{
  "index": "1998-B-5",
  "type": "NT",
  "tag": [
    "NT",
    "ALG"
  ],
  "difficulty": "",
  "question": "Let $N$ be the positive integer with 1998 decimal digits, all of them 1;\nthat is,\n\\[N=1111\\cdots 11.\\]\nFind the thousandth digit after the decimal point of $\\sqrt N$.",
  "solution": "Write $N=(10^{1998}-1)/9$.  Then\n\\begin{align*}\n\\sqrt{N} &=\\frac{10^{999}}{3}\\sqrt{1-10^{-1998}} \\\\\n&=\\frac{10^{999}}{3}\n(1-\\frac{1}{2}10^{-1998} + r),\n\\end{align*}\nwhere $r<10^{-2000}$.  Now the digits after the decimal point of\n$10^{999}/3$ are given by $.3333\\ldots$, while\nthe digits after the decimal point\nof $\\frac{1}{6}10^{-999}$ are given by $.00000\\ldots 1666666\\ldots$.\nIt follows that the first 1000 digits of $\\sqrt N$ are given\nby $.33333\\ldots 3331$;  in particular, the thousandth digit is $1$.",
  "vars": [
    "N",
    "r"
  ],
  "params": [],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "N": "allonesint",
        "r": "errorrema"
      },
      "question": "Let $allonesint$ be the positive integer with 1998 decimal digits, all of them 1;\nthat is,\n\\[allonesint=1111\\cdots 11.\\]\nFind the thousandth digit after the decimal point of $\\sqrt{allonesint}$.",
      "solution": "Write $allonesint=(10^{1998}-1)/9$.  Then\n\\begin{align*}\n\\sqrt{allonesint} &=\\frac{10^{999}}{3}\\sqrt{1-10^{-1998}} \\\\\n&=\\frac{10^{999}}{3}\n(1-\\frac{1}{2}10^{-1998} + errorrema),\n\\end{align*}\nwhere $errorrema<10^{-2000}$.  Now the digits after the decimal point of\n$10^{999}/3$ are given by $.3333\\ldots$, while\nthe digits after the decimal point\nof $\\frac{1}{6}10^{-999}$ are given by $.00000\\ldots 1666666\\ldots$.\nIt follows that the first 1000 digits of $\\sqrt{allonesint}$ are given\nby $.33333\\ldots 3331$;  in particular, the thousandth digit is $1$. "
    },
    "descriptive_long_confusing": {
      "map": {
        "N": "cylinder",
        "r": "waterfall"
      },
      "question": "Let $cylinder$ be the positive integer with 1998 decimal digits, all of them 1; that is,\n\\[\ncylinder=1111\\cdots 11.\n\\]\nFind the thousandth digit after the decimal point of $\\sqrt cylinder$.",
      "solution": "Write $cylinder=(10^{1998}-1)/9$.  Then\n\\begin{align*}\n\\sqrt{cylinder} &=\\frac{10^{999}}{3}\\sqrt{1-10^{-1998}} \\\\\n&=\\frac{10^{999}}{3}\n(1-\\frac{1}{2}10^{-1998} + waterfall),\n\\end{align*}\nwhere $waterfall<10^{-2000}$.  Now the digits after the decimal point of\n$10^{999}/3$ are given by $.3333\\ldots$, while\nthe digits after the decimal point\nof $\\frac{1}{6}10^{-999}$ are given by $.00000\\ldots 1666666\\ldots$.\nIt follows that the first 1000 digits of $\\sqrt cylinder$ are given\nby $.33333\\ldots 3331$;  in particular, the thousandth digit is $1$.} 复制输出ение 重庆时时彩彩♪♪♪  🛑  This answer seems to show extra characters at end? Actually ends after brace '}'. Ensure no extra after closing }."
