{
  "index": "2016-A-1",
  "type": "ANA",
  "tag": [
    "ANA",
    "NT"
  ],
  "difficulty": "",
  "question": "Find the smallest positive integer $j$ such that for every polynomial $p(x)$ with integer coefficients and for every integer $k$, the integer\n\\[\np^{(j)}(k) = \\left. \\frac{d^j}{dx^j} p(x) \\right|_{x=k} \n\\] \n(the $j$-th derivative of $p(x)$ at $k$) is divisible by 2016.",
  "solution": "The answer is $j=8$. First suppose that $j$ satisfies the given condition. For $p(x) = x^j$, we have $p^{(j)}(x) = j!$ and thus $j!$ is divisible by $2016$. Since $2016$ is divisible by $2^5$ and $7!$ is not, it follows that $j \\geq 8$. Conversely, we claim that $j=8$ works. Indeed, let $p(x) = \\sum_{m=0}^n a_m x^m$ be a polynomial with integer coefficients; then if $k$ is any integer, \n\\begin{align*}\np^{(8)}(k) &= \\sum_{m=8}^n m(m-1)\\cdots (m-7) a_m k^{m-8} \\\\\n&= \\sum_{m=8}^n {m\\choose 8} 8! a_m k^{m-8}\n\\end{align*}\nis divisible by $8! = 20 \\cdot 2016$, and so $p^{(8)}(k)$ is divisible by $2016$.\n\n\\noindent\n\\textbf{Remark:}\nBy the same reasoning, if one replaces $2016$ in the problem by a general integer $N$,\nthen the minimum value of $j$ is the smallest one for which $N$ divides $j!$.\nThis can be deduced from P\\'olya's observation that the set of integer-valued polynomials is the free $\\ZZ$-module generated by the binomial polynomials $\\binom{x}{n}$ for $n=0,1,\\dots$. That statement can be extended to polynomials evaluated on a subset of a Dedekind domain using Bhargava's method of \\emph{$P$-orderings}; we do not know if this generalization can be adapted to the analogue of this problem, where one considers polynomials whose $j$-th derivatives take integral values on a prescribed subset.",
  "vars": [
    "j",
    "x",
    "k",
    "m",
    "n"
  ],
  "params": [
    "p",
    "a_m",
    "N"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "j": "derivindex",
        "x": "varinput",
        "k": "integerk",
        "m": "summand",
        "n": "degree",
        "p": "polynomial",
        "a_m": "coeffsummand",
        "N": "divisor"
      },
      "question": "Find the smallest positive integer derivindex such that for every polynomial polynomial(varinput) with integer coefficients and for every integer integerk, the integer\n\\[\npolynomial^{(\\text{derivindex})}(integerk) = \\left. \\frac{d^{\\text{derivindex}}}{d{varinput}^{\\text{derivindex}}} polynomial(varinput) \\right|_{varinput=integerk} \n\\] \n(the derivindex-th derivative of polynomial(varinput) at integerk) is divisible by 2016.",
      "solution": "The answer is derivindex=8. First suppose that derivindex satisfies the given condition. For polynomial(varinput)=varinput^{\\text{derivindex}}, we have polynomial^{(\\text{derivindex})}(varinput)=derivindex! and thus derivindex! is divisible by 2016. Since 2016 is divisible by 2^5 and 7! is not, it follows that derivindex\\ge8. Conversely, we claim that derivindex=8 works. Indeed, let polynomial(varinput)=\\sum_{summand=0}^{degree} coeffsummand\\,varinput^{summand} be a polynomial with integer coefficients; then if integerk is any integer,\n\\begin{align*}\npolynomial^{(8)}(integerk) &= \\sum_{summand=8}^{degree} summand(summand-1)\\cdots(summand-7)\\,coeffsummand\\,integerk^{summand-8} \\\n&= \\sum_{summand=8}^{degree} {summand\\choose 8}\\,8!\\,coeffsummand\\,integerk^{summand-8}\n\\end{align*}\nis divisible by 8!=20\\cdot2016, and so polynomial^{(8)}(integerk) is divisible by 2016.\n\nRemark:\nBy the same reasoning, if one replaces 2016 in the problem by a general integer divisor, then the minimum value of derivindex is the smallest one for which divisor divides derivindex!. This can be deduced from P\\'olya's observation that the set of integer-valued polynomials is the free $\\ZZ$-module generated by the binomial polynomials $\\binom{varinput}{degree}$ for $degree=0,1,\\dots$. That statement can be extended to polynomials evaluated on a subset of a Dedekind domain using Bhargava's method of \\emph{$P$-orderings}; we do not know if this generalization can be adapted to the analogue of this problem, where one considers polynomials whose derivindex-th derivatives take integral values on a prescribed subset."
