{
  "index": "2016-A-2",
  "type": "NT",
  "tag": [
    "NT",
    "ALG"
  ],
  "difficulty": "",
  "question": "Given a positive integer $n$, let $M(n)$ be the largest integer $m$ such that \n\\[\n\\binom{m}{n-1} > \\binom{m-1}{n}.\n\\]\nEvaluate \n\\[\n\\lim_{n \\to \\infty} \\frac{M(n)}{n}.\n\\]",
  "solution": "The answer is $\\frac{3+\\sqrt{5}}{2}$. Note that for $m > n+1$, both binomial coefficients are nonzero and their ratio is\n\\begin{align*}\n{m\\choose n-1}/{m-1\\choose n} &= \\frac{m!n!(m-n-1)!}{(m-1)!(n-1)!(m-n+1)!} \\\\\n&= \\frac{mn}{(m-n+1)(m-n)}.\n\\end{align*}\nThus the condition ${m\\choose{n-1}} > {{m-1}\\choose n}$ is equivalent to $(m-n+1)(m-n)-mn < 0$. The left hand side of this last inequality is a quadratic function of $m$ with roots\n\\begin{align*}\n\\alpha(n) &= \\frac{3n-1+\\sqrt{5n^2-2n+1}}{2}, \\\\\n\\beta(n) &= \\frac{3n-1-\\sqrt{5n^2-2n+1}}{2},\n\\end{align*}\nboth of which are real since $5n^2-2n+1 = 4n^2+(n-1)^2 > 0$; it follows that $m$ satisfies the given inequality if and only if $\\beta(n) < m < \\alpha(n)$. (Note in particular that since $\\alpha(n)-\\beta(n) = \\sqrt{5n^2-2n+1} > 1$, there is always some integer $m$ between $\\beta(n)$ and $\\alpha(n)$.)\n\n\nWe conclude that $M(n)$ is the greatest integer strictly less than $\\alpha(n)$, and thus that \n$\\alpha(n)-1 \\leq M(n) < \\alpha(n)$. Now\n\\[\n\\lim_{n\\to\\infty} \\frac{\\alpha(n)}{n} = \\lim_{n\\to\\infty} \\frac{3-\\frac{1}{n}+\\sqrt{5-\\frac{2}{n}+\\frac{1}{n^2}}}{2}\n= \\frac{3+\\sqrt{5}}{2}\n\\]\nand similarly $\\lim_{n\\to\\infty} \\frac{\\alpha(n)-1}{n} = \\frac{3+\\sqrt{5}}{2}$, and so by the sandwich theorem, $\\lim_{n\\to\\infty} \\frac{M(n)}{n} = \\frac{3+\\sqrt{5}}{2}$.",
  "vars": [
    "m",
    "n",
    "M"
  ],
  "params": [],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "m": "searchint",
        "n": "indexvar",
        "M": "maxfunc"
      },
      "question": "Given a positive integer $indexvar$, let $maxfunc(indexvar)$ be the largest integer $searchint$ such that \n\\[\n\\binom{searchint}{indexvar-1} > \\binom{searchint-1}{indexvar}.\n\\]\nEvaluate \n\\[\n\\lim_{indexvar \\to \\infty} \\frac{maxfunc(indexvar)}{indexvar}.\n\\]",
      "solution": "The answer is $\\frac{3+\\sqrt{5}}{2}$. Note that for $searchint > indexvar+1$, both binomial coefficients are nonzero and their ratio is\n\\begin{align*}\n{searchint\\choose indexvar-1}/{searchint-1\\choose indexvar} &= \\frac{searchint!indexvar!(searchint-indexvar-1)!}{(searchint-1)!(indexvar-1)!(searchint-indexvar+1)!} \\\\\n&= \\frac{searchint\\,indexvar}{(searchint-indexvar+1)(searchint-indexvar)}.