    },
    "descriptive_long_misleading": {
      "map": {
        "N": "emptiness",
        "r": "gargantua"
      },
      "question": "Let $emptiness$ be the positive integer with 1998 decimal digits, all of them 1; that is,\n\\[emptiness=1111\\cdots 11.\\]\nFind the thousandth digit after the decimal point of $\\sqrt emptiness$.",
      "solution": "Write $emptiness=(10^{1998}-1)/9$.  Then\n\\begin{align*}\n\\sqrt{emptiness} &=\\frac{10^{999}}{3}\\sqrt{1-10^{-1998}} \\\\\n&=\\frac{10^{999}}{3}\n(1-\\frac{1}{2}10^{-1998} + gargantua),\n\\end{align*}\nwhere $gargantua<10^{-2000}$.  Now the digits after the decimal point of\n$10^{999}/3$ are given by $.3333\\ldots$, while\nthe digits after the decimal point\nof $\\frac{1}{6}10^{-999}$ are given by $.00000\\ldots 1666666\\ldots$.\nIt follows that the first 1000 digits of $\\sqrt emptiness$ are given\nby $.33333\\ldots 3331$;  in particular, the thousandth digit is $1$. "
    },
    "garbled_string": {
      "map": {
        "N": "qzxwvtnp",
        "r": "hjgrksla"
      },
      "question": "Let $qzxwvtnp$ be the positive integer with 1998 decimal digits, all of them 1; that is,\n\\[qzxwvtnp=1111\\cdots 11.\\]\nFind the thousandth digit after the decimal point of $\\sqrt{qzxwvtnp}$.",
      "solution": "Write $qzxwvtnp=(10^{1998}-1)/9$.  Then\n\\begin{align*}\n\\sqrt{qzxwvtnp} &=\\frac{10^{999}}{3}\\sqrt{1-10^{-1998}} \\\\\n&=\\frac{10^{999}}{3}\n(1-\\frac{1}{2}10^{-1998} + hjgrksla),\n\\end{align*}\nwhere $hjgrksla<10^{-2000}$.  Now the digits after the decimal point of\n$10^{999}/3$ are given by $.3333\\ldots$, while\nthe digits after the decimal point\nof $\\frac{1}{6}10^{-999}$ are given by $.00000\\ldots 1666666\\ldots$.\nIt follows that the first 1000 digits of $\\sqrt{qzxwvtnp}$ are given\nby $.33333\\ldots 3331$;  in particular, the thousandth digit is $1$. "
    },
    "kernel_variant": {
      "question": "Let $N$ be the positive integer whose decimal representation consists of  \n\n\\[\n7\\,204\\ \\text{consecutive }1\\text{'s},\\qquad  \nN \\;=\\; \\underbrace{111\\ldots 1}_{7\\,204\\text{ digits}}\n\\;=\\;\\frac{10^{7\\,204}-1}{9}.\n\\]\n\nDefine the real number  \n\n\\[\nT \\;=\\;\\sqrt{N}\\;+\\;\\frac1{\\sqrt{N}}\\;+\\;\\sqrt{\\frac{N}{9}}.\n\\]\n\nDetermine the $3\\,603$-rd digit after the decimal point of $T$ (equivalently, the digit in the $10^{-3\\,603}$–place).\n\n——————————————————————————————————————",
      "solution": "Throughout we put  \n\n\\[\n2k=7\\,204 \\;\\Longrightarrow\\; k=3\\,602,\n\\]\nso the requested digit is the one in the $10^{-(k+1)}$–place.\n\nStep 1.  A two–term expansion for $\\sqrt{N}$  \nSince $N=\\dfrac{10^{2k}-1}{9}$,\n\n\\[\n\\sqrt{N}= \\frac{10^{k}}{3}\\,\\sqrt{1-10^{-2k}}.\n\\]\n\nWith $\\varepsilon:=10^{-2k}$ ($|\\varepsilon|\\ll1$) and  \n$\\sqrt{1-\\varepsilon}=1-\\tfrac12\\varepsilon-\\tfrac18\\varepsilon^{2}-\\cdots$,\n\n\\[\n\\boxed{\\;\n\\sqrt{N}= \\frac{10^{k}}{3}-\\frac16\\,10^{-k}+R_{1}},\\qquad\n|R_{1}|<10^{-3k}.\n\\]\n\nStep 2.  An expansion for $1/\\sqrt{N}$  \nWrite $\\sqrt{N}=A(1-\\eta)$ where  \n\n\\[\nA:=\\frac{10^{k}}{3},\\qquad\n\\eta:=\\frac{ \\frac16\\,10^{-k}-R_{1}}{A}=\\frac12\\,10^{-2k}+O(10^{-4k}).\n\\]\n\nBecause $k=3\\,602$,  \n\n\\[\n|\\eta|<10^{-7\\,204}.\n\\]\n\nHence  \n\n\\[\n\\frac1{\\sqrt{N}}\n=\\frac1{A}\\,(1-\\eta)^{-1}\n=3\\,10^{-k}\\bigl(1+\\eta+\\eta^{2}+\\cdots\\bigr)\n=3\\,10^{-k}+R_{2},\n\\qquad |R_{2}|<2\\,10^{-3k}.