{
  "index": "2024-B-3",
  "type": "ANA",
  "tag": [
    "ANA",
    "NT",
    "ALG"
  ],
  "difficulty": "",
  "question": "Let $r_n$ be the $n$th smallest positive solution to $\\tan x = x$, where the argument of tangent is in radians. Prove that\n\\[\n0 < r_{n+1} - r_n - \\pi < \\frac{1}{(n^2+n)\\pi}\n\\]\nfor $n \\geq 1$.",
  "solution": "\\noindent\n\\textbf{First solution.} (by Bjorn Poonen)\nLet $\\tan^{-1} \\colon \\RR \\to (-\\frac{\\pi}{2}, \\frac{\\pi}{2})$ be the principal branch of the arctangent function, and set $t(x) := x - \\tan^{-1}(x)$. Then $t(0) = 0$ and\n\\[\n\\frac{dt}{dx} = 1 - \\frac{1}{1+x^2} = \\frac{x^2}{1+x^2} > 0 \\qquad (x \\neq 0),\n\\]\nso $t(x)$ is strictly increasing.\nWe have $\\tan x = x$ if and only if $x = \\tan^{-1} x + n\\pi$ for some $n \\in \\mathbb{Z}$;\nfrom the previous analysis it follows that $r_n$ is the unique solution of $t(x) = n \\pi$.\n\nLet $x(t)$ be the inverse function of $t(x)$, so that $r_n = x(n\\pi)$. We compute that\n\\begin{align*}\n\\frac{dx}{dt} - 1 &= \\frac{1}{dt/dx} - 1 = \\frac{1}{x^2} \\\\\n\\frac{dx}{dt} - 1 - \\frac{1}{t^2} &= \\frac{1}{x^2} - \\frac{1}{t^2}.\n\\end{align*}\nFrom this we deduce that $x(t) - t$ is strictly increasing for $t > 0$ (as then $x(t) > 0$)\nand $x(t) - t + \\frac{1}{t}$ is strictly decreasing for $t > 0$ (as then $\\tan^{-1}(x(t)) > 0$ and so $t < x(t)$).  Evaluating at $t = n\\pi$ and $t = (n+1)\\pi$, we obtain \n\\begin{align*}\nr_n - n\\pi &< r_{n+1} - (n+1) \\pi \\\\\nr_n - n\\pi + \\frac{1}{n\\pi} &> r_{n+1} - (n+1)\\pi + \\frac{1}{(n+1)\\pi},\n\\end{align*}\nwhich are the desired inequalities.\n\n\\noindent\n\\textbf{Second solution.}\nDefine the function \n\\[\nf(x) := \\tan x - x.\n\\]\nWe then have $f'(x) = \\tan^2 x$.\nBy induction on $k$, $f^{(k)}(x)$ is a polynomial of degree $k+1$ in $\\tan x$\nwith leading coefficient $k!$ and all coefficients nonnegative. In particular, on each of the intervals\n\\[\nI_n := \\left(n \\pi, n \\pi + \\frac{\\pi}{2} \\right)  \\qquad (n=0,1,\\dots),\n\\]\n$\\tan x$ is positive\nand so $f^{(k)}(x)$ is positive for each $k \\geq 1$; replacing $k$ with $k+1$, we deduce that each $f^{(k)}(x)$ is strictly increasing on $I_n$ for $k \\geq 0$.\n\nWe now analyze $f$ more closely on $I_n$.\nAs $x \\to n\\pi^+$ for $n>0$, $f(x)$ tends to $f(n\\pi) = -n\\pi < 0$;\nby contrast, as $x \\to 0^+$, $f(x)$ tends to 0 via positive values.\nIn either case, as $x \\to (n \\pi + \\frac{\\pi}{2})^-$, $f(x) \\to \\infty$.\nSince $f(x)$ is strictly increasing on $I_n$, we deduce using the intermediate value theorem that:\n\\begin{itemize}\n\\item\n$f(x)$ has no zero in $I_0$;\n\\item\nfor $n > 0$, $f(x)$ has a unique zero in $I_n$.\n\\end{itemize}\nSince $f(x)$ also has no zero between $I_n$ and $I_{n+1}$ (as it takes exclusively negative values there), we deduce that\n\\[\nn\\pi < r_n < n \\pi + \\frac{\\pi}{2}.\n\\]\nThis already suffices to prove the claimed lower bound: since\n$f(r_n+\\pi) = - \\pi < 0$ and $f$ is strictly increasing on $I_{n+1}$, \nthe quantity $\\delta := r_{n+1} - (r_n + \\pi)$ is positive.\n\nTo prove the upper bound, note that for $k \\geq 1$,\nfor $0 < x < n\\pi + \\frac{\\pi}{2}-r_n$, we have\n\\begin{align*}\nf^{(k)}(x) & \\geq f^{(k)}(r_n + \\pi) = f^{(k)}(r_n) \\\\\n&\\geq k! r_n^{k+1} > k! n^{k+1} \\pi^{k+1}.\n\\end{align*}\nFor each $k \\geq 2$, we may apply the mean value theorem with remainder to deduce that for $x$ in the same range,\n\\[\nf(r_n+\\pi+x)\\geq f(r_n+\\pi) + \\sum_{i=1}^k f^{(i)}(r_n+\\pi) \\frac{x^i}{i!}.\n\\]\nTaking the limit as $k \\to \\infty$ yields\n\\begin{align*}\nf(r_n + \\pi + x) &\\geq f(r_n+\\pi) + \\sum_{i=1}^\\infty f^{(i)}(r_n+\\pi) \\frac{x^i}{i!} \\\\\n& > -\\pi + \\sum_{i=1}^k n^{i+1} \\pi^{i+1} x^i \\\\\n&> - \\pi + \\frac{n^2\\pi^2 x}{1-n \\pi x};\n\\end{align*}\ntaking $x = \\delta$ yields\n\\[\n0 > -\\pi + n \\pi \\left(\\frac{1}{1-n \\pi \\delta} - 1\\right)\n\\]\nand so $\\delta < \\frac{1}{n(n+1)\\pi}$ as desired.\n\n\\noindent\n\\textbf{Remark.}\nThere is a mild subtlety hidden in the proof:\nif one first bounds the finite sum as\n\\[\nf(r_n+\\pi+x) > -\\pi+ \\sum_{i=1}^k n^{i+1} \\pi^{i+1} x^i\n\\]\nand then takes the limit as $k \\to \\infty$, the strict inequality is not preserved. One way around this is to write $f''(r_n) = 2r_n + 2 r_n^3$,\nretain the extra term $r_n x^2$ in the lower bound, take the limit as $k \\to \\infty$, and then discard the extra term to get back to a strict inequality. \n\n\\noindent\n\\textbf{Remark.}\nThe slightly weaker inequality $\\delta < \\frac{1}{n^2 \\pi}$\nfollows at once from the inequality\n\\[\nf'(r_n + \\pi) = f'(r_n) = \\tan^2 r_n = r_n^2 > n^2 \\pi^2\n\\]\nplus the mean value theorem.\n\n\\noindent\n\\textbf{Remark.}\nOne can also reach the desired upper bound by comparing $r_{n+1}$ to $r_n + \\pi$ using the addition formula for tangents:\n\\[\n\\tan(x+y) = \\frac{\\tan x - \\tan y}{1 + \\tan x \\tan y}.\n\\]\nNamely, one then gets\n\\begin{align*}\n\\delta &< \\tan \\delta = \\frac{\\tan r_{n+1}  - \\tan (r_n+\\pi)}{1 + \\tan r_{n+1} \\tan (r_n+\\pi)} \\\\\n&= \\frac{r_{n+1}-r_n}{1 + r_n r_{n+1}} = \\frac{\\pi + \\delta}{1 + r_n r_{n+1}}\n\\end{align*}\nand hence\n\\[\n\\delta < \\frac{\\pi}{r_n r_{n+1}} < \\frac{\\pi}{(n\\pi)((n+1)\\pi)} = \\frac{1}{(n^2+n)\\pi}.\n\\]",
  "vars": [
    "x",
    "t",
    "\\\\delta"
  ],
  "params": [
    "n",
    "k",
    "r_n",
    "r_n+1",
    "f",
    "i"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "unknownx",
        "t": "auxvalue",
        "\\delta": "smallgap",
        "n": "givenindex",
        "k": "givenorder",
        "r_n": "seqrootn",
        "r_n+1": "seqrootnplusone",
        "r_{n+1}": "seqrootnplusone",
        "f": "functionf",
        "i": "sumindex"
      },
      "question": "Let seqrootn be the givenindexth smallest positive solution to \\tan unknownx = unknownx, where the argument of tangent is in radians. Prove that\n\\[\n0 < seqrootnplusone - seqrootn - \\pi < \\frac{1}{(givenindex^2+givenindex)\\pi}\n\\]\nfor givenindex \\geq 1.",
      "solution": "\\noindent\n\\textbf{First solution.} (by Bjorn Poonen)\nLet $\\tan^{-1} \\colon \\RR \\to (-\\frac{\\pi}{2}, \\frac{\\pi}{2})$ be the principal branch of the arctangent function, and set $\\auxvalue(unknownx) := unknownx - \\tan^{-1}(unknownx)$. Then $\\auxvalue(0) = 0$ and\n\\[\n\\frac{d\\auxvalue}{d unknownx} = 1 - \\frac{1}{1+unknownx^2} = \\frac{unknownx^2}{1+unknownx^2} > 0 \\qquad (unknownx \\neq 0),\n\\]\nso $\\auxvalue(unknownx)$ is strictly increasing.\nWe have $\\tan unknownx = unknownx$ if and only if $unknownx = \\tan^{-1} unknownx + givenindex\\pi$ for some $givenindex \\in \\mathbb{Z}$;\nfrom the previous analysis it follows that $\\seqrootn$ is the unique solution of $\\auxvalue(unknownx) = givenindex \\pi$.\n\nLet $unknownx(\\auxvalue)$ be the inverse function of $\\auxvalue(unknownx)$, so that $\\seqrootn = unknownx(givenindex\\pi)$. We compute that\n\\begin{align*}\n\\frac{d unknownx}{d\\auxvalue} - 1 &= \\frac{1}{d\\auxvalue/d unknownx} - 1 = \\frac{1}{unknownx^2} \\\\\n\\frac{d unknownx}{d\\auxvalue} - 1 - \\frac{1}{\\auxvalue^2} &= \\frac{1}{unknownx^2} - \\frac{1}{\\auxvalue^2}.