{
  "index": "1972-A-6",
  "type": "ANA",
  "tag": [
    "ANA"
  ],
  "difficulty": "",
  "question": "A-6. Let \\( f(x) \\) be an integrable function in \\( 0 \\leqq x \\leqq 1 \\) and suppose \\( \\int_{0}^{1} f(x) d x=0, \\int_{0}^{1} x f(x) d x=0, \\cdots \\), \\( \\int_{0}^{1} x^{n-1} f(x) d x=0 \\) and \\( \\int_{0}^{1} x^{n} f(x) d x=1 \\). Show that \\( |f(x)| \\geqq 2^{n}(n+1) \\) in a set of positive measure.",
  "solution": "A-6 The conditions imply \\( \\int_{0}^{1}\\left(x-\\frac{1}{2}\\right)^{n} f(x) d x=1 \\). Suppose \\( |f(x)|<2^{n}(n+1) \\) except for a set of measure 0 .\n\nThen \\( 1=\\int_{0}^{1}\\left(x-\\frac{1}{2}\\right)^{n} f(x) d x<2^{n}(n+1) \\int_{0}^{1}\\left|x-\\frac{1}{2}\\right|^{n} d x=1 \\), a contradiction.",
  "vars": [
    "x"
  ],
  "params": [
    "f",
    "n"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "variable",
        "f": "function",
        "n": "degree"
      },
      "question": "A-6. Let \\( function(variable) \\) be an integrable function in \\( 0 \\leqq variable \\leqq 1 \\) and suppose \\( \\int_{0}^{1} function(variable) d variable=0, \\int_{0}^{1} variable\\, function(variable) d variable=0, \\cdots \\), \\( \\int_{0}^{1} variable^{degree-1} function(variable) d variable=0 \\) and \\( \\int_{0}^{1} variable^{degree} function(variable) d variable=1 \\). Show that \\( |function(variable)| \\geqq 2^{degree}(degree+1) \\) in a set of positive measure.",
      "solution": "A-6 The conditions imply \\( \\int_{0}^{1}\\left(variable-\\frac{1}{2}\\right)^{degree} function(variable) d variable=1 \\). Suppose \\( |function(variable)|<2^{degree}(degree+1) \\) except for a set of measure 0 .\n\nThen \\( 1=\\int_{0}^{1}\\left(variable-\\frac{1}{2}\\right)^{degree} function(variable) d variable<2^{degree}(degree+1) \\int_{0}^{1}\\left|variable-\\frac{1}{2}\\right|^{degree} d variable=1 \\), a contradiction."
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "marzipans",
        "f": "candlewax",
        "n": "shoelaces"
      },
      "question": "A-6. Let \\( candlewax(marzipans) \\) be an integrable function in \\( 0 \\leqq marzipans \\leqq 1 \\) and suppose \\( \\int_{0}^{1} candlewax(marzipans) d marzipans=0, \\int_{0}^{1} marzipans candlewax(marzipans) d marzipans=0, \\cdots \\), \\( \\int_{0}^{1} marzipans^{shoelaces-1} candlewax(marzipans) d marzipans=0 \\) and \\( \\int_{0}^{1} marzipans^{shoelaces} candlewax(marzipans) d marzipans=1 \\). Show that \\( |candlewax(marzipans)| \\geqq 2^{shoelaces}(shoelaces+1) \\) in a set of positive measure.",
      "solution": "A-6 The conditions imply \\( \\int_{0}^{1}\\left(marzipans-\\frac{1}{2}\\right)^{shoelaces} candlewax(marzipans) d marzipans=1 \\). Suppose \\( |candlewax(marzipans)|<2^{shoelaces}(shoelaces+1) \\) except for a set of measure 0 .\n\nThen \\( 1=\\int_{0}^{1}\\left(marzipans-\\frac{1}{2}\\right)^{shoelaces} candlewax(marzipans) d marzipans<2^{shoelaces}(shoelaces+1) \\int_{0}^{1}\\left|marzipans-\\frac{1}{2}\\right|^{shoelaces} d marzipans=1 \\), a contradiction."
