\]\nthe number of survivors not exceeding $N$; thus $\\rho(a_n)=n$ and $a_n$ is\nthe least integer whose rank equals $n$.\n\n------------------------------------------------------------------\n1.  Properties of the successive gaps\n------------------------------------------------------------------\n\n(a)  Between two consecutive squares $t^{2}$ and $(t+1)^{2}$ there are\nexactly $2t$ nonsquares, so at most one deleted integer can lie between two\nsurvivors.  Therefore $\\Delta_n\\in\\{1,2\\}$.\n\n(b)  The gap $\\Delta_n$ equals $2$ precisely when the missing integer\n$a_n-1$ is a perfect square.  Put $a_n-1=t^{2}$.  Because $t^{2}$ itself\nis deleted, its rank satisfies  \n\\[\n\\rho(t^{2})=\\rho(a_n-1)=n-1 .\n\\]\nBut $\\rho(t^{2})=t^{2}-t$, hence  \n\\[\nn-1=t^{2}-t\\qquad\\Longrightarrow\\qquad n=t^{2}-t+1. \\tag{1}\n\\]\nSince $t-1<\\sqrt n<t$, we have $\\lfloor\\sqrt n\\rfloor=t-1$, and consequently  \n\\[\nn+\\lfloor\\sqrt n\\rfloor\n   =(t^{2}-t+1)+(t-1)=t^{2},\n\\]\na perfect square.  \n\nConversely, if $n+\\lfloor\\sqrt n\\rfloor=v^{2}$ is a square, write\n$v=\\lfloor\\sqrt n\\rfloor+u$ with $u\\in\\{0,1\\}$.  Reversing the above\nsteps shows that $n$ satisfies (1), hence $a_n-1=v^{2}$ is removed and\n$\\Delta_n=2$.  The equivalence follows.\n\n------------------------------------------------------------------\n2.  Closed formula for $a_n$ \\;($\\star$)\n------------------------------------------------------------------\n\nLet  \n\\[\nm:=\\Bigl\\lfloor\\sqrt n+\\tfrac12\\Bigr\\rfloor,\\qquad x:=n+m.\n\\]\nBecause  \n\\[\nm^{2}-m+\\tfrac14<n<m^{2}+m+\\tfrac14,\n\\]\nwe have $m^{2}<x<m^{2}+2m+1=(m+1)^{2}$, so that\n$\\lfloor\\sqrt x\\rfloor=m$ and $\\rho(x)=x-m=n$.  Since $a_n$ is the minimal\ninteger of rank $n$, it must equal $x$; thus $a_n=n+m$, which is\nprecisely formula ($\\star$).\n\n------------------------------------------------------------------\n3.  A closed form for $S_n$ \\;($\\star\\star$)\n------------------------------------------------------------------\n\nFor every integer $t\\ge 1$ set  \n\\[\nI_t:=\\{k\\in\\mathbb N:\\;t^{2}-t+1\\le k\\le t^{2}+t\\},\\qquad |I_t|=2t .\\tag{2}\n\\]\nOn each interval $I_t$ the quantity\n$\\bigl\\lfloor\\sqrt k+\\tfrac12\\bigr\\rfloor$ is constant and equal to $t$.\n\nWith $m$ as above, we have $n=m^{2}-m+r$ $(0\\le r\\le 2m)$, so that\n$\\{1,\\dots ,n\\}$ consists of  \n\n(i) the full intervals $I_1,\\dots,I_{m-1}$, and  \n(ii) the first $r$ elements of $I_m$.\n\nHence\n\\[\n\\sum_{k=1}^{n}\\Bigl\\lfloor\\sqrt k+\\tfrac12\\Bigr\\rfloor\n   =\\sum_{t=1}^{m-1}(2t)\\cdot t+m\\,r\n   =2\\sum_{t=1}^{m-1}t^{2}+m\\,r\n   =\\frac{m(m-1)(2m-1)}{3}+m\\,r .\\tag{3}\n\\]\nAdding $\\sum_{k=1}^{n}k=n(n+1)/2$ gives ($\\star\\star$).\n\n------------------------------------------------------------------\n4.  The polynomial $P_n$\n------------------------------------------------------------------\n\nBy definition\n\\[\nP_n=S_n-\\frac{n(n+1)}{2}\n     =\\frac{m(m-1)(2m-1)}{3}+m\\,r .\n\\]\nSubstituting $r=n-m^{2}+m$ and simplifying yields  \n\\[\nP_n=mn-\\frac{m\\bigl(m^{2}-1\\bigr)}{3}.\n\\]\n\n------------------------------------------------------------------\n5.  Asymptotics of $S_n$ \\;($\\star\\star\\star$)\n------------------------------------------------------------------\n\nWrite $n=m^{2}+\\theta$ with $-m<\\theta\\le m$ (so that\n$|\\theta|\\le\\sqrt n$) and recall $r=\\theta+m$.  