\n\\end{align*}\nFrom this we deduce that $unknownx(\\auxvalue) - \\auxvalue$ is strictly increasing for $\\auxvalue > 0$ (as then $unknownx(\\auxvalue) > 0$)\nand $unknownx(\\auxvalue) - \\auxvalue + \\frac{1}{\\auxvalue}$ is strictly decreasing for $\\auxvalue > 0$ (as then $\\tan^{-1}(unknownx(\\auxvalue)) > 0$ and so $\\auxvalue < unknownx(\\auxvalue)$).  Evaluating at $\\auxvalue = givenindex\\pi$ and $\\auxvalue = (givenindex+1)\\pi$, we obtain \n\\begin{align*}\n\\seqrootn - givenindex\\pi &< \\seqrootnplusone - (givenindex+1) \\pi \\\\\n\\seqrootn - givenindex\\pi + \\frac{1}{givenindex\\pi} &> \\seqrootnplusone - (givenindex+1)\\pi + \\frac{1}{(givenindex+1)\\pi},\n\\end{align*}\nwhich are the desired inequalities.\n\n\\noindent\n\\textbf{Second solution.}\nDefine the function \n\\[\n\\functionf(unknownx) := \\tan unknownx - unknownx.\n\\]\nWe then have $\\functionf'(unknownx) = \\tan^2 unknownx$.\nBy induction on $givenorder$, $\\functionf^{(givenorder)}(unknownx)$ is a polynomial of degree $givenorder+1$ in $\\tan unknownx$\nwith leading coefficient $givenorder!$ and all coefficients nonnegative. In particular, on each of the intervals\n\\[\nI_{givenindex} := \\left(givenindex \\pi, givenindex \\pi + \\frac{\\pi}{2} \\right)  \\qquad (givenindex=0,1,\\dots),\n\\]\n$\\tan unknownx$ is positive\nand so $\\functionf^{(givenorder)}(unknownx)$ is positive for each $givenorder \\geq 1$; replacing $givenorder$ with $givenorder+1$, we deduce that each $\\functionf^{(givenorder)}(unknownx)$ is strictly increasing on $I_{givenindex}$ for $givenorder \\geq 0$.\n\nWe now analyze $\\functionf$ more closely on $I_{givenindex}$.\nAs $unknownx \\to givenindex\\pi^+$ for $givenindex>0$, $\\functionf(unknownx)$ tends to $\\functionf(givenindex\\pi) = -givenindex\\pi < 0$;\nby contrast, as $unknownx \\to 0^+$, $\\functionf(unknownx)$ tends to 0 via positive values.\nIn either case, as $unknownx \\to (givenindex \\pi + \\frac{\\pi}{2})^- $, $\\functionf(unknownx) \\to \\infty$.\nSince $\\functionf(unknownx)$ is strictly increasing on $I_{givenindex}$, we deduce using the intermediate value theorem that:\n\\begin{itemize}\n\\item\n$\\functionf(unknownx)$ has no zero in $I_0$;\n\\item\nfor $givenindex > 0$, $\\functionf(unknownx)$ has a unique zero in $I_{givenindex}$.\n\\end{itemize}\nSince $\\functionf(unknownx)$ also has no zero between $I_{givenindex}$ and $I_{givenindex+1}$ (as it takes exclusively negative values there), we deduce that\n\\[\ngivenindex\\pi < \\seqrootn < givenindex \\pi + \\frac{\\pi}{2}.\n\\]\nThis already suffices to prove the claimed lower bound: since\n$\\functionf(\\seqrootn+\\pi) = - \\pi < 0$ and $\\functionf$ is strictly increasing on $I_{givenindex+1}$, \nthe quantity $\\smallgap := \\seqrootnplusone - (\\seqrootn + \\pi)$ is positive.\n\nTo prove the upper bound, note that for $givenorder \\geq 1$,\nfor $0 < unknownx < givenindex\\pi + \\frac{\\pi}{2}-\\seqrootn$, we have\n\\begin{align*}\n\\functionf^{(givenorder)}(unknownx) & \\geq \\functionf^{(givenorder)}(\\seqrootn + \\pi) = \\functionf^{(givenorder)}(\\seqrootn) \\\\\n&\\geq givenorder! \\seqrootn^{givenorder+1} > givenorder! givenindex^{givenorder+1} \\pi^{givenorder+1}.\n\\end{align*}\nFor each $givenorder \\geq 2$, we may apply the mean value theorem with remainder to deduce that for unknownx in the same range,\n\\[\n\\functionf(\\seqrootn+\\pi+unknownx)\\geq \\functionf(\\seqrootn+\\pi) + \\sum_{\\sumindex=1}^{givenorder} \\functionf^{(\\sumindex)}(\\seqrootn+\\pi) \\frac{unknownx^{\\sumindex}}{\\sumindex!}.\n\\]\nTaking the limit as $givenorder \\to \\infty$ yields\n\\begin{align*}\n\\functionf(\\seqrootn + \\pi + unknownx) &\\geq \\functionf(\\seqrootn+\\pi) + \\sum_{\\sumindex=1}^{\\infty} \\functionf^{(\\sumindex)}(\\seqrootn+\\pi) \\frac{unknownx^{\\sumindex}}{\\sumindex!} \\\\\n& > -\\pi + \\sum_{\\sumindex=1}^{\\infty} givenindex^{\\sumindex+1} \\pi^{\\sumindex+1} unknownx^{\\sumindex} \\\\\n&> - \\pi + \\frac{givenindex^2\\pi^2 unknownx}{1-givenindex \\pi unknownx};\n\\end{align*}\ntaking $unknownx = \\smallgap$ yields\n\\[\n0 > -\\pi + givenindex \\pi \\left(\\frac{1}{1-givenindex \\pi \\smallgap} - 1\\right)\n\\]\nand so $\\smallgap < \\frac{1}{givenindex(givenindex+1)\\pi}$ as desired.\n\n\\noindent\n\\textbf{Remark.}\nThere is a mild subtlety hidden in the proof:\nif one first bounds the finite sum as\n\\[\n\\functionf(\\seqrootn+\\pi+unknownx) > -\\pi+ \\sum_{\\sumindex=1}^{givenorder} givenindex^{\\sumindex+1} \\pi^{\\sumindex+1} unknownx^{\\sumindex}\n\\]\nand then takes the limit as $givenorder \\to \\infty$, the strict inequality is not preserved. One way around this is to write $\\functionf''(\\seqrootn) = 2\\seqrootn + 2 \\seqrootn^3$,\nretain the extra term $\\seqrootn unknownx^2$ in the lower bound, take the limit as $givenorder \\to \\infty$, and then discard the extra term to get back to a strict inequality. \n\n\\noindent\n\\textbf{Remark.}\nThe slightly weaker inequality $\\smallgap < \\frac{1}{givenindex^2 \\pi}$\nfollows at once from the inequality\n\\[\n\\functionf'(\\seqrootn + \\pi) = \\functionf'(\\seqrootn) = \\tan^2 \\seqrootn = \\seqrootn^2 > givenindex^2 \\pi^2\n\\]\nplus the mean value theorem.\n\n\\noindent\n\\textbf{Remark.}\nOne can also reach the desired upper bound by comparing $\\seqrootnplusone$ to $\\seqrootn + \\pi$ using the addition formula for tangents:\n\\[\n\\tan(unknownx+y) = \\frac{\\tan unknownx - \\tan y}{1 + \\tan unknownx \\tan y}.\n\\]\nNamely, one then gets\n\\begin{align*}\n\\smallgap &< \\tan \\smallgap = \\frac{\\tan \\seqrootnplusone  - \\tan (\\seqrootn+\\pi)}{1 + \\tan \\seqrootnplusone \\tan (\\seqrootn+\\pi)} \\\\\n&= \\frac{\\seqrootnplusone-\\seqrootn}{1 + \\seqrootn \\seqrootnplusone} = \\frac{\\pi + \\smallgap}{1 + \\seqrootn \\seqrootnplusone}\n\\end{align*}\nand hence\n\\[\n\\smallgap < \\frac{\\pi}{\\seqrootn \\seqrootnplusone} < \\frac{\\pi}{(givenindex\\pi)((givenindex+1)\\pi)} = \\frac{1}{(givenindex^2+givenindex)\\pi}.\n\\]"
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "compassrose",
        "t": "meadowlark",
        "\\delta": "harpsichord",
        "n": "sailboat",
        "k": "evergreen",
        "r_n": "fireball",
        "r_{n+1}": "firebrand",
        "f": "galaxyway"
      },
      "question": "Let $fireball$ be the $sailboat$th smallest positive solution to $\\tan compassrose = compassrose$, where the argument of tangent is in radians. Prove that\n\\[\n0 < firebrand - fireball - \\pi < \\frac{1}{(sailboat^2+sailboat)\\pi}\n\\]\nfor $sailboat \\geq 1$.",