\n\\]\n\nStep 3.  $\\sqrt{N/9}$  \nSince $\\sqrt{N/9}=\\dfrac13\\sqrt{N}$, from Step 1\n\n\\[\n\\boxed{\\;\n\\sqrt{\\frac{N}{9}}=\\frac{10^{k}}{9}-\\frac1{18}\\,10^{-k}+R_{3}},\\qquad\n|R_{3}|<10^{-3k}.\n\\]\n\nStep 4.  Assemble $T$  \nAdding the three boxed formulas gives  \n\n\\[\nT=\\bigl(\\tfrac{10^{k}}{3}+\\tfrac{10^{k}}{9}\\bigr)\n+\\Bigl[\\,3\\,10^{-k}-\\tfrac16\\,10^{-k}-\\tfrac1{18}\\,10^{-k}\\Bigr]\n+(R_{1}+R_{2}+R_{3}).\n\\]\n\nThe bracket equals $3-\\tfrac16-\\tfrac1{18}=25/9$, so\n\n\\[\n\\boxed{\\;\nT=\\frac{4}{9}\\,10^{k}+\\frac{25}{9}\\,10^{-k}+R},\\qquad\n|R|<4\\,10^{-3k}.\n\\]\n\nStep 5.  Separate the dominant integer and fractional pieces  \n\n(a)  $\\dfrac{4}{9}\\,10^{k}$.  \nBecause $10^{k}=9M+1$ for $M:=\\dfrac{10^{k}-1}{9}=11\\ldots1$ ($k$ ones),\n\n\\[\n\\frac{4}{9}\\,10^{k}=4M+\\frac49.\n\\]\n\nHence its integer part is the $k$-digit string $44\\ldots4$ and its fractional\npart is $0.\\overline{4}$.\n\n(b)  $F:=\\dfrac{25}{9}\\,10^{-k}$.  \nSince $\\dfrac{25}{9}=2.\\overline{7}$, multiplying by $10^{-k}$ places  \n\na digit $2$ in the $10^{-k}$ position followed by an infinite string of $7$’s.\n\nStep 6.  Add the two fractional parts and trace the carries  \nLet  \n\n\\[\nS:=\\frac49+\\frac{25}{9}\\,10^{-k}.\n\\]\n\nMultiply by $10^{k}$:\n\n\\[\n10^{k}S=\\frac{4\\cdot10^{k}+25}{9}\n       =\\frac{4(9M+1)+25}{9}\n       =4M+\\frac{4+25}{9}\n       =4M+\\frac{29}{9}\n       =4M+3+\\frac{2}{9}.\n\\]\n\nKey observations  \n\n• The integer $4M$ ends in a $4$, so $4M+3$ ends in a $7$.  \n  Therefore the digit in the $10^{-k}$ place of $S$ (the units digit of\n  $10^{k}S$) is $7$.\n\n• The fractional part of $10^{k}S$ is $\\tfrac29=0.\\overline{2}$, so the digit\n  in the $10^{-(k+1)}$ place of $S$ is $2$.\n\nStep 7.  Effect of the remainder $R$  \nBecause $|R|<4\\,10^{-3k}$ the earliest decimal place it can influence is\n$10^{-3k}$, far to the right of $10^{-(k+1)}$ (since $3k=10\\,806\\gg k+1=3\\,603$).\nHence the digits found in Step 6 are final.\n\nStep 8.  Conclusion  \n\nImmediately after its decimal point $T$ begins  \n\n\\[\nT=0.444\\ldots4\\,7\\,2\\,2\\,2\\ldots,\n\\qquad\n\\begin{array}{c}\n\\!\\!\\!\\!\\!\\uparrow\\\\[-2pt]\n10^{-k}\n\\end{array}\n\\qquad\n\\begin{array}{c}\n\\!\\!\\!\\!\\!\\uparrow\\\\[-2pt]\n10^{-(k+1)}\n\\end{array}\n\\]\n\nso the $3\\,603$-rd digit after the decimal point—the $10^{-(k+1)}$ digit—is  \n\n\\[\n\\boxed{2}.\n\\]\n\n——————————————————————————————————————",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.762931",
        "was_fixed": false,
        "difficulty_analysis": "1. Multiple interacting terms.  Unlike the original single-root question, this problem forces the solver to handle √N, its reciprocal, and a scalar multiple √(N/9) simultaneously; each behaves on a different magnitude scale, so careful bookkeeping of powers of 10 is essential.\n\n2. Position-sensitive carries without cancellation.  \nThe digit sought lies precisely where three separate series overlap:  \n • the repeating 4’s coming from 10^{k}/3 + 10^{k}/9,  \n • an isolated “2” generated by (25/9)·10^{-k}, and  \n • possible carries produced by the infinite tails 0.\\overline{7} and the error bounds.  \nDeciding that no carry occurs and that later tails cannot change the target digit demands explicit upper bounds on all neglected terms well beyond those needed in the original problem.