\n\\end{align*}\nThus the condition ${searchint\\choose{indexvar-1}} > {{searchint-1}\\choose indexvar}$ is equivalent to $(searchint-indexvar+1)(searchint-indexvar)-searchint\\,indexvar < 0$. The left-hand side of this last inequality is a quadratic function of $searchint$ with roots\n\\begin{align*}\n\\alpha(indexvar) &= \\frac{3indexvar-1+\\sqrt{5indexvar^2-2indexvar+1}}{2}, \\\\\n\\beta(indexvar) &= \\frac{3indexvar-1-\\sqrt{5indexvar^2-2indexvar+1}}{2},\n\\end{align*}\nboth of which are real since $5indexvar^2-2indexvar+1 = 4indexvar^2+(indexvar-1)^2 > 0$; it follows that $searchint$ satisfies the given inequality if and only if $\\beta(indexvar) < searchint < \\alpha(indexvar)$. (Note in particular that since $\\alpha(indexvar)-\\beta(indexvar) = \\sqrt{5indexvar^2-2indexvar+1} > 1$, there is always some integer $searchint$ between $\\beta(indexvar)$ and $\\alpha(indexvar)$.)\n\nWe conclude that $maxfunc(indexvar)$ is the greatest integer strictly less than $\\alpha(indexvar)$, and thus that \n$\\alpha(indexvar)-1 \\leq maxfunc(indexvar) < \\alpha(indexvar)$. Now\n\\[\n\\lim_{indexvar\\to\\infty} \\frac{\\alpha(indexvar)}{indexvar} = \\lim_{indexvar\\to\\infty} \\frac{3-\\frac{1}{indexvar}+\\sqrt{5-\\frac{2}{indexvar}+\\frac{1}{indexvar^2}}}{2} = \\frac{3+\\sqrt{5}}{2}\n\\]\nand similarly $\\lim_{indexvar\\to\\infty} \\frac{\\alpha(indexvar)-1}{indexvar} = \\frac{3+\\sqrt{5}}{2}$, and so by the sandwich theorem, $\\lim_{indexvar\\to\\infty} \\frac{maxfunc(indexvar)}{indexvar} = \\frac{3+\\sqrt{5}}{2}$."
    },
    "descriptive_long_confusing": {
      "map": {
        "m": "sandcastle",
        "n": "cloudless",
        "M": "astronomy"
      },
      "question": "Given a positive integer $cloudless$, let $astronomy(cloudless)$ be the largest integer $sandcastle$ such that \n\\[\n\\binom{sandcastle}{cloudless-1} > \\binom{sandcastle-1}{cloudless}.\n\\]\nEvaluate \n\\[\n\\lim_{cloudless \\to \\infty} \\frac{astronomy(cloudless)}{cloudless}.\n\\]",
      "solution": "The answer is $\\frac{3+\\sqrt{5}}{2}$. Note that for $sandcastle > cloudless+1$, both binomial coefficients are nonzero and their ratio is\n\\begin{align*}\n{sandcastle\\choose cloudless-1}/{sandcastle-1\\choose cloudless} &= \\frac{sandcastle!cloudless!(sandcastle-cloudless-1)!}{(sandcastle-1)!(cloudless-1)!(sandcastle-cloudless+1)!} \\\n&= \\frac{sandcastle\\,cloudless}{(sandcastle-cloudless+1)(sandcastle-cloudless)}.