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "constantval",
        "f": "nonfunction",
        "n": "continuous"
      },
      "question": "A-6. Let \\( nonfunction(constantval) \\) be an integrable function in \\( 0 \\leqq constantval \\leqq 1 \\) and suppose \\( \\int_{0}^{1} nonfunction(constantval) d constantval=0, \\int_{0}^{1} constantval nonfunction(constantval) d constantval=0, \\cdots \\), \\( \\int_{0}^{1} constantval^{continuous-1} nonfunction(constantval) d constantval=0 \\) and \\( \\int_{0}^{1} constantval^{continuous} nonfunction(constantval) d constantval=1 \\). Show that \\( |nonfunction(constantval)| \\geqq 2^{continuous}(continuous+1) \\) in a set of positive measure.",
      "solution": "A-6 The conditions imply \\( \\int_{0}^{1}\\left(constantval-\\frac{1}{2}\\right)^{continuous} nonfunction(constantval) d constantval=1 \\). Suppose \\( |nonfunction(constantval)|<2^{continuous}(continuous+1) \\) except for a set of measure 0 .\n\nThen \\( 1=\\int_{0}^{1}\\left(constantval-\\frac{1}{2}\\right)^{continuous} nonfunction(constantval) d constantval<2^{continuous}(continuous+1) \\int_{0}^{1}\\left|constantval-\\frac{1}{2}\\right|^{continuous} d constantval=1 \\), a contradiction."
    },
    "garbled_string": {
      "map": {
        "x": "qzxwvtnp",
        "f": "hjgrksla",
        "n": "rckpldvu"
      },
      "question": "A-6. Let \\( hjgrksla(qzxwvtnp) \\) be an integrable function in \\( 0 \\leqq qzxwvtnp \\leqq 1 \\) and suppose \\( \\int_{0}^{1} hjgrksla(qzxwvtnp) d qzxwvtnp=0, \\int_{0}^{1} qzxwvtnp hjgrksla(qzxwvtnp) d qzxwvtnp=0, \\cdots \\), \\( \\int_{0}^{1} qzxwvtnp^{rckpldvu-1} hjgrksla(qzxwvtnp) d qzxwvtnp=0 \\) and \\( \\int_{0}^{1} qzxwvtnp^{rckpldvu} hjgrksla(qzxwvtnp) d qzxwvtnp=1 \\). Show that \\( |hjgrksla(qzxwvtnp)| \\geqq 2^{rckpldvu}(rckpldvu+1) \\) in a set of positive measure.",
      "solution": "A-6 The conditions imply \\( \\int_{0}^{1}\\left(qzxwvtnp-\\frac{1}{2}\\right)^{rckpldvu} hjgrksla(qzxwvtnp) d qzxwvtnp=1 \\). Suppose \\( |hjgrksla(qzxwvtnp)|<2^{rckpldvu}(rckpldvu+1) \\) except for a set of measure 0 .\n\nThen \\( 1=\\int_{0}^{1}\\left(qzxwvtnp-\\frac{1}{2}\\right)^{rckpldvu} hjgrksla(qzxwvtnp) d qzxwvtnp<2^{rckpldvu}(rckpldvu+1) \\int_{0}^{1}\\left|qzxwvtnp-\\frac{1}{2}\\right|^{rckpldvu} d qzxwvtnp=1 \\), a contradiction."