    },
    "descriptive_long_confusing": {
      "map": {
        "j": "butterfly",
        "x": "pineapple",
        "k": "bookshelf",
        "m": "lemonade",
        "n": "chocolate",
        "p": "notebook",
        "a_m": "marshmallow",
        "N": "raincloud"
      },
      "question": "Find the smallest positive integer $butterfly$ such that for every polynomial $notebook(pineapple)$ with integer coefficients and for every integer $bookshelf$, the integer\n\\[\nnotebook^{(butterfly)}(bookshelf) = \\left. \\frac{d^{butterfly}}{d pineapple^{butterfly}} notebook(pineapple) \\right|_{pineapple=bookshelf} \n\\]\n(the $butterfly$-th derivative of $notebook(pineapple)$ at $bookshelf$) is divisible by 2016.",
      "solution": "The answer is $butterfly=8$. First suppose that $butterfly$ satisfies the given condition. For $notebook(pineapple) = pineapple^{butterfly}$, we have $notebook^{(butterfly)}(pineapple) = butterfly!$ and thus $butterfly!$ is divisible by $2016$. Since $2016$ is divisible by $2^5$ and $7!$ is not, it follows that $butterfly \\ge 8$. Conversely, we claim that $butterfly=8$ works. Indeed, let $notebook(pineapple) = \\sum_{lemonade=0}^{chocolate} marshmallow\\, pineapple^{lemonade}$ be a polynomial with integer coefficients; then if $bookshelf$ is any integer,\n\\begin{align*}\nnotebook^{(8)}(bookshelf) &= \\sum_{lemonade=8}^{chocolate} lemonade(lemonade-1)\\cdots (lemonade-7)\\, marshmallow\\, bookshelf^{lemonade-8} \\\\\n&= \\sum_{lemonade=8}^{chocolate} {lemonade\\choose 8}\\, 8!\\, marshmallow\\, bookshelf^{lemonade-8}\n\\end{align*}\nis divisible by $8! = 20 \\cdot 2016$, and so $notebook^{(8)}(bookshelf)$ is divisible by 2016.\n\n\\noindent\n\\textbf{Remark:}\nBy the same reasoning, if one replaces 2016 in the problem by a general integer $raincloud$, then the minimum value of $butterfly$ is the smallest one for which $raincloud$ divides $butterfly!$. This can be deduced from P\\'olya's observation that the set of integer-valued polynomials is the free $\\ZZ$-module generated by the binomial polynomials $\\binom{pineapple}{chocolate}$ for $chocolate=0,1,\\dots$. That statement can be extended to polynomials evaluated on a subset of a Dedekind domain using Bhargava's method of \\emph{$P$-orderings}; we do not know if this generalization can be adapted to the analogue of this problem, where one considers polynomials whose $butterfly$-th derivatives take integral values on a prescribed subset."