\n\\end{align*}\nThus the condition ${sandcastle\\choose{cloudless-1}} > {{sandcastle-1}\\choose cloudless}$ is equivalent to $(sandcastle-cloudless+1)(sandcastle-cloudless)-sandcastle\\,cloudless < 0$. The left hand side of this last inequality is a quadratic function of $sandcastle$ with roots\n\\begin{align*}\n\\alpha(cloudless) &= \\frac{3cloudless-1+\\sqrt{5cloudless^2-2cloudless+1}}{2}, \\\\\n\\beta(cloudless) &= \\frac{3cloudless-1-\\sqrt{5cloudless^2-2cloudless+1}}{2},\n\\end{align*}\nboth of which are real since $5cloudless^2-2cloudless+1 = 4cloudless^2+(cloudless-1)^2 > 0$; it follows that $sandcastle$ satisfies the given inequality if and only if $\\beta(cloudless) < sandcastle < \\alpha(cloudless)$. (Note in particular that since $\\alpha(cloudless)-\\beta(cloudless) = \\sqrt{5cloudless^2-2cloudless+1} > 1$, there is always some integer $sandcastle$ between $\\beta(cloudless)$ and $\\alpha(cloudless)$.)\n\nWe conclude that $astronomy(cloudless)$ is the greatest integer strictly less than $\\alpha(cloudless)$, and thus that \n$\\alpha(cloudless)-1 \\leq astronomy(cloudless) < \\alpha(cloudless)$. Now\n\\[\n\\lim_{cloudless\\to\\infty} \\frac{\\alpha(cloudless)}{cloudless} = \\lim_{cloudless\\to\\infty} \\frac{3-\\frac{1}{cloudless}+\\sqrt{5-\\frac{2}{cloudless}+\\frac{1}{cloudless^2}}}{2}\n= \\frac{3+\\sqrt{5}}{2}\n\\]\nand similarly $\\lim_{cloudless\\to\\infty} \\frac{\\alpha(cloudless)-1}{cloudless} = \\frac{3+\\sqrt{5}}{2}$, and so by the sandwich theorem, $\\lim_{cloudless\\to\\infty} \\frac{astronomy(cloudless)}{cloudless} = \\frac{3+\\sqrt{5}}{2}$. "
    },
    "descriptive_long_misleading": {
      "map": {
        "m": "tinycount",
        "n": "negativevalue",
        "M": "smallestnum"
      },
      "question": "Given a positive integer $negativevalue$, let $smallestnum(negativevalue)$ be the largest integer $tinycount$ such that \n\\[\n\\binom{tinycount}{negativevalue-1} > \\binom{tinycount-1}{negativevalue}.\n\\]\nEvaluate \n\\[\n\\lim_{negativevalue \\to \\infty} \\frac{smallestnum(negativevalue)}{negativevalue}.\n\\]",
      "solution": "The answer is $\\frac{3+\\sqrt{5}}{2}$. Note that for $tinycount > negativevalue+1$, both binomial coefficients are nonzero and their ratio is\n\\begin{align*}\n{tinycount\\choose negativevalue-1}/{tinycount-1\\choose negativevalue} &= \\frac{tinycount! \\, negativevalue! \\, (tinycount-negativevalue-1)!}{(tinycount-1)! \\, (negativevalue-1)! \\, (tinycount-negativevalue+1)!} \\\\\n&= \\frac{tinycount \\, negativevalue}{(tinycount-negativevalue+1)(tinycount-negativevalue)}.