      "solution": "\\noindent\n\\textbf{First solution.} (by Bjorn Poonen)\nLet $\\tan^{-1} \\colon \\RR \\to (-\\frac{\\pi}{2}, \\frac{\\pi}{2})$ be the principal branch of the arctangent function, and set $meadowlark(compassrose) := compassrose - \\tan^{-1}(compassrose)$. Then $meadowlark(0) = 0$ and\n\\[\n\\frac{d meadowlark}{d compassrose} = 1 - \\frac{1}{1+compassrose^2} = \\frac{compassrose^2}{1+compassrose^2} > 0 \\qquad (compassrose \\neq 0),\n\\]\nso $meadowlark(compassrose)$ is strictly increasing.\nWe have $\\tan compassrose = compassrose$ if and only if $compassrose = \\tan^{-1} compassrose + sailboat\\pi$ for some $sailboat \\in \\mathbb{Z}$;\nfrom the previous analysis it follows that $fireball$ is the unique solution of $meadowlark(compassrose) = sailboat \\pi$.\n\nLet $compassrose(meadowlark)$ be the inverse function of $meadowlark(compassrose)$, so that $fireball = compassrose(sailboat\\pi)$. We compute that\n\\begin{align*}\n\\frac{d compassrose}{d meadowlark} - 1 &= \\frac{1}{d meadowlark/d compassrose} - 1 = \\frac{1}{compassrose^2} \\\\\n\\frac{d compassrose}{d meadowlark} - 1 - \\frac{1}{meadowlark^2} &= \\frac{1}{compassrose^2} - \\frac{1}{meadowlark^2}.\n\\end{align*}\nFrom this we deduce that $compassrose(meadowlark) - meadowlark$ is strictly increasing for $meadowlark > 0$ (as then $compassrose(meadowlark) > 0$)\nand $compassrose(meadowlark) - meadowlark + \\frac{1}{meadowlark}$ is strictly decreasing for $meadowlark > 0$ (as then $\\tan^{-1}(compassrose(meadowlark)) > 0$ and so $meadowlark < compassrose(meadowlark)$).  Evaluating at $meadowlark = sailboat\\pi$ and $meadowlark = (sailboat+1)\\pi$, we obtain \n\\begin{align*}\nfireball - sailboat\\pi &< firebrand - (sailboat+1) \\pi \\\\\nfireball - sailboat\\pi + \\frac{1}{sailboat\\pi} &> firebrand - (sailboat+1)\\pi + \\frac{1}{(sailboat+1)\\pi},\n\\end{align*}\nwhich are the desired inequalities.\n\n\\noindent\n\\textbf{Second solution.}\nDefine the function \n\\[\ngalaxyway(compassrose) := \\tan compassrose - compassrose.\n\\]\nWe then have $galaxyway'(compassrose) = \\tan^2 compassrose$.\nBy induction on $evergreen$, $galaxyway^{(evergreen)}(compassrose)$ is a polynomial of degree $evergreen+1$ in $\\tan compassrose$\nwith leading coefficient $evergreen!$ and all coefficients nonnegative. In particular, on each of the intervals\n\\[\nI_{sailboat} := \\left(sailboat \\pi, sailboat \\pi + \\frac{\\pi}{2} \\right)  \\qquad (sailboat=0,1,\\dots),\n\\]\n$\\tan compassrose$ is positive\nand so $galaxyway^{(evergreen)}(compassrose)$ is positive for each $evergreen \\geq 1$; replacing $evergreen$ with $evergreen+1$, we deduce that each $galaxyway^{(evergreen)}(compassrose)$ is strictly increasing on $I_{sailboat}$ for $evergreen \\geq 0$.\n\nWe now analyze $galaxyway$ more closely on $I_{sailboat}$.\nAs $compassrose \\to sailboat\\pi^+$ for $sailboat>0$, $galaxyway(compassrose)$ tends to $galaxyway(sailboat\\pi) = -sailboat\\pi < 0$;\nby contrast, as $compassrose \\to 0^+$, $galaxyway(compassrose)$ tends to 0 via positive values.\nIn either case, as $compassrose \\to (sailboat \\pi + \\frac{\\pi}{2})^-$, $galaxyway(compassrose) \\to \\infty$.\nSince $galaxyway(compassrose)$ is strictly increasing on $I_{sailboat}$, we deduce using the intermediate value theorem that:\n\\begin{itemize}\n\\item\n$galaxyway(compassrose)$ has no zero in $I_0$;\n\\item\nfor $sailboat > 0$, $galaxyway(compassrose)$ has a unique zero in $I_{sailboat}$.\n\\end{itemize}\nSince $galaxyway(compassrose)$ also has no zero between $I_{sailboat}$ and $I_{sailboat+1}$ (as it takes exclusively negative values there), we deduce that\n\\[\nsailboat\\pi < fireball < sailboat \\pi + \\frac{\\pi}{2}.\n\\]\nThis already suffices to prove the claimed lower bound: since\n$galaxyway(fireball+\\pi) = - \\pi < 0$ and $galaxyway$ is strictly increasing on $I_{sailboat+1}$, \nthe quantity $harpsichord := firebrand - (fireball + \\pi)$ is positive.\n\nTo prove the upper bound, note that for $evergreen \\geq 1$,\nfor $0 < compassrose < sailboat\\pi + \\frac{\\pi}{2}-fireball$, we have\n\\begin{align*}\ngalaxyway^{(evergreen)}(compassrose) & \\geq galaxyway^{(evergreen)}(fireball + \\pi) = galaxyway^{(evergreen)}(fireball) \\\\\n&\\geq evergreen! \\, fireball^{evergreen+1} > evergreen! \\, sailboat^{evergreen+1} \\pi^{evergreen+1}.\n\\end{align*}\nFor each $evergreen \\geq 2$, we may apply the mean value theorem with remainder to deduce that for $compassrose$ in the same range,\n\\[\ngalaxyway(fireball+\\pi+compassrose)\\geq galaxyway(fireball+\\pi) + \\sum_{i=1}^{evergreen} galaxyway^{(i)}(fireball+\\pi) \\frac{compassrose^i}{i!}.\n\\]\nTaking the limit as $evergreen \\to \\infty$ yields\n\\begin{align*}\ngalaxyway(fireball + \\pi + compassrose) &\\geq galaxyway(fireball+\\pi) + \\sum_{i=1}^\\infty galaxyway^{(i)}(fireball+\\pi) \\frac{compassrose^i}{i!} \\\\\n& > -\\pi + \\sum_{i=1}^\\infty sailboat^{i+1} \\pi^{i+1} compassrose^i \\\\\n&> - \\pi + \\frac{sailboat^2\\pi^2 compassrose}{1- sailboat \\pi compassrose};\n\\end{align*}\ntaking $compassrose = harpsichord$ yields\n\\[\n0 > -\\pi + sailboat \\pi \\left(\\frac{1}{1- sailboat \\pi harpsichord} - 1\\right)\n\\]\nand so $harpsichord < \\frac{1}{sailboat(sailboat+1)\\pi}$ as desired.\n\n\\noindent\n\\textbf{Remark.}\nThere is a mild subtlety hidden in the proof:\nif one first bounds the finite sum as\n\\[\ngalaxyway(fireball+\\pi+compassrose) > -\\pi+ \\sum_{i=1}^{evergreen} sailboat^{i+1} \\pi^{i+1} compassrose^i\n\\]\nand then takes the limit as $evergreen \\to \\infty$, the strict inequality is not preserved. One way around this is to write $galaxyway''(fireball) = 2fireball + 2 fireball^3$,\nretain the extra term $fireball compassrose^2$ in the lower bound, take the limit as $evergreen \\to \\infty$, and then discard the extra term to get back to a strict inequality. \n\n\\noindent\n\\textbf{Remark.}\nThe slightly weaker inequality $harpsichord < \\frac{1}{sailboat^2 \\pi}$\nfollows at once from the inequality\n\\[\ngalaxyway'(fireball + \\pi) = galaxyway'(fireball) = \\tan^2 fireball = fireball^2 > sailboat^2 \\pi^2\n\\]\nplus the mean value theorem.\n\n\\noindent\n\\textbf{Remark.}\nOne can also reach the desired upper bound by comparing $firebrand$ to $fireball + \\pi$ using the addition formula for tangents:\n\\[\n\\tan(compassrose + y) = \\frac{\\tan compassrose - \\tan y}{1 + \\tan compassrose \\tan y}.\n\\]\nNamely, one then gets\n\\begin{align*}\nharpsichord &< \\tan harpsichord = \\frac{\\tan firebrand  - \\tan (fireball+\\pi)}{1 + \\tan firebrand \\tan (fireball+\\pi)} \\\\\n&= \\frac{firebrand-fireball}{1 + fireball firebrand} = \\frac{\\pi + harpsichord}{1 + fireball firebrand}\n\\end{align*}\nand hence\n\\[\nharpsichord < \\frac{\\pi}{fireball firebrand} < \\frac{\\pi}{(sailboat\\pi)((sailboat+1)\\pi)} = \\frac{1}{(sailboat^2+sailboat)\\pi}.\n\\]"
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "knownvalue",
        "t": "timelessness",
        "\\\\delta": "largesum",
        "n": "continuum",
        "k": "staticity",
        "r_n": "surfacevalue",
        "r_n+1": "surfacevaluenext",
        "f": "constantvalue",
        "i": "singularity"
      },
      "question": "Let $surfacevalue$ be the $continuum$th smallest positive solution to $\\tan knownvalue = knownvalue$, where the argument of tangent is in radians. Prove that\n\\[\n0 < surfacevaluenext - surfacevalue - \\pi < \\frac{1}{(continuum^2+continuum)\\pi}\n\\]\nfor $continuum \\geq 1$.",