    },
    "kernel_variant": {
      "question": "Let $d\\ge 2$ and $n\\ge 1$ be fixed integers.  \nLet  \n\\[\nf:[0,1]^{d}\\longrightarrow\\mathbb R\n\\]\nbe an integrable function that satisfies the moment conditions  \n\n\\[\n\\int_{[0,1]^{d}}x_{1}^{\\alpha_{1}}\\cdots x_{d}^{\\alpha_{d}}\\,\n      f(x_{1},\\dots ,x_{d})\\,\n      dx_{1}\\dots dx_{d}=0\n\\qquad\\text{for every multi–index }\\alpha=(\\alpha_{1},\\dots ,\\alpha_{d})\n\\text{ with }|\\alpha|:=\\alpha_{1}+\\dots +\\alpha_{d}\\le nd-1,\n\\]\n\n\\[\n\\int_{[0,1]^{d}}x_{1}^{n}\\,x_{2}^{n}\\dots x_{d}^{n}\\,\n      f(x_{1},\\dots ,x_{d})\\,\n      dx_{1}\\dots dx_{d}=1 .\n\\]\n\nProve that the set  \n\\[\nE_{\\ast}:=\\bigl\\{(x_{1},\\dots ,x_{d})\\in[0,1]^{d}:\n            \\,|f(x_{1},\\dots ,x_{d})|\\ge 2^{nd}(n+1)^{d}\\bigr\\}\n\\]\nhas strictly positive Lebesgue measure.  In particular,  \n\n\\[\n\\operatorname*{ess\\,sup}_{(x_{1},\\dots ,x_{d})\\in[0,1]^{d}}\n      |f(x_{1},\\dots ,x_{d})|\n      \\;\\ge\\;2^{nd}(n+1)^{d}.\n\\]",
      "solution": "Step 1 – A degree–$nd$ test polynomial whose integral against $f$ equals $1$.  \nDefine  \n\\[\nP(x_{1},\\dots ,x_{d}):=\\prod_{i=1}^{d}\\bigl(x_{i}-\\tfrac12\\bigr)^{n}.\n\\tag{1}\n\\]\nExpanding one factor yields  \n\\[\n\\bigl(x_{i}-\\tfrac12\\bigr)^{n}=\\sum_{k=0}^{n}\\binom{n}{k}\n      \\bigl(-\\tfrac12\\bigr)^{\\,n-k}x_{i}^{k}.\n\\tag{2}\n\\]\nHence $P$ is a linear combination of monomials\n$x_{1}^{k_{1}}\\cdots x_{d}^{k_{d}}$ with total degree\n$k_{1}+\\dots +k_{d}\\le nd$.  \nThe coefficient of the unique top–degree monomial\n$x_{1}^{n}\\dots x_{d}^{n}$ is $1$ (every factor contributes the $x_{i}^{n}$–term\nwith coefficient $1$ and any other choice lowers at least one exponent).\n\nApplying the hypotheses to the expansion of $P$ gives  \n\\[\n\\int_{[0,1]^{d}}P\\,f\n      =\\int_{[0,1]^{d}}x_{1}^{n}\\dots x_{d}^{n}\\,f\n      =1,\n\\tag{3}\n\\]\nbecause all lower–degree terms vanish by the first moment condition.\n\nStep 2 – The $L^{1}$–norm of $P$.  \nBy Fubini and symmetry,\n\\[\n\\int_{[0,1]^{d}}|P|\n    =\\prod_{i=1}^{d}\\int_{0}^{1}\n        \\bigl|x_{i}-\\tfrac12\\bigr|^{n}\\,dx_{i}.\n\\tag{4}\n\\]\nFor a single factor,\n\\[\n\\int_{0}^{1}\\bigl|x-\\tfrac12\\bigr|^{n}\\,dx\n    =2\\int_{0}^{1/2}\\bigl(\\tfrac12-x\\bigr)^{n}\\,dx\n    =\\frac1{2^{\\,n}(n+1)}.\n\\tag{5}\n\\]\nTaking the product of $d$ identical factors,\n\\[\n\\int_{[0,1]^{d}}|P|\n      =\\Bigl(\\frac1{2^{\\,n}(n+1)}\\Bigr)^{d}\n      =\\frac1{2^{\\,nd}(n+1)^{d}}.\n\\tag{6}\n\\]\n\nStep 3 – Assuming $|f|$ is too small leads to a contradiction.  \nSuppose, contrary to what we want to prove, that  \n\\[\n|f(x_{1},\\dots ,x_{d})|<2^{\\,nd}(n+1)^{d}\n\\quad\\text{for almost every }(x_{1},\\dots ,x_{d})\\in[0,1]^{d}.\n\\tag{7}\n\\]\nThen by Hölder’s inequality (here, plain absolute convergence suffices) and\nequations (3)–(6),\n\\[\n1=\\Bigl|\\int_{[0,1]^{d}}P\\,f\\Bigr|\n  \\le \\int_{[0,1]^{d}}|P|\\,|f|\n  < 2^{\\,nd}(n+1)^{d}\\cdot\\frac1{2^{\\,nd}(n+1)^{d}}\n  =1,\n\\]\na contradiction.  Hence assumption (7) fails.\n\nStep 4 – Positivity of the critical level set.  \nSince (7) is false, the complement of the set  \n\\[\nE_{\\ast}:=\\bigl\\{x\\in[0,1]^{d}:\\,\n        |f(x)|\\ge 2^{\\,nd}(n+1)^{d}\\bigr\\}\n\\]\ncannot have full measure.  Therefore $\\mu(E_{\\ast})>0$, which is exactly the\ndesired statement.  In particular,\n\\[\n\\operatorname*{ess\\,sup}_{x\\in[0,1]^{d}}|f(x)|\n      \\ge 2^{\\,nd}(n+1)^{d}.\n\\]",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.605356",