\n\n3. Higher-order error control.  \nBounding R₁, R₂, R₃ simultaneously and proving they are far too small to influence the 10^{-(k+1)} digit requires two binomial/reciprocal expansions and a uniform inequality, not just a single first-order estimate.\n\n4. Magnitude management across 7 200 decimal places.  \nKeeping track of which term first becomes visible in which decimal position, then proving that nothing sooner or larger disturbs it, is conceptually and computationally heavier than the pattern-matching argument that solved the original.\n\nIn short, the problem demands a longer chain of approximations, accurate error bounding, and subtle carry analysis across three interacting series—making it markedly more intricate than both the original and the current kernel variant."
      }
    },
    "original_kernel_variant": {
      "question": "Let N be the positive integer whose decimal representation consists of  \n\n  7 204 consecutive 1’s; that is  \n\n   N = 111…1 (7 204 digits) = (10^{7 204} – 1)/9.\n\nDefine  \n\n   T = √N + 1/√N + √(N/9).\n\nDetermine the 3 603-rd digit after the decimal point of T (that is, the digit in the 10^{-3 603}-place).\n\n--------------------------------------------------------------------",
      "solution": "Throughout we write  \n\n  2k = 7 204 ⇒ k = 3 602.\n\nThus the requested digit is the one in the 10^{-(k+1)}–place.\n\nStep 1.  A two–term expansion for √N  \nSince N = (10^{2k} – 1)/9,\n\n √N = 10^{k}/3·√(1 – 10^{-2k}).                                       (1)\n\nPut ε := 10^{-2k} (|ε| ≪ 1).  Using √(1 – ε)=1 – ½ε – ⅛ε² – …,\n\n √N = 10^{k}/3 – (1/6)·10^{-k} + R₁,                                 (2)\n\nwith |R₁| < 10^{-3k}.  (The first neglected term is O(ε²10^{k}).)\n\nStep 2.  An expansion for 1/√N with a sharp error bound  \nWrite √N = A(1 – η) where  \n\n A := 10^{k}/3, η := [(1/6)·10^{-k} – R₁]/A  \n    = ½ε + O(ε²) = ½·10^{-2k} + O(10^{-4k}).                     (3)\n\nBecause k = 3 602, |η| < 0.6·10^{-2k} < 10^{-7 200}.  Hence\n\n 1/√N = A^{-1}(1 – η)^{-1}  \n   = 3·10^{-k}(1 + η + η² + η³+⋯).                              (4)\n\nThe tail R₂ := 3·10^{-k}(η + η²+⋯) satisfies  \n\n |R₂| ≤ 3·10^{-k}·|η|/(1 – |η|)  \n   < 3·10^{-k}·0.6·10^{-2k}·1.001                                  (5)\n   < 2·10^{-3k}.                                                    (6)\n\nThus  \n\n 1/√N = 3·10^{-k} + R₂, |R₂|<2·10^{-3k}.                            (7)\n\nStep 3.  The term √(N/9) = (1/3)√N   \nFrom (2),\n\n √(N/9) = 10^{k}/9 – (1/18)·10^{-k} + R₃, |R₃|<10^{-3k}.            (8)\n\nStep 4.  Assemble T  \nAdding (2), (7) and (8) gives  \n\nT = (10^{k}/3 + 10^{k}/9)                        (integer–type terms)  \n + [3·10^{-k} – (1/6)10^{-k} – (1/18)10^{-k}]                       (9)\n + (R₁+R₂+R₃).\n\nThe bracket equals  \n\n 3 – 1/6 – 1/18 = 50/18 = 25/9.\n\nHence  \n\n T = (10^{k}/3 + 10^{k}/9) + F + R,                                (10)\n\nwith  \n\n F := (25/9)·10^{-k}, |R| := |R₁+R₂+R₃| < (1+2+1)·10^{-3k} = 4·10^{-3k}. (11)\n\nStep 5.  The two principal fractional pieces\n\n(a) 10^{k}/3 + 10^{k}/9.  \n • Its integer part ends in k identical 4’s.  \n • Its fractional part is 0.\\overline{4}.\n\n(b) F = (25/9)·10^{-k}.  \n Because 25/9 = 2.