      "solution": "\\noindent\n\\textbf{First solution.} (by Bjorn Poonen)\nLet $\\tan^{-1} \\colon \\RR \\to (-\\frac{\\pi}{2}, \\frac{\\pi}{2})$ be the principal branch of the arctangent function, and set $timelessness(knownvalue) := knownvalue - \\tan^{-1}(knownvalue)$. Then $timelessness(0) = 0$ and\n\\[\n\\frac{dtimelessness}{dknownvalue} = 1 - \\frac{1}{1+knownvalue^2} = \\frac{knownvalue^2}{1+knownvalue^2} > 0 \\qquad (knownvalue \\neq 0),\n\\]\nso $timelessness(knownvalue)$ is strictly increasing.\nWe have $\\tan knownvalue = knownvalue$ if and only if $knownvalue = \\tan^{-1} knownvalue + continuum\\pi$ for some $continuum \\in \\mathbb{Z}$;\nfrom the previous analysis it follows that $surfacevalue$ is the unique solution of $timelessness(knownvalue) = continuum \\pi$.\n\nLet $knownvalue(timelessness)$ be the inverse function of $timelessness(knownvalue)$, so that $surfacevalue = knownvalue(continuum\\pi)$. We compute that\n\\begin{align*}\n\\frac{dknownvalue}{dtimelessness} - 1 &= \\frac{1}{dtimelessness/dknownvalue} - 1 = \\frac{1}{knownvalue^2} \\\\\n\\frac{dknownvalue}{dtimelessness} - 1 - \\frac{1}{timelessness^2} &= \\frac{1}{knownvalue^2} - \\frac{1}{timelessness^2}.\n\\end{align*}\nFrom this we deduce that $knownvalue(timelessness) - timelessness$ is strictly increasing for $timelessness > 0$ (as then $knownvalue(timelessness) > 0$)\nand $knownvalue(timelessness) - timelessness + \\frac{1}{timelessness}$ is strictly decreasing for $timelessness > 0$ (as then $\\tan^{-1}(knownvalue(timelessness)) > 0$ and so $timelessness < knownvalue(timelessness)$).  Evaluating at $timelessness = continuum\\pi$ and $timelessness = (continuum+1)\\pi$, we obtain \n\\begin{align*}\nsurfacevalue - continuum\\pi &< surfacevaluenext - (continuum+1) \\pi \\\\\nsurfacevalue - continuum\\pi + \\frac{1}{continuum\\pi} &> surfacevaluenext - (continuum+1)\\pi + \\frac{1}{(continuum+1)\\pi},\n\\end{align*}\nwhich are the desired inequalities.\n\n\\noindent\n\\textbf{Second solution.}\nDefine the function \n\\[\nconstantvalue(knownvalue) := \\tan knownvalue - knownvalue.\n\\]\nWe then have $constantvalue'(knownvalue) = \\tan^2 knownvalue$.\nBy induction on $staticity$, $constantvalue^{(staticity)}(knownvalue)$ is a polynomial of degree $staticity+1$ in $\\tan knownvalue$\nwith leading coefficient $staticity!$ and all coefficients nonnegative. In particular, on each of the intervals\n\\[\nI_{continuum} := \\left(continuum \\pi, continuum \\pi + \\frac{\\pi}{2} \\right)  \\qquad (continuum=0,1,\\dots),\n\\]\n$\\tan knownvalue$ is positive\nand so $constantvalue^{(staticity)}(knownvalue)$ is positive for each $staticity \\geq 1$; replacing $staticity$ with $staticity+1$, we deduce that each $constantvalue^{(staticity)}(knownvalue)$ is strictly increasing on $I_{continuum}$ for $staticity \\geq 0$.\n\nWe now analyze $constantvalue$ more closely on $I_{continuum}$.\nAs $knownvalue \\to continuum\\pi^+$ for $continuum>0$, $constantvalue(knownvalue)$ tends to $constantvalue(continuum\\pi) = -continuum\\pi < 0$;\nby contrast, as $knownvalue \\to 0^+$, $constantvalue(knownvalue)$ tends to 0 via positive values.\nIn either case, as $knownvalue \\to (continuum \\pi + \\frac{\\pi}{2})^-$, $constantvalue(knownvalue) \\to \\infty$.\nSince $constantvalue(knownvalue)$ is strictly increasing on $I_{continuum}$, we deduce using the intermediate value theorem that:\n\\begin{itemize}\n\\item\n$constantvalue(knownvalue)$ has no zero in $I_0$;\n\\item\nfor $continuum > 0$, $constantvalue(knownvalue)$ has a unique zero in $I_{continuum}$.\n\\end{itemize}\nSince $constantvalue(knownvalue)$ also has no zero between $I_{continuum}$ and $I_{continuum+1}$ (as it takes exclusively negative values there), we deduce that\n\\[\ncontinuum\\pi < surfacevalue < continuum \\pi + \\frac{\\pi}{2}.\n\\]\nThis already suffices to prove the claimed lower bound: since\n$constantvalue(surfacevalue+\\pi) = - \\pi < 0$ and $constantvalue$ is strictly increasing on $I_{continuum+1}$, \nthe quantity $largesum := surfacevaluenext - (surfacevalue + \\pi)$ is positive.\n\nTo prove the upper bound, note that for $staticity \\geq 1$,\nfor $0 < knownvalue < continuum\\pi + \\frac{\\pi}{2}-surfacevalue$, we have\n\\begin{align*}\nconstantvalue^{(staticity)}(knownvalue) & \\geq constantvalue^{(staticity)}(surfacevalue + \\pi) = constantvalue^{(staticity)}(surfacevalue) \\\\\n&\\geq staticity! \\, surfacevalue^{staticity+1} > staticity! \\, continuum^{staticity+1} \\pi^{staticity+1}.\n\\end{align*}\nFor each $staticity \\geq 2$, we may apply the mean value theorem with remainder to deduce that for $knownvalue$ in the same range,\n\\[\nconstantvalue(surfacevalue+\\pi+knownvalue)\\geq constantvalue(surfacevalue+\\pi) + \\sum_{singularity=1}^{staticity} constantvalue^{(singularity)}(surfacevalue+\\pi) \\frac{knownvalue^{singularity}}{singularity!}.\n\\]\nTaking the limit as $staticity \\to \\infty$ yields\n\\begin{align*}\nconstantvalue(surfacevalue + \\pi + knownvalue) &\\geq constantvalue(surfacevalue+\\pi) + \\sum_{singularity=1}^{\\infty} constantvalue^{(singularity)}(surfacevalue+\\pi) \\frac{knownvalue^{singularity}}{singularity!} \\\\\n& > -\\pi + \\sum_{singularity=1}^{staticity} continuum^{singularity+1} \\pi^{singularity+1} knownvalue^{singularity} \\\\\n&> - \\pi + \\frac{continuum^2\\pi^2 knownvalue}{1-continuum \\pi knownvalue};\n\\end{align*}\ntaking $knownvalue = largesum$ yields\n\\[\n0 > -\\pi + continuum \\pi \\left(\\frac{1}{1-continuum \\pi largesum} - 1\\right)\n\\]\nand so $largesum < \\frac{1}{continuum(continuum+1)\\pi}$ as desired.\n\n\\noindent\n\\textbf{Remark.}\nThere is a mild subtlety hidden in the proof:\nif one first bounds the finite sum as\n\\[\nconstantvalue(surfacevalue+\\pi+knownvalue) > -\\pi+ \\sum_{singularity=1}^{staticity} continuum^{singularity+1} \\pi^{singularity+1} knownvalue^{singularity}\n\\]\nand then takes the limit as $staticity \\to \\infty$, the strict inequality is not preserved. One way around this is to write $constantvalue''(surfacevalue) = 2surfacevalue + 2 surfacevalue^3$,\nretain the extra term $surfacevalue knownvalue^2$ in the lower bound, take the limit as $staticity \\to \\infty$, and then discard the extra term to get back to a strict inequality. \n\n\\noindent\n\\textbf{Remark.}\nThe slightly weaker inequality $largesum < \\frac{1}{continuum^2 \\pi}$\nfollows at once from the inequality\n\\[\nconstantvalue'(surfacevalue + \\pi) = constantvalue'(surfacevalue) = \\tan^2 surfacevalue = surfacevalue^2 > continuum^2 \\pi^2\n\\]\nplus the mean value theorem.\n\n\\noindent\n\\textbf{Remark.}\nOne can also reach the desired upper bound by comparing $surfacevaluenext$ to $surfacevalue + \\pi$ using the addition formula for tangents:\n\\[\n\\tan(knownvalue+knownvalue) = \\frac{\\tan knownvalue - \\tan knownvalue}{1 + \\tan knownvalue \\tan knownvalue}.\n\\]\nNamely, one then gets\n\\begin{align*}\nlargesum &< \\tan largesum = \\frac{\\tan surfacevaluenext  - \\tan (surfacevalue+\\pi)}{1 + \\tan surfacevaluenext \\tan (surfacevalue+\\pi)} \\\\\n&= \\frac{surfacevaluenext-surfacevalue}{1 + surfacevalue surfacevaluenext} = \\frac{\\pi + largesum}{1 + surfacevalue surfacevaluenext}\n\\end{align*}\nand hence\n\\[\nlargesum < \\frac{\\pi}{surfacevalue surfacevaluenext} < \\frac{\\pi}{(continuum\\pi)((continuum+1)\\pi)} = \\frac{1}{(continuum^2+continuum)\\pi}.\n\\]"
    },
    "garbled_string": {
      "map": {
        "x": "qzxwvtnp",
        "t": "hjgrksla",
        "\\delta": "znvritoh",
        "n": "skjfdmca",
        "k": "blatufre",
        "r_n": "vmxokped",
        "r_{n+1}": "gqrsadim",
        "f": "ujlopreb",
        "i": "cxswlqpo"
      },
      "question": "Let $vmxokped$ be the $skjfdmca$th smallest positive solution to $\\tan qzxwvtnp = qzxwvtnp$, where the argument of tangent is in radians. Prove that\n\\[\n0 < gqrsadim - vmxokped - \\pi < \\frac{1}{(skjfdmca^2+skjfdmca)\\pi}\n\\]\nfor $skjfdmca \\geq 1$.",