Formula ($\\star\\star$)\nthen becomes  \n\\[\n\\begin{aligned}\nS_n\n&=\\frac{n(n+1)}{2}\n  +\\frac{m(m-1)(2m-1)}{3}+m\\,r\\\\[2mm]\n&=\\frac{n(n+1)}{2}\n  +\\frac{2}{3}m^{3}-m^{2}+\\frac13 m\n  +m(\\theta+m).\n\\end{aligned}\n\\]\nExpanding $n(n+1)/2$ and collecting like terms we obtain the **exact**\nidentity  \n\\[\n\\boxed{\\;\nS_n\n=\\frac{1}{2}n^{2}\n +\\frac{2}{3}m^{3}\n +\\frac{1}{2}m^{2}\n +\\theta\\Bigl(m+\\frac12\\Bigr)\n +\\frac{1}{3}m\n\\;} .\\tag{4}\n\\]\n\nSince $m=\\sqrt n+O(1)$ and $|\\theta|\\le\\sqrt n$, each of the last\nthree summands in (4) is $O(n)$:\n\\[\n\\frac{2}{3}m^{3}=\\frac{2}{3}n^{3/2}+O(n),\\qquad\n\\frac{1}{2}m^{2}=O(n),\\qquad\n\\theta\\Bigl(m+\\frac12\\Bigr)+\\frac{1}{3}m=O(n).\n\\]\nTherefore  \n\\[\nS_n=\\frac12\\,n^{2}+\\frac23\\,n^{3/2}+O(n),\n\\]\nestablishing ($\\star\\star\\star$).  Subtracting $n(n+1)/2$ finally yields  \n\\[\nP_n=\\frac23\\,n^{3/2}+O(n).\n\\]\n\n------------------------------------------------------------------\nAll five items are now proved rigorously, and the exact term\n$\\tfrac13 m$—previously omitted—has been reinstated in formula (4),\nmaking the derivation completely correct.",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.566091",
        "was_fixed": false,
        "difficulty_analysis": "Compared with the original kernel problem, which only required the\nsingle explicit description aₙ = n+⌊√n+½⌋, the enhanced variant asks\nfor\n\n• an analysis of the successive gaps (still needed, but used purely as\n  a tool);\n\n• a derivation of a *second* non-trivial closed formula (★★) for the\n  cumulative sum Sₙ;\n\n• the extraction of a cubic expression for Pₙ;\n\n• a complete asymptotic expansion of Sₙ up to order n, including the\n  exact leading constant.\n\nHandling the intervals (5) and summing both first- and second-degree\npolynomials over a variable number of blocks forces the solver to use\nblock-partition arguments, weighted summations of squares, and careful\nbook-keeping of the remainder r.  The asymptotic part demands an\nadditional layer of estimation and big-O manipulation that was totally\nabsent from the original problem.  Each added requirement introduces a\nnew conceptual ingredient (finite calculus, polynomial summation,\nasymptotic analysis), making the enhanced task substantially more\ntechnically involved and decidedly harder than either the original or\nthe current kernel variant."
      }
    },
    "original_kernel_variant": {
      "question": "Consider the ordinary list of positive integers  \n\\[\n1,2,3,4,5,6,\\dots\n\\]\nand erase every perfect square.  \nLet  \n\\[\na_1,a_2,a_3,\\dots\\qquad (a_1=2,\\;a_2=3,\\;a_3=5,\\dots)\n\\]\nbe the surviving integers, and introduce the auxiliary sequences  \n\n\\[\n\\Delta_n:=a_n-a_{\\,n-1}\\quad (n\\ge 2),\\qquad\nS_n:=\\sum_{k=1}^{n} a_k ,\\qquad\nP_n:=\\sum_{k=1}^{n}\\bigl(a_k-k\\bigr).\n\\]\n\n1.  (a)  Prove that $\\Delta_n\\in\\{1,2\\}$ for every $n\\ge 2$.  \n    (b)  Show that  \n    \\[\n        \\Delta_n=2\\;\\Longleftrightarrow\\;\n        n+\\lfloor\\sqrt n\\rfloor\\text{ is a perfect square}.\n    \\]\n\n2.  Establish the explicit formula  \n    \\[\n        \\boxed{\\;a_n=n+\\Bigl\\lfloor\\sqrt n+\\tfrac12\\Bigr\\rfloor\\;} .\\tag{$\\star$}\n    \\]\n\n3.  