\n\\end{align*}\nThus the condition ${tinycount\\choose{negativevalue-1}} > {{tinycount-1}\\choose negativevalue}$ is equivalent to $(tinycount-negativevalue+1)(tinycount-negativevalue)-tinycount\\,negativevalue < 0$. The left hand side of this last inequality is a quadratic function of $tinycount$ with roots\n\\begin{align*}\n\\alpha(negativevalue) &= \\frac{3\\,negativevalue-1+\\sqrt{5\\,negativevalue^2-2\\,negativevalue+1}}{2}, \\\\\n\\beta(negativevalue) &= \\frac{3\\,negativevalue-1-\\sqrt{5\\,negativevalue^2-2\\,negativevalue+1}}{2},\n\\end{align*}\nboth of which are real since $5\\,negativevalue^2-2\\,negativevalue+1 = 4\\,negativevalue^2+(negativevalue-1)^2 > 0$; it follows that $tinycount$ satisfies the given inequality if and only if $\\beta(negativevalue) < tinycount < \\alpha(negativevalue)$. (Note in particular that since $\\alpha(negativevalue)-\\beta(negativevalue) = \\sqrt{5\\,negativevalue^2-2\\,negativevalue+1} > 1$, there is always some integer $tinycount$ between $\\beta(negativevalue)$ and $\\alpha(negativevalue)$.)\n\nWe conclude that $smallestnum(negativevalue)$ is the greatest integer strictly less than $\\alpha(negativevalue)$, and thus that \n$\\alpha(negativevalue)-1 \\leq smallestnum(negativevalue) < \\alpha(negativevalue)$. Now\n\\[\n\\lim_{negativevalue\\to\\infty} \\frac{\\alpha(negativevalue)}{negativevalue} = \\lim_{negativevalue\\to\\infty} \\frac{3-\\frac{1}{negativevalue}+\\sqrt{5-\\frac{2}{negativevalue}+\\frac{1}{negativevalue^2}}}{2}\n= \\frac{3+\\sqrt{5}}{2}\n\\]\nand similarly $\\lim_{negativevalue\\to\\infty} \\frac{\\alpha(negativevalue)-1}{negativevalue} = \\frac{3+\\sqrt{5}}{2}$, and so by the sandwich theorem, $\\lim_{negativevalue\\to\\infty} \\frac{smallestnum(negativevalue)}{negativevalue} = \\frac{3+\\sqrt{5}}{2}$. "
    },
    "garbled_string": {
      "map": {
        "m": "hjgrksla",
        "n": "qzxwvtnp",
        "M": "vbfjdksa"
      },
      "question": "Given a positive integer $qzxwvtnp$, let $vbfjdksa(qzxwvtnp)$ be the largest integer $hjgrksla$ such that \n\\[\n\\binom{hjgrksla}{qzxwvtnp-1} > \\binom{hjgrksla-1}{qzxwvtnp}.\n\\]\nEvaluate \n\\[\n\\lim_{qzxwvtnp \\to \\infty} \\frac{vbfjdksa(qzxwvtnp)}{qzxwvtnp}.\n\\]",
      "solution": "The answer is $\\frac{3+\\sqrt{5}}{2}$. Note that for $hjgrksla > qzxwvtnp+1$, both binomial coefficients are nonzero and their ratio is\n\\begin{align*}\n{hjgrksla\\choose qzxwvtnp-1}/{hjgrksla-1\\choose qzxwvtnp} &= \\frac{hjgrksla!qzxwvtnp!(hjgrksla-qzxwvtnp-1)!}{(hjgrksla-1)!(qzxwvtnp-1)!(hjgrksla-qzxwvtnp+1)!} \\\\\n&= \\frac{hjgrksla qzxwvtnp}{(hjgrksla-qzxwvtnp+1)(hjgrksla-qzxwvtnp)}.