      "solution": "\\noindent\n\\textbf{First solution.} (by Bjorn Poonen)\nLet $\\tan^{-1} \\colon \\RR \\to (-\\frac{\\pi}{2}, \\frac{\\pi}{2})$ be the principal branch of the arctangent function, and set $hjgrksla(qzxwvtnp) := qzxwvtnp - \\tan^{-1}(qzxwvtnp)$. Then $hjgrksla(0) = 0$ and\n\\[\n\\frac{d hjgrksla}{d qzxwvtnp} = 1 - \\frac{1}{1+qzxwvtnp^2} = \\frac{qzxwvtnp^2}{1+qzxwvtnp^2} > 0 \\qquad (qzxwvtnp \\neq 0),\n\\]\nso $hjgrksla(qzxwvtnp)$ is strictly increasing.\nWe have $\\tan qzxwvtnp = qzxwvtnp$ if and only if $qzxwvtnp = \\tan^{-1} qzxwvtnp + skjfdmca\\pi$ for some $skjfdmca \\in \\mathbb{Z}$;\nfrom the previous analysis it follows that $vmxokped$ is the unique solution of $hjgrksla(qzxwvtnp) = skjfdmca \\pi$.\n\nLet $qzxwvtnp(hjgrksla)$ be the inverse function of $hjgrksla(qzxwvtnp)$, so that $vmxokped = qzxwvtnp(skjfdmca\\pi)$. We compute that\n\\begin{align*}\n\\frac{d qzxwvtnp}{d hjgrksla} - 1 &= \\frac{1}{d hjgrksla/d qzxwvtnp} - 1 = \\frac{1}{qzxwvtnp^2} \\\\\n\\frac{d qzxwvtnp}{d hjgrksla} - 1 - \\frac{1}{hjgrksla^2} &= \\frac{1}{qzxwvtnp^2} - \\frac{1}{hjgrksla^2}.\n\\end{align*}\nFrom this we deduce that $qzxwvtnp(hjgrksla) - hjgrksla$ is strictly increasing for $hjgrksla > 0$ (as then $qzxwvtnp(hjgrksla) > 0$)\nand $qzxwvtnp(hjgrksla) - hjgrksla + \\frac{1}{hjgrksla}$ is strictly decreasing for $hjgrksla > 0$ (as then $\\tan^{-1}(qzxwvtnp(hjgrksla)) > 0$ and so $hjgrksla < qzxwvtnp(hjgrksla)$).  Evaluating at $hjgrksla = skjfdmca\\pi$ and $hjgrksla = (skjfdmca+1)\\pi$, we obtain \n\\begin{align*}\nvmxokped - skjfdmca\\pi &< gqrsadim - (skjfdmca+1) \\pi \\\\\nvmxokped - skjfdmca\\pi + \\frac{1}{skjfdmca\\pi} &> gqrsadim - (skjfdmca+1)\\pi + \\frac{1}{(skjfdmca+1)\\pi},\n\\end{align*}\nwhich are the desired inequalities.\n\n\\noindent\n\\textbf{Second solution.}\nDefine the function \n\\[\nujlopreb(qzxwvtnp) := \\tan qzxwvtnp - qzxwvtnp.\n\\]\nWe then have $ujlopreb'(qzxwvtnp) = \\tan^2 qzxwvtnp$.\nBy induction on $blatufre$, $ujlopreb^{(blatufre)}(qzxwvtnp)$ is a polynomial of degree $blatufre+1$ in $\\tan qzxwvtnp$\nwith leading coefficient $blatufre!$ and all coefficients nonnegative. In particular, on each of the intervals\n\\[\nI_{skjfdmca} := \\left(skjfdmca \\pi, skjfdmca \\pi + \\frac{\\pi}{2} \\right)  \\qquad (skjfdmca=0,1,\\dots),\n\\]\n$\\tan qzxwvtnp$ is positive\nand so $ujlopreb^{(blatufre)}(qzxwvtnp)$ is positive for each $blatufre \\geq 1$; replacing $blatufre$ with $blatufre+1$, we deduce that each $ujlopreb^{(blatufre)}(qzxwvtnp)$ is strictly increasing on $I_{skjfdmca}$ for $blatufre \\geq 0$.\n\nWe now analyze $ujlopreb$ more closely on $I_{skjfdmca}$.\nAs $qzxwvtnp \\to skjfdmca\\pi^+$ for $skjfdmca>0$, $ujlopreb(qzxwvtnp)$ tends to $ujlopreb(skjfdmca\\pi) = -skjfdmca\\pi < 0$;\nby contrast, as $qzxwvtnp \\to 0^+$, $ujlopreb(qzxwvtnp)$ tends to 0 via positive values.\nIn either case, as $qzxwvtnp \\to (skjfdmca \\pi + \\frac{\\pi}{2})^-$, $ujlopreb(qzxwvtnp) \\to \\infty$.\nSince $ujlopreb(qzxwvtnp)$ is strictly increasing on $I_{skjfdmca}$, we deduce using the intermediate value theorem that:\n\\begin{itemize}\n\\item\n$ujlopreb(qzxwvtnp)$ has no zero in $I_0$;\n\\item\nfor $skjfdmca > 0$, $ujlopreb(qzxwvtnp)$ has a unique zero in $I_{skjfdmca}$.\n\\end{itemize}\nSince $ujlopreb(qzxwvtnp)$ also has no zero between $I_{skjfdmca}$ and $I_{skjfdmca+1}$ (as it takes exclusively negative values there), we deduce that\n\\[\nskjfdmca\\pi < vmxokped < skjfdmca \\pi + \\frac{\\pi}{2}.\n\\]\nThis already suffices to prove the claimed lower bound: since\n$ujlopreb(vmxokped+\\pi) = - \\pi < 0$ and $ujlopreb$ is strictly increasing on $I_{skjfdmca+1}$, \nthe quantity $znvritoh := gqrsadim - (vmxokped + \\pi)$ is positive.\n\nTo prove the upper bound, note that for $blatufre \\geq 1$,\nfor $0 < qzxwvtnp < skjfdmca\\pi + \\frac{\\pi}{2}-vmxokped$, we have\n\\begin{align*}\nujlopreb^{(blatufre)}(qzxwvtnp) & \\geq ujlopreb^{(blatufre)}(vmxokped + \\pi) = ujlopreb^{(blatufre)}(vmxokped) \\\\\n&\\geq blatufre! \\, vmxokped^{blatufre+1} > blatufre! \\, skjfdmca^{blatufre+1} \\pi^{blatufre+1}.\n\\end{align*}\nFor each $blatufre \\geq 2$, we may apply the mean value theorem with remainder to deduce that for $qzxwvtnp$ in the same range,\n\\[\nujlopreb(vmxokped+\\pi+qzxwvtnp)\\geq ujlopreb(vmxokped+\\pi) + \\sum_{cxswlqpo=1}^{blatufre} ujlopreb^{(cxswlqpo)}(vmxokped+\\pi) \\frac{qzxwvtnp^{cxswlqpo}}{cxswlqpo!}.\n\\]\nTaking the limit as $blatufre \\to \\infty$ yields\n\\begin{align*}\nujlopreb(vmxokped + \\pi + qzxwvtnp) &\\geq ujlopreb(vmxokped+\\pi) + \\sum_{cxswlqpo=1}^\\infty ujlopreb^{(cxswlqpo)}(vmxokped+\\pi) \\frac{qzxwvtnp^{cxswlqpo}}{cxswlqpo!} \\\\\n& > -\\pi + \\sum_{cxswlqpo=1}^{blatufre} skjfdmca^{cxswlqpo+1} \\pi^{cxswlqpo+1} qzxwvtnp^{cxswlqpo} \\\\\n&> - \\pi + \\frac{skjfdmca^2\\pi^2 qzxwvtnp}{1-skjfdmca \\pi qzxwvtnp};\n\\end{align*}\ntaking $qzxwvtnp = znvritoh$ yields\n\\[\n0 > -\\pi + skjfdmca \\pi \\left(\\frac{1}{1-skjfdmca \\pi znvritoh} - 1\\right)\n\\]\nand so $znvritoh < \\frac{1}{skjfdmca(skjfdmca+1)\\pi}$ as desired.\n\n\\noindent\n\\textbf{Remark.}\nThere is a mild subtlety hidden in the proof:\nif one first bounds the finite sum as\n\\[\nujlopreb(vmxokped+\\pi+qzxwvtnp) > -\\pi+ \\sum_{cxswlqpo=1}^{blatufre} skjfdmca^{cxswlqpo+1} \\pi^{cxswlqpo+1} qzxwvtnp^{cxswlqpo}\n\\]\nand then takes the limit as $blatufre \\to \\infty$, the strict inequality is not preserved. One way around this is to write $ujlopreb''(vmxokped) = 2vmxokped + 2 vmxokped^3$,\nretain the extra term $vmxokped qzxwvtnp^2$ in the lower bound, take the limit as $blatufre \\to \\infty$, and then discard the extra term to get back to a strict inequality. \n\n\\noindent\n\\textbf{Remark.}\nThe slightly weaker inequality $znvritoh < \\frac{1}{skjfdmca^2 \\pi}$\nfollows at once from the inequality\n\\[\nujlopreb'(vmxokped + \\pi) = ujlopreb'(vmxokped) = \\tan^2 vmxokped = vmxokped^2 > skjfdmca^2 \\pi^2\n\\]\nplus the mean value theorem.\n\n\\noindent\n\\textbf{Remark.}\nOne can also reach the desired upper bound by comparing $gqrsadim$ to $vmxokped + \\pi$ using the addition formula for tangents:\n\\[\n\\tan(qzxwvtnp+y) = \\frac{\\tan qzxwvtnp - \\tan y}{1 + \\tan qzxwvtnp \\tan y}.\n\\]\nNamely, one then gets\n\\begin{align*}\nznvritoh &< \\tan znvritoh = \\frac{\\tan gqrsadim  - \\tan (vmxokped+\\pi)}{1 + \\tan gqrsadim \\tan (vmxokped+\\pi)} \\\\\n&= \\frac{gqrsadim-vmxokped}{1 + vmxokped gqrsadim} = \\frac{\\pi + znvritoh}{1 + vmxokped gqrsadim}\n\\end{align*}\nand hence\n\\[\nznvritoh < \\frac{\\pi}{vmxokped gqrsadim} < \\frac{\\pi}{(skjfdmca\\pi)((skjfdmca+1)\\pi)} = \\frac{1}{(skjfdmca^2+skjfdmca)\\pi}.\n\\]"
    },
    "kernel_variant": {