For the remainder of the problem write  \n    \\[\n        m:=\\Bigl\\lfloor\\sqrt n+\\tfrac12\\Bigr\\rfloor,\n        \\qquad\n        r:=n-m^{2}+m ,\n    \\]\n    so that $0\\le r\\le 2m$.  \n    Prove the identity  \n    \\[\n        \\boxed{\\;\n        S_n=\\frac{n(n+1)}{2}+\\frac{m(m-1)(2m-1)}{3}+m\\,r\n        \\;} .\\tag{$\\star\\star$}\n    \\]\n\n4.  Deduce an explicit expression for the “excess–sum’’\n    \\[\n        P_n\n        =S_n-\\frac{n(n+1)}{2}\n        =\\frac{m(m-1)(2m-1)}{3}+m\\,r\n        =mn-\\frac{m\\bigl(m^{2}-1\\bigr)}{3}.\n    \\]\n\n5.  Finally, prove the asymptotic expansion  \n    \\[\n        \\boxed{\\;\n        S_n=\\frac12\\,n^{2}+\\frac23\\,n^{3/2}+O(n)\n        \\;} .\\tag{$\\star\\star\\star$}\n    \\]\n    (and therefore $P_n=\\frac23\\,n^{3/2}+O(n)$).\n\nA complete proof of every item is required.",
      "solution": "Throughout we set  \n\\[\n\\rho(N):=N-\\lfloor\\sqrt N\\rfloor ,\n\\]\nthe number of survivors not exceeding $N$; thus $\\rho(a_n)=n$ and $a_n$ is\nthe least integer whose rank equals $n$.\n\n------------------------------------------------------------------\n1.  Properties of the successive gaps\n------------------------------------------------------------------\n\n(a)  Between two consecutive squares $t^{2}$ and $(t+1)^{2}$ there are\nexactly $2t$ nonsquares, so at most one deleted integer can lie between two\nsurvivors.  Therefore $\\Delta_n\\in\\{1,2\\}$.\n\n(b)  The gap $\\Delta_n$ equals $2$ precisely when the missing integer\n$a_n-1$ is a perfect square.  Put $a_n-1=t^{2}$.  Because $t^{2}$ itself\nis deleted, its rank satisfies  \n\\[\n\\rho(t^{2})=\\rho(a_n-1)=n-1 .\n\\]\nBut $\\rho(t^{2})=t^{2}-t$, hence  \n\\[\nn-1=t^{2}-t\\qquad\\Longrightarrow\\qquad n=t^{2}-t+1. \\tag{1}\n\\]\nSince $t-1<\\sqrt n<t$, we have $\\lfloor\\sqrt n\\rfloor=t-1$, and consequently  \n\\[\nn+\\lfloor\\sqrt n\\rfloor\n   =(t^{2}-t+1)+(t-1)=t^{2},\n\\]\na perfect square.  \n\nConversely, if $n+\\lfloor\\sqrt n\\rfloor=v^{2}$ is a square, write\n$v=\\lfloor\\sqrt n\\rfloor+u$ with $u\\in\\{0,1\\}$.  Reversing the above\nsteps shows that $n$ satisfies (1), hence $a_n-1=v^{2}$ is removed and\n$\\Delta_n=2$.  The equivalence follows.\n\n------------------------------------------------------------------\n2.  Closed formula for $a_n$ \\;($\\star$)\n------------------------------------------------------------------\n\nLet  \n\\[\nm:=\\Bigl\\lfloor\\sqrt n+\\tfrac12\\Bigr\\rfloor,\\qquad x:=n+m.\n\\]\nBecause  \n\\[\nm^{2}-m+\\tfrac14<n<m^{2}+m+\\tfrac14,\n\\]\nwe have $m^{2}<x<m^{2}+2m+1=(m+1)^{2}$, so that\n$\\lfloor\\sqrt x\\rfloor=m$ and $\\rho(x)=x-m=n$.  Since $a_n$ is the minimal\ninteger of rank $n$, it must equal $x$; thus $a_n=n+m$, which is\nprecisely formula ($\\star$).\n\n------------------------------------------------------------------\n3.  A closed form for $S_n$ \\;($\\star\\star$)\n------------------------------------------------------------------\n\nFor every integer $t\\ge 1$ set  \n\\[\nI_t:=\\{k\\in\\mathbb N:\\;t^{2}-t+1\\le k\\le t^{2}+t\\},\\qquad |I_t|=2t .\\tag{2}\n\\]\nOn each interval $I_t$ the quantity\n$\\bigl\\lfloor\\sqrt k+\\tfrac12\\bigr\\rfloor$ is constant and equal to $t$.\n\nWith $m$ as above, we have $n=m^{2}-m+r$ $(0\\le r\\le 2m)$, so that\n$\\{1,\\dots ,n\\}$ consists of  \n\n(i) the full intervals $I_1,\\dots,I_{m-1}$, and  \n(ii) the first $r$ elements of $I_m$.