\n\\end{align*}\nThus the condition ${hjgrksla\\choose{qzxwvtnp-1}} > {{hjgrksla-1}\\choose qzxwvtnp}$ is equivalent to $(hjgrksla-qzxwvtnp+1)(hjgrksla-qzxwvtnp)-hjgrksla qzxwvtnp < 0$. The left hand side of this last inequality is a quadratic function of $hjgrksla$ with roots\n\\begin{align*}\n\\alpha(qzxwvtnp) &= \\frac{3qzxwvtnp-1+\\sqrt{5qzxwvtnp^2-2qzxwvtnp+1}}{2}, \\\\\n\\beta(qzxwvtnp) &= \\frac{3qzxwvtnp-1-\\sqrt{5qzxwvtnp^2-2qzxwvtnp+1}}{2},\n\\end{align*}\nboth of which are real since $5qzxwvtnp^2-2qzxwvtnp+1 = 4qzxwvtnp^2+(qzxwvtnp-1)^2 > 0$; it follows that $hjgrksla$ satisfies the given inequality if and only if $\\beta(qzxwvtnp) < hjgrksla < \\alpha(qzxwvtnp)$. (Note in particular that since $\\alpha(qzxwvtnp)-\\beta(qzxwvtnp) = \\sqrt{5qzxwvtnp^2-2qzxwvtnp+1} > 1$, there is always some integer $hjgrksla$ between $\\beta(qzxwvtnp)$ and $\\alpha(qzxwvtnp)$.)\n\n\nWe conclude that $vbfjdksa(qzxwvtnp)$ is the greatest integer strictly less than $\\alpha(qzxwvtnp)$, and thus that \n$\\alpha(qzxwvtnp)-1 \\leq vbfjdksa(qzxwvtnp) < \\alpha(qzxwvtnp)$. Now\n\\[\n\\lim_{qzxwvtnp\\to\\infty} \\frac{\\alpha(qzxwvtnp)}{qzxwvtnp} = \\lim_{qzxwvtnp\\to\\infty} \\frac{3-\\frac{1}{qzxwvtnp}+\\sqrt{5-\\frac{2}{qzxwvtnp}+\\frac{1}{qzxwvtnp^2}}}{2}\n= \\frac{3+\\sqrt{5}}{2}\n\\]\nand similarly $\\lim_{qzxwvtnp\\to\\infty} \\frac{\\alpha(qzxwvtnp)-1}{qzxwvtnp} = \\frac{3+\\sqrt{5}}{2}$, and so by the sandwich theorem, $\\lim_{qzxwvtnp\\to\\infty} \\frac{vbfjdksa(qzxwvtnp)}{qzxwvtnp} = \\frac{3+\\sqrt{5}}{2}$. "
    },
    "kernel_variant": {
      "question": "For every integer n ≥ 3 let M(n) be the greatest integer m satisfying  \n                      𝑪(m, n − 2) > 𝑪(m − 2, n),  \nwhere 𝑪(a, b)= {a choose b}.  Evaluate  \n          lim_{n→∞} M(n)/n.",
      "solution": "(≈ 170 words, two paragraphs)  \nWrite x = m − n.  A routine factorial manipulation gives  \n 𝑪(m, n−2)/𝑪(m−2, n)= n(n−1)m(m−1)/[(m−n+2)(m−n+1)(m−n)(m−n−1)],  \nso the inequality becomes  \n F_n(x):=(x+n)(x+n−1)n(n−1) − (x+2)(x+1)x(x−1) > 0, x ≥ 0.  \nBecause F_n(0)=n^2(n−1)^2>0 while the coefficient of x^4 is −1, F_n decreases from positive values, crosses the axis exactly once, and thereafter stays negative.  Denote this unique positive root by α(n).  The condition F_n(x)>0 is equivalent to 0 ≤ x<α(n), hence  \n M(n)=n+⌊α(n)−1⌋.  \n\nExpanding gives F_n(x)=−x^4+(n^2+n+5)x^2+(2n^2−2n−6)x+n^4−n^3−n^2+n.  Put t=x/n and divide by n^4; at t=α(n)/n the resulting equation is  \n −t^4+(1+1/n)^2t^2+2(1−1/n)t/n+1−1/n=0.  \nLetting n→∞ removes the O(1/n) terms and leaves t^4=(1+t)^2.  For t>0 this reduces to t^2−t−1=0, whose positive solution is the golden ratio φ=(1+√5)/2≈1.618.  Thus α(n)=φ n+O(1), giving φ n−1<M(n)<φ n; by the squeeze theorem  \n          lim_{n→∞} M(n)/n = φ.",
      "_replacement_note": {
        "replaced_at": "2025-07-05T22:17:12.154110",
        "reason": "Original kernel variant was too easy compared to the original problem"
      }
    }
  },
  "checked": true,
  "problem_type": "calculation"
}