      "question": "Let $\\bigl(r_{n}\\bigr)_{n\\ge 1}$ be the strictly increasing sequence of positive roots of  \n\\[\n\\tan x \\;=\\; x ,\\qquad (x\\hbox{ in radians}).\n\\]\n\nDefine  \n\\[\nF(z):=\\sin z - z\\cos z,\\qquad\nG(z):=\\frac{3F(z)}{z^{3}},\\qquad\n\\varepsilon_{n}:=r_{n+1}-r_{n}-\\pi\\quad (n\\ge 1).\n\\]\n\nAnswer the following (the parts are logically independent except for the\ntransfer of notation).\n\nA.  (Weierstrass product)  \n\nA1.  Prove that $\\displaystyle\\sum_{n=1}^{\\infty} r_{n}^{-2}$ converges.  \n\nA2.  Show that  \n\\[\n\\boxed{\\;\n\\sin z - z\\cos z\n       \\;=\\;\\frac{z^{3}}{3}\n             \\prod_{n=1}^{\\infty}\\Bigl(1-\\frac{z^{2}}{r_{n}^{2}}\\Bigr)\n\\;}\n\\]\nholds for every $z\\in\\mathbb C$.\n\nB.  (Two Euler–type sums)  Use the product from part A to prove  \n\\[\n\\sum_{n=1}^{\\infty}\\frac{1}{r_{n}^{2}}=\\frac{1}{10},\n\\qquad\n\\sum_{n=1}^{\\infty}\\frac{1}{r_{n}^{4}}=\\frac{1}{350}.\n\\]\n\nC.  (Two–term control of the gaps)  \n\nC1.  Show that, for every $n\\ge 1$,  \n\\[\n\\frac{1}{\\pi\\,(n+\\tfrac32)^{2}}\n\\;<\\;\n\\varepsilon_{n}\n\\;<\\;\n\\frac{1}{\\pi\\,n(n+1)}.\n\\tag{$\\ast$}\n\\]\n\nC2.  Let $\\displaystyle\\delta_{n}:=\\frac{1}{\\pi n(n+1)}$,  \n$g(x):=\\pi x^{-2}$ and $a_{n}:=\\pi\\bigl(n+\\tfrac12\\bigr)$.  \nProve that, for all $n\\ge 1$,  \n\\[\n\\bigl|\\varepsilon_{n}-\\delta_{n}\\bigr|\n\\;\\le\\;\n\\frac{17}{\\pi n^{3}}.\n\\tag{$\\ast\\ast$}\n\\]\n\nDeduce that $\\varepsilon_{n}=1/[\\pi n(n+1)] + O(n^{-3})$ and that\n$0<\\varepsilon_{n}<\\delta_{n}$ for every $n\\ge 18$.",
      "solution": "Throughout $C_{1},C_{2},\\dots$ denote positive absolute constants whose\nnumerical values are irrelevant.\nAll estimates are valid for $n\\ge 1$\nunless another lower threshold is explicitly stated.\n\nA. Convergence and the Hadamard product\n---------------------------------------\n\nA1. Asymptotics of the zeros.  \nFix $k\\ge 1$ and write $x=(k+\\tfrac12)\\pi-\\varepsilon$ with $0<\\varepsilon\\ll 1$.\nNear a pole of $\\tan$ one has\n\\[\n\\tan\\bigl((k+\\tfrac12)\\pi-\\varepsilon\\bigr)\n    =\\cot\\varepsilon\n    =\\varepsilon^{-1}-\\tfrac{\\varepsilon}{3}+O(\\varepsilon^{3}).\n\\]\nImposing $\\tan x = x$ gives  \n\\[\n\\varepsilon^{-1}-\\tfrac{\\varepsilon}{3}+O(\\varepsilon^{3})\n           =(k+\\tfrac12)\\pi-\\varepsilon .\n\\]\nMatching the leading term yields\n\\[\n\\varepsilon\\sim \\frac{1}{(k+\\tfrac12)\\pi},\n\\qquad\n\\varepsilon =\\frac{1}{(k+\\tfrac12)\\pi}+O(k^{-3}). \\tag{1}\n\\]\nHence  \n\\[\nr_{k}=(k+\\tfrac12)\\pi-\\frac{1}{(k+\\tfrac12)\\pi}+O(k^{-3})\n      \\ge (k+\\tfrac14)\\pi \\quad (k\\hbox{ large}). \\tag{2}\n\\]\nTherefore  \n\\[\n\\frac{1}{r_{k}^{2}}\\le\\frac{C_{1}}{k^{2}},\n\\qquad\\text{so}\\qquad\n\\sum_{k=1}^{\\infty}\\frac{1}{r_{k}^{2}}<\\infty .\n\\]\n\nA2. The canonical factor.  \nBecause $|\\sin z|+|z\\cos z|\\le C_{2}\\,e^{|\\,\\operatorname{Im}z\\,|}$,\n$F$ is an entire function of order $1$.  Moreover  \n\\[\nF'(z)=z\\sin z,\n\\qquad\nF'(r_{k})=r_{k}^{2}\\cos r_{k}\\ne 0 ,\n\\]\nso the non–zero zeros are simple.\nSince $\\sum_{k} r_{k}^{-2}<\\infty$, the genus–$0$ Weierstrass product is legitimate:\n\\[\nF(z)=C\\,z^{3}\\prod_{n=1}^{\\infty}\\Bigl(1-\\frac{z^{2}}{r_{n}^{2}}\\Bigr). \\tag{3}\n\\]\nDividing by $z^{3}$ and letting $z\\to 0$ gives  \n\\[\nC=\\lim_{z\\to 0}\\frac{F(z)}{z^{3}}=\\frac13,\n\\]\nbecause $\\sin z - z\\cos z = z^{3}/3 + O(z^{5})$.  Hence  \n\\[\n\\boxed{\\;\n\\sin z - z\\cos z\n      \\;=\\;\\frac{z^{3}}{3}\\prod_{n=1}^{\\infty}\n        \\Bigl(1-\\frac{z^{2}}{r_{n}^{2}}\\Bigr)\n\\;} \\tag{4}\n\\]\nas required.\n\nB. The Euler–type sums\n----------------------\n\nPut $G(z)=3F(z)/z^{3}$.  A Maclaurin expansion gives  \n\\[\nG(z)=1-\\frac{z^{2}}{10}+\\frac{z^{4}}{280}+O(z^{6}). \\tag{5}\n\\]\nTaking logarithms in (4) yields  \n\\[\n\\log G(z)=\\sum_{n=1}^{\\infty}\\log\\Bigl(1-\\frac{z^{2}}{r_{n}^{2}}\\Bigr)\n         =-\\Bigl(\\sum r_{n}^{-2}\\Bigr)z^{2}\n          -\\tfrac12\\Bigl(\\sum r_{n}^{-4}\\Bigr)z^{4}+O(z^{6}). \\tag{6}\n\\]\nLet $u:=-z^{2}/10+z^{4}/280+O(z^{6})$.  Then  \n\\[\n\\log G(z)=u-\\tfrac12u^{2}+O(u^{3})\n         =-\\frac{z^{2}}{10}-\\frac{z^{4}}{700}+O(z^{6}). \\tag{7}\n\\]\nComparing the coefficients of $z^{2}$ and $z^{4}$ in (6)–(7) gives  \n\\[\n\\boxed{\\;\n\\sum_{n=1}^{\\infty}\\frac{1}{r_{n}^{2}}=\\frac{1}{10},\n\\quad\n\\sum_{n=1}^{\\infty}\\frac{1}{r_{n}^{4}}=\\frac{1}{350}\n\\;} \\tag{8}\n\\]\ncompleting part B.\n\nC. A two–term estimate for the gaps\n------------------------------------\n\nC1. The inequalities $(\\ast)$.\n\nUpper bound (right–hand inequality).  \nExactly as in the original solution, the tangent addition formula yields  \n\\[\n\\varepsilon_{n}\n   < \\frac{1}{\\pi\\,n(n+1)}. \\tag{9}\n\\]\n\nLower bound (left–hand inequality).  \nDefine  \n\\[\nf(x)=\\tan x - x,\n\\qquad\nf'(x)=\\tan^{2}x.\n\\]\nThe function $f$ is strictly increasing on each interval\n$\\bigl(m\\pi,\\,m\\pi+\\tfrac{\\pi}{2}\\bigr)$, $m\\in\\mathbb N$,\nand satisfies $f(r_{n}+\\pi)=-\\pi<0<f(r_{n+1})$.\nThe mean–value theorem applied to $f$ on\n$[\\,r_{n}+\\pi,\\;r_{n+1}]$ gives a point  \n\\[\n\\xi_{n}\\in\\bigl(r_{n}+\\pi,\\;r_{n+1}\\bigr)\n\\quad\\text{such that}\\quad\nf'(\\xi_{n})\n    =\\frac{f(r_{n+1})-f(r_{n}+\\pi)}{\\varepsilon_{n}}\n    =\\frac{\\pi}{\\varepsilon_{n}}. \\tag{10}\n\\]\nBecause $f'(x)=\\tan^{2}x$ and $\\tan x$ is increasing on the interval,\n$\\tan\\xi_{n}<\\tan r_{n+1}=r_{n+1}$.  Consequently  \n\\[\n\\varepsilon_{n}\n   =\\frac{\\pi}{\\tan^{2}\\xi_{n}}\n   >\\frac{\\pi}{r_{n+1}^{2}}. \\tag{11}\n\\]\nEach zero lies in the interior of its half–period, so  \n\\[\nr_{n+1}<(n+\\tfrac32)\\pi. \\tag{12}\n\\]\nInserting (12) into (11) yields the stated lower bound  \n\\[\n\\varepsilon_{n}>\\frac{1}{\\pi\\,(n+\\tfrac32)^{2}}. \\tag{13}\n\\]\nTogether, (9) and (13) give $(\\ast)$.\n\nC2. A refined $O(n^{-3})$ error term $(\\ast\\ast)$.\n\nWe keep the notation  \n\\[\nh(x):=\\frac{\\pi}{\\tan^{2}x},\n\\qquad\ng(x):=\\frac{\\pi}{x^{2}},\n\\qquad\n\\varepsilon_{n}=h(\\xi_{n})\\quad\\text{from \\eqref{10}}.\n\\]\n\nStep 1. Replacing $\\tan x$ by $x$ at $\\xi_{n}$.  \nRewrite  \n\\[\n|h(\\xi_{n})-g(\\xi_{n})|\n      =\\pi\\Bigl|\\frac{1}{\\tan^{2}\\xi_{n}}-\\frac{1}{\\xi_{n}^{2}}\\Bigr|\n      =\\pi\\,\n        \\frac{|\\tan\\xi_{n}-\\xi_{n}|\\,\n              (\\tan\\xi_{n}+\\xi_{n})}\n             {\\xi_{n}^{2}\\tan^{2}\\xi_{n}}. \\tag{14}\n\\]\n\nWe need suitable bounds for the three factors on the right–hand side.\n\n(i) On the interval $\\bigl(r_{n}+\\pi,r_{n+1}\\bigr)$ the function\n$f(x)=\\tan x - x$ ranges between $-\\pi$ and $0$; hence\n\\[\n|\\tan\\xi_{n}-\\xi_{n}|\\le\\pi. \\tag{15}\n\\]\n\n(ii)  Since $\\tan\\xi_{n}-\\xi_{n}\\ge -\\pi$, we have\n\\[\n\\tan\\xi_{n}+\\xi_{n}\n   =\\bigl(\\tan\\xi_{n}-\\xi_{n}\\bigr)+2\\xi_{n}\n   \\le\\pi+2\\xi_{n}.                                     \\tag{16}\n\\]\n\n(iii)  Denominator estimate.  \nFrom \\eqref{10} we know $\\tan^{2}\\xi_{n}=\\pi/\\varepsilon_{n}$.\nUsing the already proved upper bound \\eqref{9},\n\\[\n\\tan^{2}\\xi_{n}=\\frac{\\pi}{\\varepsilon_{n}}\n          >\\pi^{2}n(n+1). \\tag{17}\n\\]\nBecause $\\xi_{n}<r_{n+1}<(n+\\tfrac32)\\pi$,\n\\[\n\\xi_{n}^{2}<\\bigl(n+\\tfrac32\\bigr)^{2}\\pi^{2}. \\tag{18}\n\\]\nA direct computation shows that, for every $n\\ge 1$,\n\\[\n4n(n+1)\\;\\ge\\;\\bigl(n+\\tfrac32\\bigr)^{2}. \\tag{19}\n\\]\nCombining (17)–(19) gives the convenient bound\n\\[\n\\tan^{2}\\xi_{n}\\ge\\frac{\\xi_{n}^{2}}{4}. \\tag{20}\n\\]\n\nInsert the bounds (15)–(16) and (20) into (14) and use\n$\\xi_{n}\\ge r_{n}\\ge (n+\\tfrac14)\\pi$:\n\\[\n|h(\\xi_{n})-g(\\xi_{n})|\n     \\le \\pi\\,\\frac{\\pi(\\pi+2\\xi_{n})}{\\xi_{n}^{2}\\cdot(\\tfrac14)\\xi_{n}^{2}}\n     =\\frac{4\\pi^{2}(\\pi+2\\xi_{n})}{\\xi_{n}^{4}}\n     \\le\\frac{12\\pi^{2}}{\\xi_{n}^{3}}\n     \\le\\frac{12}{\\pi n^{3}}.                           \\tag{21}\n\\]\nThus  \n\\[\n\\boxed{|h(\\xi_{n})-g(\\xi_{n})|\\le\\frac{12}{\\pi n^{3}}}. \\tag{22}\n\\]\n\nStep 2. Stability of $g$ on a short interval.  \nSince $g'(x)=-2\\pi x^{-3}$, for $x\\ge\\pi n$ one has\n$|g'(x)|\\le 2/(\\pi^{2}n^{3})$.\nFurthermore  \n\\[\n|\\,\\xi_{n}-a_{n}|\n      \\le |\\xi_{n}-r_{n+1}|+|r_{n+1}-a_{n}|\n      <\\varepsilon_{n}+\\pi\n      <2\\pi .\n\\]\nHence  \n\\[\n|g(\\xi_{n})-g(a_{n})|\n   \\le |g'(\\zeta)|\\,|\\,\\xi_{n}-a_{n}|\n   \\le \\frac{4}{\\pi n^{3}}. \\tag{23}\n\\]\n\nStep 3. Taylor expansion of $g(a_{n})-\\delta_{n}$.  \nA direct calculation gives  \n\\[\ng(a_{n})-\\delta_{n}\n      =\\frac{\\pi}{\\bigl[\\pi(n+\\tfrac12)\\bigr]^{2}}\n        -\\frac{1}{\\pi n(n+1)}\n      =-\\frac{1}{4\\pi\\,(n+\\tfrac12)^{2}n(n+1)}\n      =O(n^{-4}). \\tag{24}\n\\]\nSo, for $n\\ge 1$,\n\\[\n|g(a_{n})-\\delta_{n}|\\le\\frac{1}{\\pi n^{4}}\n           \\le\\frac{1}{\\pi n^{3}}. \\tag{25}\n\\]\n\nStep 4. Collecting the pieces.  \nCombining (22)–(25) with\n$\\varepsilon_{n}=h(\\xi_{n})$ yields  \n\\[\n\\bigl|\\varepsilon_{n}-\\delta_{n}\\bigr|\n  \\le |h(\\xi_{n})-g(\\xi_{n})|\n       +|g(\\xi_{n})-g(a_{n})|\n       +|g(a_{n})-\\delta_{n}|\n  \\le \\frac{12}{\\pi n^{3}}\n       +\\frac{4}{\\pi n^{3}}\n       +\\frac{1}{\\pi n^{3}}\n  <  \\frac{17}{\\pi n^{3}}, \\tag{26}\n\\]\nwhich is $(\\ast\\ast)$.\n\nBecause $\\delta_{n}\\asymp n^{-2}$ whereas\n$17/(\\pi n^{3})=o(\\delta_{n})$, we have  \n\\[\n\\boxed{\\;\n\\varepsilon_{n}=\\frac{1}{\\pi n(n+1)}+O(n^{-3})\n\\;} .\n\\]\nFinally, $17/(\\pi n^{3})<\\delta_{n}$ is equivalent to\n$n^{2}-17n-17>0$, i.e. $n\\ge 18$.  Hence (26) implies  \n\\[\n0<\\varepsilon_{n}<\\delta_{n}\\qquad(n\\ge 18),\n\\]\nand the finitely many smaller indices can be checked numerically.\n\n\\qedsymbol",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.884139",
        "was_fixed": false,
        "difficulty_analysis": "•  Extra mathematical structures The problem now calls for Weierstrass’ factor–\n   ization theorem, infinite products, and manipulation of entire functions,\n   none of which is needed in the original statement.\n\n•  Higher theoretical requirements Part B forces the contestant to connect\n   coefficients in a Maclaurin series with coefficients in a logarithmic\n   product, a classical technique from complex analysis (comparable to the\n   derivation of Euler’s product for \\(\\sin\\)).  Moreover two different sums\n   of reciprocal powers have to be computed exactly.\n\n•  Deeper interaction of ideas To solve Part C one must combine:  \n   (i) asymptotics coming from Newton’s method for \\(f(x)=\\tan x-x\\);  \n   (ii) analytic information extracted in Part B; and  \n   (iii) careful error propagation, culminating in an explicit second–order\n   inequality rather than the first–order one asked for originally.\n\n•  More steps and subtler estimates The original problem needs only a\n   first–order Taylor argument.  The enhanced variant demands a two–stage\n   expansion, an appeal to complex function theory, and the control of an\n   error term of order \\(n^{-3}\\).\n\nAltogether the new kernel variant is substantially harder than both the\noriginal and the previous kernel version, requiring advanced tools from\nanalysis, infinite products, and asymptotic approximation rather than a\nsingle elementary inequality."
      }
    },
    "original_kernel_variant": {
      "question": "Let $\\bigl(r_{n}\\bigr)_{n\\ge 1}$ be the strictly increasing sequence of positive roots of  \n\\[\n\\tan x \\;=\\; x ,\\qquad (x\\hbox{ in radians}).\n\\]\n\nDefine  \n\\[\nF(z):=\\sin z - z\\cos z,\\qquad\nG(z):=\\frac{3F(z)}{z^{3}},\\qquad\n\\varepsilon_{n}:=r_{n+1}-r_{n}-\\pi\\quad (n\\ge 1).\n\\]\n\nAnswer the following (the parts are logically independent except for the\ntransfer of notation).\n\nA.  (Weierstrass product)  \n\nA1.  Prove that $\\displaystyle\\sum_{n=1}^{\\infty} r_{n}^{-2}$ converges.  \n\nA2.  Show that  \n\\[\n\\boxed{\\;\n\\sin z - z\\cos z\n       \\;=\\;\\frac{z^{3}}{3}\n             \\prod_{n=1}^{\\infty}\\Bigl(1-\\frac{z^{2}}{r_{n}^{2}}\\Bigr)\n\\;}\n\\]\nholds for every $z\\in\\mathbb C$.\n\nB.  (Two Euler–type sums)  Use the product from part A to prove  \n\\[\n\\sum_{n=1}^{\\infty}\\frac{1}{r_{n}^{2}}=\\frac{1}{10},\n\\qquad\n\\sum_{n=1}^{\\infty}\\frac{1}{r_{n}^{4}}=\\frac{1}{350}.\n\\]\n\nC.  (Two–term control of the gaps)  \n\nC1.  Show that, for every $n\\ge 1$,  \n\\[\n\\frac{1}{\\pi\\,(n+\\tfrac32)^{2}}\n\\;<\\;\n\\varepsilon_{n}\n\\;<\\;\n\\frac{1}{\\pi\\,n(n+1)}.\n\\tag{$\\ast$}\n\\]\n\nC2.  Let $\\displaystyle\\delta_{n}:=\\frac{1}{\\pi n(n+1)}$,  \n$g(x):=\\pi x^{-2}$ and $a_{n}:=\\pi\\bigl(n+\\tfrac12\\bigr)$.  \nProve that, for all $n\\ge 1$,  \n\\[\n\\bigl|\\varepsilon_{n}-\\delta_{n}\\bigr|\n\\;\\le\\;\n\\frac{17}{\\pi n^{3}}.\n\\tag{$\\ast\\ast$}\n\\]\n\nDeduce that $\\varepsilon_{n}=1/[\\pi n(n+1)] + O(n^{-3})$ and that\n$0<\\varepsilon_{n}<\\delta_{n}$ for every $n\\ge 18$.",
      "solution": "Throughout $C_{1},C_{2},\\dots$ denote positive absolute constants whose\nnumerical values are irrelevant.\nAll estimates are valid for $n\\ge 1$\nunless another lower threshold is explicitly stated.\n\nA. Convergence and the Hadamard product\n---------------------------------------\n\nA1. Asymptotics of the zeros.  \nFix $k\\ge 1$ and write $x=(k+\\tfrac12)\\pi-\\varepsilon$ with $0<\\varepsilon\\ll 1$.\nNear a pole of $\\tan$ one has\n\\[\n\\tan\\bigl((k+\\tfrac12)\\pi-\\varepsilon\\bigr)\n    =\\cot\\varepsilon\n    =\\varepsilon^{-1}-\\tfrac{\\varepsilon}{3}+O(\\varepsilon^{3}).\n\\]\nImposing $\\tan x = x$ gives  \n\\[\n\\varepsilon^{-1}-\\tfrac{\\varepsilon}{3}+O(\\varepsilon^{3})\n           =(k+\\tfrac12)\\pi-\\varepsilon .\n\\]\nMatching the leading term yields\n\\[\n\\varepsilon\\sim \\frac{1}{(k+\\tfrac12)\\pi},\n\\qquad\n\\varepsilon =\\frac{1}{(k+\\tfrac12)\\pi}+O(k^{-3}). \\tag{1}\n\\]\nHence  \n\\[\nr_{k}=(k+\\tfrac12)\\pi-\\frac{1}{(k+\\tfrac12)\\pi}+O(k^{-3})\n      \\ge (k+\\tfrac14)\\pi \\quad (k\\hbox{ large}). \\tag{2}\n\\]\nTherefore  \n\\[\n\\frac{1}{r_{k}^{2}}\\le\\frac{C_{1}}{k^{2}},\n\\qquad\\text{so}\\qquad\n\\sum_{k=1}^{\\infty}\\frac{1}{r_{k}^{2}}<\\infty .\n\\]\n\nA2. The canonical factor.  \nBecause $|\\sin z|+|z\\cos z|\\le C_{2}\\,e^{|\\,\\operatorname{Im}z\\,|}$,\n$F$ is an entire function of order $1$.  Moreover  \n\\[\nF'(z)=z\\sin z,\n\\qquad\nF'(r_{k})=r_{k}^{2}\\cos r_{k}\\ne 0 ,\n\\]\nso the non–zero zeros are simple.\nSince $\\sum_{k} r_{k}^{-2}<\\infty$, the genus–$0$ Weierstrass product is legitimate:\n\\[\nF(z)=C\\,z^{3}\\prod_{n=1}^{\\infty}\\Bigl(1-\\frac{z^{2}}{r_{n}^{2}}\\Bigr). \\tag{3}\n\\]\nDividing by $z^{3}$ and letting $z\\to 0$ gives  \n\\[\nC=\\lim_{z\\to 0}\\frac{F(z)}{z^{3}}=\\frac13,\n\\]\nbecause $\\sin z - z\\cos z = z^{3}/3 + O(z^{5})$.  Hence  \n\\[\n\\boxed{\\;\n\\sin z - z\\cos z\n      \\;=\\;\\frac{z^{3}}{3}\\prod_{n=1}^{\\infty}\n        \\Bigl(1-\\frac{z^{2}}{r_{n}^{2}}\\Bigr)\n\\;} \\tag{4}\n\\]\nas required.\n\nB. The Euler–type sums\n----------------------\n\nPut $G(z)=3F(z)/z^{3}$.  A Maclaurin expansion gives  \n\\[\nG(z)=1-\\frac{z^{2}}{10}+\\frac{z^{4}}{280}+O(z^{6}). \\tag{5}\n\\]\nTaking logarithms in (4) yields  \n\\[\n\\log G(z)=\\sum_{n=1}^{\\infty}\\log\\Bigl(1-\\frac{z^{2}}{r_{n}^{2}}\\Bigr)\n         =-\\Bigl(\\sum r_{n}^{-2}\\Bigr)z^{2}\n          -\\tfrac12\\Bigl(\\sum r_{n}^{-4}\\Bigr)z^{4}+O(z^{6}). \\tag{6}\n\\]\nLet $u:=-z^{2}/10+z^{4}/280+O(z^{6})$.  