        "was_fixed": false,
        "difficulty_analysis": "• Dimension jump:  The problem moves from a single interval to a d-dimensional cube, requiring multi–variable integration, multi-indices and Fubini.  \n• Exponent explosion:  Instead of vanishing moments up to degree n–1, the function must annihilate every monomial of total degree ≤ nd−1; the number of separate constraints grows combinatorially with both n and d.  \n• High-degree target polynomial:  The crucial polynomial P now has degree nd and depends on every coordinate, so computing its L¹–norm forces the solver to separate variables, handle products, and evaluate Beta–function type integrals.  \n• Sharper constant:  The required lower bound is 2^{nd}(n+1)^{d}, exponentially larger than in the original, and its derivation hinges on keeping track of dimension-dependent factors.  \n• Same core idea, deeper execution:  While the contradiction principle mirrors the original, carrying it out demands considerably more bookkeeping, higher-dimensional reasoning, and comfort with multi-index notation and product measures."
      }
    },
    "original_kernel_variant": {
      "question": "Let $d\\ge 2$ and $n\\ge 1$ be fixed integers.  \nLet  \n\\[\nf:[0,1]^{d}\\longrightarrow\\mathbb R\n\\]\nbe an integrable function that satisfies the moment conditions  \n\n\\[\n\\int_{[0,1]^{d}}x_{1}^{\\alpha_{1}}\\cdots x_{d}^{\\alpha_{d}}\\,\n      f(x_{1},\\dots ,x_{d})\\,\n      dx_{1}\\dots dx_{d}=0\n\\qquad\\text{for every multi–index }\\alpha=(\\alpha_{1},\\dots ,\\alpha_{d})\n\\text{ with }|\\alpha|:=\\alpha_{1}+\\dots +\\alpha_{d}\\le nd-1,\n\\]\n\n\\[\n\\int_{[0,1]^{d}}x_{1}^{n}\\,x_{2}^{n}\\dots x_{d}^{n}\\,\n      f(x_{1},\\dots ,x_{d})\\,\n      dx_{1}\\dots dx_{d}=1 .\n\\]\n\nProve that the set  \n\\[\nE_{\\ast}:=\\bigl\\{(x_{1},\\dots ,x_{d})\\in[0,1]^{d}:\n            \\,|f(x_{1},\\dots ,x_{d})|\\ge 2^{nd}(n+1)^{d}\\bigr\\}\n\\]\nhas strictly positive Lebesgue measure.  In particular,  \n\n\\[\n\\operatorname*{ess\\,sup}_{(x_{1},\\dots ,x_{d})\\in[0,1]^{d}}\n      |f(x_{1},\\dots ,x_{d})|\n      \\;\\ge\\;2^{nd}(n+1)^{d}.\n\\]",
      "solution": "Step 1 – A degree–$nd$ test polynomial whose integral against $f$ equals $1$.  \nDefine  \n\\[\nP(x_{1},\\dots ,x_{d}):=\\prod_{i=1}^{d}\\bigl(x_{i}-\\tfrac12\\bigr)^{n}.\n\\tag{1}\n\\]\nExpanding one factor yields  \n\\[\n\\bigl(x_{i}-\\tfrac12\\bigr)^{n}=\\sum_{k=0}^{n}\\binom{n}{k}\n      \\bigl(-\\tfrac12\\bigr)^{\\,n-k}x_{i}^{k}.\n\\tag{2}\n\\]\nHence $P$ is a linear combination of monomials\n$x_{1}^{k_{1}}\\cdots x_{d}^{k_{d}}$ with total degree\n$k_{1}+\\dots +k_{d}\\le nd$.  \nThe coefficient of the unique top–degree monomial\n$x_{1}^{n}\\dots x_{d}^{n}$ is $1$ (every factor contributes the $x_{i}^{n}$–term\nwith coefficient $1$ and any other choice lowers at least one exponent).\n\nApplying the hypotheses to the expansion of $P$ gives  \n\\[\n\\int_{[0,1]^{d}}P\\,f\n      =\\int_{[0,1]^{d}}x_{1}^{n}\\dots x_{d}^{n}\\,f\n      =1,\n\\tag{3}\n\\]\nbecause all lower–degree terms vanish by the first moment condition.\n\nStep 2 – The $L^{1}$–norm of $P$.  \nBy Fubini and symmetry,\n\\[\n\\int_{[0,1]^{d}}|P|\n    =\\prod_{i=1}^{d}\\int_{0}^{1}\n        \\bigl|x_{i}-\\tfrac12\\bigr|^{n}\\,dx_{i}.\n\\tag{4}\n\\]\nFor a single factor,\n\\[\n\\int_{0}^{1}\\bigl|x-\\tfrac12\\bigr|^{n}\\,dx\n    =2\\int_{0}^{1/2}\\bigl(\\tfrac12-x\\bigr)^{n}\\,dx\n    =\\frac1{2^{\\,n}(n+1)}.\n\\tag{5}\n\\]\nTaking the product of $d$ identical factors,\n\\[\n\\int_{[0,1]^{d}}|P|\n      =\\Bigl(\\frac1{2^{\\,n}(n+1)}\\Bigr)^{d}\n      =\\frac1{2^{\\,nd}(n+1)^{d}}.\n\\tag{6}\n\\]\n\nStep 3 – Assuming $|f|$ is too small leads to a contradiction.  \nSuppose, contrary to what we want to prove, that  \n\\[\n|f(x_{1},\\dots ,x_{d})|<2^{\\,nd}(n+1)^{d}\n\\quad\\text{for almost every }(x_{1},\\dots ,x_{d})\\in[0,1]^{d}.\n\\tag{7}\n\\]\nThen by Hölder’s inequality (here, plain absolute convergence suffices) and\nequations (3)–(6),\n\\[\n1=\\Bigl|\\int_{[0,1]^{d}}P\\,f\\Bigr|\n  \\le \\int_{[0,1]^{d}}|P|\\,|f|\n  < 2^{\\,nd}(n+1)^{d}\\cdot\\frac1{2^{\\,nd}(n+1)^{d}}\n  =1,\n\\]\na contradiction.  Hence assumption (7) fails.\n\nStep 4 – Positivity of the critical level set.  \nSince (7) is false, the complement of the set  \n\\[\nE_{\\ast}:=\\bigl\\{x\\in[0,1]^{d}:\\,\n        |f(x)|\\ge 2^{\\,nd}(n+1)^{d}\\bigr\\}\n\\]\ncannot have full measure.  Therefore $\\mu(E_{\\ast})>0$, which is exactly the\ndesired statement.  In particular,\n\\[\n\\operatorname*{ess\\,sup}_{x\\in[0,1]^{d}}|f(x)|\n      \\ge 2^{\\,nd}(n+1)^{d}.\n\\]",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.484242",
        "was_fixed": false,
        "difficulty_analysis": "• Dimension jump:  The problem moves from a single interval to a d-dimensional cube, requiring multi–variable integration, multi-indices and Fubini.  \n• Exponent explosion:  Instead of vanishing moments up to degree n–1, the function must annihilate every monomial of total degree ≤ nd−1; the number of separate constraints grows combinatorially with both n and d.  \n• High-degree target polynomial:  The crucial polynomial P now has degree nd and depends on every coordinate, so computing its L¹–norm forces the solver to separate variables, handle products, and evaluate Beta–function type integrals.  \n• Sharper constant:  The required lower bound is 2^{nd}(n+1)^{d}, exponentially larger than in the original, and its derivation hinges on keeping track of dimension-dependent factors.  \n• Same core idea, deeper execution:  While the contradiction principle mirrors the original, carrying it out demands considerably more bookkeeping, higher-dimensional reasoning, and comfort with multi-index notation and product measures."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}