\n\nHence\n\\[\n\\sum_{k=1}^{n}\\Bigl\\lfloor\\sqrt k+\\tfrac12\\Bigr\\rfloor\n   =\\sum_{t=1}^{m-1}(2t)\\cdot t+m\\,r\n   =2\\sum_{t=1}^{m-1}t^{2}+m\\,r\n   =\\frac{m(m-1)(2m-1)}{3}+m\\,r .\\tag{3}\n\\]\nAdding $\\sum_{k=1}^{n}k=n(n+1)/2$ gives ($\\star\\star$).\n\n------------------------------------------------------------------\n4.  The polynomial $P_n$\n------------------------------------------------------------------\n\nBy definition\n\\[\nP_n=S_n-\\frac{n(n+1)}{2}\n     =\\frac{m(m-1)(2m-1)}{3}+m\\,r .\n\\]\nSubstituting $r=n-m^{2}+m$ and simplifying yields  \n\\[\nP_n=mn-\\frac{m\\bigl(m^{2}-1\\bigr)}{3}.\n\\]\n\n------------------------------------------------------------------\n5.  Asymptotics of $S_n$ \\;($\\star\\star\\star$)\n------------------------------------------------------------------\n\nWrite $n=m^{2}+\\theta$ with $-m<\\theta\\le m$ (so that\n$|\\theta|\\le\\sqrt n$) and recall $r=\\theta+m$.  Formula ($\\star\\star$)\nthen becomes  \n\\[\n\\begin{aligned}\nS_n\n&=\\frac{n(n+1)}{2}\n  +\\frac{m(m-1)(2m-1)}{3}+m\\,r\\\\[2mm]\n&=\\frac{n(n+1)}{2}\n  +\\frac{2}{3}m^{3}-m^{2}+\\frac13 m\n  +m(\\theta+m).\n\\end{aligned}\n\\]\nExpanding $n(n+1)/2$ and collecting like terms we obtain the **exact**\nidentity  \n\\[\n\\boxed{\\;\nS_n\n=\\frac{1}{2}n^{2}\n +\\frac{2}{3}m^{3}\n +\\frac{1}{2}m^{2}\n +\\theta\\Bigl(m+\\frac12\\Bigr)\n +\\frac{1}{3}m\n\\;} .\\tag{4}\n\\]\n\nSince $m=\\sqrt n+O(1)$ and $|\\theta|\\le\\sqrt n$, each of the last\nthree summands in (4) is $O(n)$:\n\\[\n\\frac{2}{3}m^{3}=\\frac{2}{3}n^{3/2}+O(n),\\qquad\n\\frac{1}{2}m^{2}=O(n),\\qquad\n\\theta\\Bigl(m+\\frac12\\Bigr)+\\frac{1}{3}m=O(n).\n\\]\nTherefore  \n\\[\nS_n=\\frac12\\,n^{2}+\\frac23\\,n^{3/2}+O(n),\n\\]\nestablishing ($\\star\\star\\star$).  Subtracting $n(n+1)/2$ finally yields  \n\\[\nP_n=\\frac23\\,n^{3/2}+O(n).\n\\]\n\n------------------------------------------------------------------\nAll five items are now proved rigorously, and the exact term\n$\\tfrac13 m$—previously omitted—has been reinstated in formula (4),\nmaking the derivation completely correct.",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.462216",
        "was_fixed": false,
        "difficulty_analysis": "Compared with the original kernel problem, which only required the\nsingle explicit description aₙ = n+⌊√n+½⌋, the enhanced variant asks\nfor\n\n• an analysis of the successive gaps (still needed, but used purely as\n  a tool);\n\n• a derivation of a *second* non-trivial closed formula (★★) for the\n  cumulative sum Sₙ;\n\n• the extraction of a cubic expression for Pₙ;\n\n• a complete asymptotic expansion of Sₙ up to order n, including the\n  exact leading constant.\n\nHandling the intervals (5) and summing both first- and second-degree\npolynomials over a variable number of blocks forces the solver to use\nblock-partition arguments, weighted summations of squares, and careful\nbook-keeping of the remainder r.  The asymptotic part demands an\nadditional layer of estimation and big-O manipulation that was totally\nabsent from the original problem.  Each added requirement introduces a\nnew conceptual ingredient (finite calculus, polynomial summation,\nasymptotic analysis), making the enhanced task substantially more\ntechnically involved and decidedly harder than either the original or\nthe current kernel variant."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}