Then  \n\\[\n\\log G(z)=u-\\tfrac12u^{2}+O(u^{3})\n         =-\\frac{z^{2}}{10}-\\frac{z^{4}}{700}+O(z^{6}). \\tag{7}\n\\]\nComparing the coefficients of $z^{2}$ and $z^{4}$ in (6)–(7) gives  \n\\[\n\\boxed{\\;\n\\sum_{n=1}^{\\infty}\\frac{1}{r_{n}^{2}}=\\frac{1}{10},\n\\quad\n\\sum_{n=1}^{\\infty}\\frac{1}{r_{n}^{4}}=\\frac{1}{350}\n\\;} \\tag{8}\n\\]\ncompleting part B.\n\nC. A two–term estimate for the gaps\n------------------------------------\n\nC1. The inequalities $(\\ast)$.\n\nUpper bound (right–hand inequality).  \nExactly as in the original solution, the tangent addition formula yields  \n\\[\n\\varepsilon_{n}\n   < \\frac{1}{\\pi\\,n(n+1)}. \\tag{9}\n\\]\n\nLower bound (left–hand inequality).  \nDefine  \n\\[\nf(x)=\\tan x - x,\n\\qquad\nf'(x)=\\tan^{2}x.\n\\]\nThe function $f$ is strictly increasing on each interval\n$\\bigl(m\\pi,\\,m\\pi+\\tfrac{\\pi}{2}\\bigr)$, $m\\in\\mathbb N$,\nand satisfies $f(r_{n}+\\pi)=-\\pi<0<f(r_{n+1})$.\nThe mean–value theorem applied to $f$ on\n$[\\,r_{n}+\\pi,\\;r_{n+1}]$ gives a point  \n\\[\n\\xi_{n}\\in\\bigl(r_{n}+\\pi,\\;r_{n+1}\\bigr)\n\\quad\\text{such that}\\quad\nf'(\\xi_{n})\n    =\\frac{f(r_{n+1})-f(r_{n}+\\pi)}{\\varepsilon_{n}}\n    =\\frac{\\pi}{\\varepsilon_{n}}. \\tag{10}\n\\]\nBecause $f'(x)=\\tan^{2}x$ and $\\tan x$ is increasing on the interval,\n$\\tan\\xi_{n}<\\tan r_{n+1}=r_{n+1}$.  Consequently  \n\\[\n\\varepsilon_{n}\n   =\\frac{\\pi}{\\tan^{2}\\xi_{n}}\n   >\\frac{\\pi}{r_{n+1}^{2}}. \\tag{11}\n\\]\nEach zero lies in the interior of its half–period, so  \n\\[\nr_{n+1}<(n+\\tfrac32)\\pi. \\tag{12}\n\\]\nInserting (12) into (11) yields the stated lower bound  \n\\[\n\\varepsilon_{n}>\\frac{1}{\\pi\\,(n+\\tfrac32)^{2}}. \\tag{13}\n\\]\nTogether, (9) and (13) give $(\\ast)$.\n\nC2. A refined $O(n^{-3})$ error term $(\\ast\\ast)$.\n\nWe keep the notation  \n\\[\nh(x):=\\frac{\\pi}{\\tan^{2}x},\n\\qquad\ng(x):=\\frac{\\pi}{x^{2}},\n\\qquad\n\\varepsilon_{n}=h(\\xi_{n})\\quad\\text{from \\eqref{10}}.\n\\]\n\nStep 1. Replacing $\\tan x$ by $x$ at $\\xi_{n}$.  \nRewrite  \n\\[\n|h(\\xi_{n})-g(\\xi_{n})|\n      =\\pi\\Bigl|\\frac{1}{\\tan^{2}\\xi_{n}}-\\frac{1}{\\xi_{n}^{2}}\\Bigr|\n      =\\pi\\,\n        \\frac{|\\tan\\xi_{n}-\\xi_{n}|\\,\n              (\\tan\\xi_{n}+\\xi_{n})}\n             {\\xi_{n}^{2}\\tan^{2}\\xi_{n}}. \\tag{14}\n\\]\n\nWe need suitable bounds for the three factors on the right–hand side.\n\n(i) On the interval $\\bigl(r_{n}+\\pi,r_{n+1}\\bigr)$ the function\n$f(x)=\\tan x - x$ ranges between $-\\pi$ and $0$; hence\n\\[\n|\\tan\\xi_{n}-\\xi_{n}|\\le\\pi. \\tag{15}\n\\]\n\n(ii)  Since $\\tan\\xi_{n}-\\xi_{n}\\ge -\\pi$, we have\n\\[\n\\tan\\xi_{n}+\\xi_{n}\n   =\\bigl(\\tan\\xi_{n}-\\xi_{n}\\bigr)+2\\xi_{n}\n   \\le\\pi+2\\xi_{n}.                                     \\tag{16}\n\\]\n\n(iii)  Denominator estimate.  \nFrom \\eqref{10} we know $\\tan^{2}\\xi_{n}=\\pi/\\varepsilon_{n}$.\nUsing the already proved upper bound \\eqref{9},\n\\[\n\\tan^{2}\\xi_{n}=\\frac{\\pi}{\\varepsilon_{n}}\n          >\\pi^{2}n(n+1). \\tag{17}\n\\]\nBecause $\\xi_{n}<r_{n+1}<(n+\\tfrac32)\\pi$,\n\\[\n\\xi_{n}^{2}<\\bigl(n+\\tfrac32\\bigr)^{2}\\pi^{2}. \\tag{18}\n\\]\nA direct computation shows that, for every $n\\ge 1$,\n\\[\n4n(n+1)\\;\\ge\\;\\bigl(n+\\tfrac32\\bigr)^{2}. \\tag{19}\n\\]\nCombining (17)–(19) gives the convenient bound\n\\[\n\\tan^{2}\\xi_{n}\\ge\\frac{\\xi_{n}^{2}}{4}. \\tag{20}\n\\]\n\nInsert the bounds (15)–(16) and (20) into (14) and use\n$\\xi_{n}\\ge r_{n}\\ge (n+\\tfrac14)\\pi$:\n\\[\n|h(\\xi_{n})-g(\\xi_{n})|\n     \\le \\pi\\,\\frac{\\pi(\\pi+2\\xi_{n})}{\\xi_{n}^{2}\\cdot(\\tfrac14)\\xi_{n}^{2}}\n     =\\frac{4\\pi^{2}(\\pi+2\\xi_{n})}{\\xi_{n}^{4}}\n     \\le\\frac{12\\pi^{2}}{\\xi_{n}^{3}}\n     \\le\\frac{12}{\\pi n^{3}}.                           \\tag{21}\n\\]\nThus  \n\\[\n\\boxed{|h(\\xi_{n})-g(\\xi_{n})|\\le\\frac{12}{\\pi n^{3}}}. \\tag{22}\n\\]\n\nStep 2. Stability of $g$ on a short interval.  \nSince $g'(x)=-2\\pi x^{-3}$, for $x\\ge\\pi n$ one has\n$|g'(x)|\\le 2/(\\pi^{2}n^{3})$.\nFurthermore  \n\\[\n|\\,\\xi_{n}-a_{n}|\n      \\le |\\xi_{n}-r_{n+1}|+|r_{n+1}-a_{n}|\n      <\\varepsilon_{n}+\\pi\n      <2\\pi .\n\\]\nHence  \n\\[\n|g(\\xi_{n})-g(a_{n})|\n   \\le |g'(\\zeta)|\\,|\\,\\xi_{n}-a_{n}|\n   \\le \\frac{4}{\\pi n^{3}}. \\tag{23}\n\\]\n\nStep 3. Taylor expansion of $g(a_{n})-\\delta_{n}$.  \nA direct calculation gives  \n\\[\ng(a_{n})-\\delta_{n}\n      =\\frac{\\pi}{\\bigl[\\pi(n+\\tfrac12)\\bigr]^{2}}\n        -\\frac{1}{\\pi n(n+1)}\n      =-\\frac{1}{4\\pi\\,(n+\\tfrac12)^{2}n(n+1)}\n      =O(n^{-4}). \\tag{24}\n\\]\nSo, for $n\\ge 1$,\n\\[\n|g(a_{n})-\\delta_{n}|\\le\\frac{1}{\\pi n^{4}}\n           \\le\\frac{1}{\\pi n^{3}}. \\tag{25}\n\\]\n\nStep 4. Collecting the pieces.  \nCombining (22)–(25) with\n$\\varepsilon_{n}=h(\\xi_{n})$ yields  \n\\[\n\\bigl|\\varepsilon_{n}-\\delta_{n}\\bigr|\n  \\le |h(\\xi_{n})-g(\\xi_{n})|\n       +|g(\\xi_{n})-g(a_{n})|\n       +|g(a_{n})-\\delta_{n}|\n  \\le \\frac{12}{\\pi n^{3}}\n       +\\frac{4}{\\pi n^{3}}\n       +\\frac{1}{\\pi n^{3}}\n  <  \\frac{17}{\\pi n^{3}}, \\tag{26}\n\\]\nwhich is $(\\ast\\ast)$.\n\nBecause $\\delta_{n}\\asymp n^{-2}$ whereas\n$17/(\\pi n^{3})=o(\\delta_{n})$, we have  \n\\[\n\\boxed{\\;\n\\varepsilon_{n}=\\frac{1}{\\pi n(n+1)}+O(n^{-3})\n\\;} .\n\\]\nFinally, $17/(\\pi n^{3})<\\delta_{n}$ is equivalent to\n$n^{2}-17n-17>0$, i.e. $n\\ge 18$.  Hence (26) implies  \n\\[\n0<\\varepsilon_{n}<\\delta_{n}\\qquad(n\\ge 18),\n\\]\nand the finitely many smaller indices can be checked numerically.\n\n\\qedsymbol",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.668083",
        "was_fixed": false,
        "difficulty_analysis": "•  Extra mathematical structures The problem now calls for Weierstrass’ factor–\n   ization theorem, infinite products, and manipulation of entire functions,\n   none of which is needed in the original statement.\n\n•  Higher theoretical requirements Part B forces the contestant to connect\n   coefficients in a Maclaurin series with coefficients in a logarithmic\n   product, a classical technique from complex analysis (comparable to the\n   derivation of Euler’s product for \\(\\sin\\)).  Moreover two different sums\n   of reciprocal powers have to be computed exactly.\n\n•  Deeper interaction of ideas To solve Part C one must combine:  \n   (i) asymptotics coming from Newton’s method for \\(f(x)=\\tan x-x\\);  \n   (ii) analytic information extracted in Part B; and  \n   (iii) careful error propagation, culminating in an explicit second–order\n   inequality rather than the first–order one asked for originally.\n\n•  More steps and subtler estimates The original problem needs only a\n   first–order Taylor argument.  The enhanced variant demands a two–stage\n   expansion, an appeal to complex function theory, and the control of an\n   error term of order \\(n^{-3}\\).\n\nAltogether the new kernel variant is substantially harder than both the\noriginal and the previous kernel version, requiring advanced tools from\nanalysis, infinite products, and asymptotic approximation rather than a\nsingle elementary inequality."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}