    },
    "descriptive_long_misleading": {
      "map": {
        "j": "integralindex",
        "x": "constantvalue",
        "k": "fractionpoint",
        "m": "baseindex",
        "n": "depthindex",
        "p": "constantform",
        "a_m": "divisorvalue",
        "N": "fractionalmod"
      },
      "question": "Find the smallest positive integer $integralindex$ such that for every polynomial $constantform(constantvalue)$ with integer coefficients and for every integer $fractionpoint$, the integer\n\\[\nconstantform^{(integralindex)}(fractionpoint) = \\left. \\frac{d^{integralindex}}{dconstantvalue^{integralindex}} constantform(constantvalue) \\right|_{constantvalue=fractionpoint} \n\\] \n(the $integralindex$-th derivative of $constantform(constantvalue)$ at $fractionpoint$) is divisible by 2016.",
      "solution": "The answer is $integralindex=8$. First suppose that $integralindex$ satisfies the given condition. For $constantform(constantvalue) = constantvalue^{integralindex}$, we have $constantform^{(integralindex)}(constantvalue) = integralindex!$ and thus $integralindex!$ is divisible by 2016. Since 2016 is divisible by $2^5$ and $7!$ is not, it follows that $integralindex \\geq 8$. Conversely, we claim that $integralindex=8$ works. Indeed, let $constantform(constantvalue) = \\sum_{baseindex=0}^{depthindex} divisorvalue\\, constantvalue^{baseindex}$ be a polynomial with integer coefficients; then if $fractionpoint$ is any integer,\n\\begin{align*}\nconstantform^{(8)}(fractionpoint) &= \\sum_{baseindex=8}^{depthindex} baseindex(baseindex-1)\\cdots (baseindex-7)\\, divisorvalue\\, fractionpoint^{baseindex-8} \\\\\n&= \\sum_{baseindex=8}^{depthindex} {baseindex\\choose 8}\\, 8!\\, divisorvalue\\, fractionpoint^{baseindex-8}\n\\end{align*}\nis divisible by $8! = 20 \\cdot 2016$, and so $constantform^{(8)}(fractionpoint)$ is divisible by 2016.\n\n\\textbf{Remark:}\nBy the same reasoning, if one replaces 2016 in the problem by a general integer $fractionalmod$, then the minimum value of $integralindex$ is the smallest one for which $fractionalmod$ divides $integralindex!$. This can be deduced from P\\'olya's observation that the set of integer-valued polynomials is the free $\\ZZ$-module generated by the binomial polynomials $\\binom{constantvalue}{depthindex}$ for $depthindex=0,1,\\dots$. That statement can be extended to polynomials evaluated on a subset of a Dedekind domain using Bhargava's method of \\emph{$P$-orderings}; we do not know if this generalization can be adapted to the analogue of this problem, where one considers polynomials whose $integralindex$-th derivatives take integral values on a prescribed subset."
    },
    "garbled_string": {
      "map": {
        "j": "qzxwvtnp",
        "x": "hjgrksla",
        "k": "mprxvaet",
        "m": "ubnylsgc",
        "n": "vadqcpen",
        "p": "twsrkmab",
        "a_m": "lzwqenth",
        "N": "sdmleqyv"
      },
      "question": "Find the smallest positive integer $qzxwvtnp$ such that for every polynomial $twsrkmab(hjgrksla)$ with integer coefficients and for every integer $mprxvaet$, the integer\n\\[\n twsrkmab^{(qzxwvtnp)}(mprxvaet) = \\left. \\frac{d^{qzxwvtnp}}{d{hjgrksla}^{qzxwvtnp}} twsrkmab(hjgrksla) \\right|_{hjgrksla=mprxvaet}\n\\]\n(the $qzxwvtnp$-th derivative of $twsrkmab(hjgrksla)$ at $mprxvaet$) is divisible by 2016.",
      "solution": "The answer is $qzxwvtnp=8$. First suppose that $qzxwvtnp$ satisfies the given condition. For $twsrkmab(hjgrksla) = hjgrksla^{qzxwvtnp}$, we have $twsrkmab^{(qzxwvtnp)}(hjgrksla) = qzxwvtnp!$ and thus $qzxwvtnp!$ is divisible by 2016. Since 2016 is divisible by $2^5$ and $7!$ is not, it follows that $qzxwvtnp \\ge 8$. Conversely, we claim that $qzxwvtnp=8$ works. Indeed, let $twsrkmab(hjgrksla) = \\sum_{ubnylsgc=0}^{vadqcpen} lzwqenth_{ubnylsgc} \\, hjgrksla^{ubnylsgc}$ be a polynomial with integer coefficients; then if $mprxvaet$ is any integer,\n\\begin{align*}\n twsrkmab^{(8)}(mprxvaet) &= \\sum_{ubnylsgc=8}^{vadqcpen} ubnylsgc(ubnylsgc-1)\\cdots (ubnylsgc-7) \\, lzwqenth_{ubnylsgc} \\, mprxvaet^{ubnylsgc-8} \\\\\n &= \\sum_{ubnylsgc=8}^{vadqcpen} {ubnylsgc\\choose 8} 8! \\, lzwqenth_{ubnylsgc} \\, mprxvaet^{ubnylsgc-8}\n\\end{align*}\nis divisible by $8! = 20 \\cdot 2016$, and so $twsrkmab^{(8)}(mprxvaet)$ is divisible by 2016.\n\n\\noindent\n\\textbf{Remark:} By the same reasoning, if one replaces 2016 in the problem by a general integer $sdmleqyv$, then the minimum value of $qzxwvtnp$ is the smallest one for which $sdmleqyv$ divides $qzxwvtnp!$. This can be deduced from P\\'olya's observation that the set of integer-valued polynomials is the free $\\ZZ$-module generated by the binomial polynomials $\\binom{hjgrksla}{vadqcpen}$ for $vadqcpen=0,1,\\dots$. That statement can be extended to polynomials evaluated on a subset of a Dedekind domain using Bhargava's method of \\emph{$P$-orderings}; we do not know if this generalization can be adapted to the analogue of this problem, where one considers polynomials whose $qzxwvtnp$-th derivatives take integral values on a prescribed subset."