\\overline{7}, multiplying by 10^{-k} puts  \n\n   “2” in the 10^{-k} place and an infinite string of 7’s beginning in the 10^{-(k+1)} place:\n\n  F = 0.00…0 2 7777 7777 …                                         (12)  \n         ↑  \n      position k\n\nStep 6.  Adding 0.\\overline{4} and F — carry propagation  \nLet d = 10^{-k}.  Then\n\nS := 0.\\overline{4} + 2d + (7/9)d  \n = 4/9 + 25d/9.                                                     (13)\n\nMultiply S by 10^{k-1} so that its 10^{-k} digit becomes the units digit:\n\n10^{k-1}S = 10^{k-1}·4/9 + 25/9·10^{k-1}d  \n     = (4m+4/9) + 25/90    ( because 10^{k-1}=9m+1).            (14)\n\nIts fractional part is\n\n 4/9 + 25/90 = 8/18 + 5/18 = 13/18 = 0.722 222….                  (15)\n\nMultiplying that fractional part once more by 10,\n\n10·(0.722 222…) = 7 + 2/9.                                          (16)\n\nInterpretation  \n\n • The digit in the 10^{-k} place (units digit of 10^{k}S) is 7.  \n • The carry leaves a fractional part 2/9 = 0.\\overline{2}; hence the digit in the 10^{-(k+1)} place is 2.\n\nStep 7.  Influence of the error term R  \n|R| < 4·10^{-3k}.  The earliest decimal position it could affect is the 10^{-3k} place, which is far to the right of 10^{-(k+1)} because 3k ≫ k+1 (specifically, 3k = 10 806 while k+1 = 3 603).  Therefore the digits found in Step 6 are final.\n\nStep 8.  Conclusion  \nImmediately after the decimal point,\n\n T = 0.444…4 7 2 2 2 2 … ,  \n         ↑  \n    10^{-k} 10^{-(k+1)}\n\nHence the 3 603-rd digit after the decimal point – the 10^{-(k+1)} digit – is\n\n 2.\n\nAnswer: 2.\n\n--------------------------------------------------------------------",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.585463",
        "was_fixed": false,
        "difficulty_analysis": "1. Multiple interacting terms.  Unlike the original single-root question, this problem forces the solver to handle √N, its reciprocal, and a scalar multiple √(N/9) simultaneously; each behaves on a different magnitude scale, so careful bookkeeping of powers of 10 is essential.\n\n2. Position-sensitive carries without cancellation.  \nThe digit sought lies precisely where three separate series overlap:  \n • the repeating 4’s coming from 10^{k}/3 + 10^{k}/9,  \n • an isolated “2” generated by (25/9)·10^{-k}, and  \n • possible carries produced by the infinite tails 0.\\overline{7} and the error bounds.  \nDeciding that no carry occurs and that later tails cannot change the target digit demands explicit upper bounds on all neglected terms well beyond those needed in the original problem.\n\n3. Higher-order error control.  \nBounding R₁, R₂, R₃ simultaneously and proving they are far too small to influence the 10^{-(k+1)} digit requires two binomial/reciprocal expansions and a uniform inequality, not just a single first-order estimate.\n\n4. Magnitude management across 7 200 decimal places.  \nKeeping track of which term first becomes visible in which decimal position, then proving that nothing sooner or larger disturbs it, is conceptually and computationally heavier than the pattern-matching argument that solved the original.\n\nIn short, the problem demands a longer chain of approximations, accurate error bounding, and subtle carry analysis across three interacting series—making it markedly more intricate than both the original and the current kernel variant."
      }
    }
  },
  "checked": true,
  "problem_type": "calculation"
}