    },
    "kernel_variant": {
      "question": "Determine the least positive integer $j$ with the following property:\nfor every polynomial $p(x)$ having integer coefficients and for every integer $k$, the number\n\\[\n          p^{(j)}(k)=\\left.\\frac{d^{\\,j}}{dx^{j}}\\,p(x)\\right|_{x=k}\n\\]\nis divisible by $9450$.",
      "solution": "Write  9450 in its prime factorisation:\n9450 = 2·3^3·5^2·7.\n\n1. (A necessary condition.)\n   Suppose j satisfies the stated divisibility condition.  Evaluate it on the specific polynomial\n   q(x)= (x+1)^j.  Differentiating j times gives q^(j)(x)=j! for every x, so in particular j! must be divisible by 9450.\n\n2. (Finding the smallest such j.)  We determine the minimal j for which 9450|j! by comparing prime–power exponents.\n   • For the factor 7 we need j≥7.\n   • For 3^3 we need ⌊j/3⌋+⌊j/9⌋+⌊j/27⌋ ≥ 3, forcing j≥9.\n   • For 5^2 we need ⌊j/5⌋+⌊j/25⌋ ≥ 2, forcing j≥10.\n   • The single factor 2 causes no further increase.\n   Thus the least candidate is j=10, and indeed\n     10! = 3 628 800 = 384 × 9450,\n   so 9450|10!.  Consequently every admissible j must satisfy j≥10.\n\n3. (Sufficiency of j=10.)  Let p(x)=∑_{m=0}^n a_m x^m (a_m∈ℤ) and fix an integer k.  Repeated differentiation gives\n   p^(10)(k)=∑_{m=10}^n m(m−1)⋯(m−9) a_m k^{m−10}\n            =10! ∑_{m=10}^n (m choose 10) a_m k^{m−10}.\n   The binomial coefficient (m choose 10), the coefficients a_m, and k^{m−10} are all integers, so the right–hand side is an integer multiple of 10!.  Because 10! itself is a multiple of 9450, the expression p^(10)(k) is divisible by 9450 for every p and k.\n\n4. (Conclusion.)  The integer j=10 both is necessary and suffices, hence is minimal.\n\nBoxed answer: j=10",
      "_meta": {
        "core_steps": [
          "Use the test polynomial p(x)=x^j to force the condition N | j! (since p^{(j)}(k)=j!)",
          "Compute the prime factorization of N to find the smallest j with N | j! (lower bound on j)",
          "For that j, write the j-th derivative of a general integer-coefficient polynomial as p^{(j)}(k)=j!·Σ a_m·C(m,j)·k^{m-j}",
          "Because a_m, k, C(m,j) are integers, j! divides every such derivative (upper bound)",
          "Combine the bounds: that j is both necessary and sufficient, hence minimal"
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Modulus whose divisibility is demanded; replacing it by any positive integer leaves the argument intact",
            "original": "2016"
          },
          "slot2": {
            "description": "Specific polynomial used in the necessity step; any monomial of degree j works",
            "original": "p(x)=x^j"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}