{
  "index": "1980-B-4",
  "type": "COMB",
  "tag": [
    "COMB"
  ],
  "difficulty": "",
  "question": "Problem B-4\nLet \\( A_{1}, A_{2}, \\ldots, A_{1066} \\) be subsets of a finite set \\( X \\) suck that \\( \\left|A_{1}\\right|>\\frac{1}{2}|X| \\) for \\( 1 \\leqslant i \\leqslant 1066 \\). Prove there exist ten elements \\( x_{1}, \\ldots, x_{10} \\) of \\( X \\) such that every \\( A_{1} \\) contains at least one of \\( x_{1}, \\ldots, x_{10} \\).\n(Here \\( |S| \\) means the number of elements in the set \\( S \\).)",
  "solution": "B-4.\nThe result we are asked to prove is clearly not true if \\( |X|<10 \\). Hence we assume that \\( |X| \\geqslant 10 \\) or that the \\( A \\), are distinct, which implies that \\( |X| \\geqslant 10 \\).\n\nLet \\( X=\\left\\{x_{1}, \\ldots, x_{m}\\right\\} \\), with \\( m=|X| \\), and let \\( n_{t} \\), be the number of \\( j \\) such that \\( x_{t} \\) is in \\( A_{j} \\). Let \\( N \\) be the number of ordered pairs \\( (i, j) \\) such that \\( x_{i} \\) is in \\( A_{,} \\). Then\n\\[\nN=n_{1}+n_{2}+\\cdots+n_{m}=\\left|A_{1}\\right|+\\left|A_{2}\\right|+\\cdots+\\left|A_{1066}\\right|>1066(\\mathrm{~m} / 2)=533 \\mathrm{~m}\n\\]\n\nHence one of the \\( n_{i} \\), say \\( n_{1} \\), exceeds 533 .\nLet \\( B_{1}, \\ldots, B_{s} \\) be those sets \\( A_{j} \\) not containing \\( x_{1} \\) and \\( Y=\\left\\{x_{2}, x_{3}, \\ldots, x_{m}\\right\\} \\). Then \\( s=1066- \\) \\( n_{1} \\leqslant 532 \\) and each \\( \\left|B_{j}\\right|>|Y| / 2 \\). We can assume that \\( x_{2} \\) is in at least as many \\( B_{j} \\) as any other \\( x_{i} \\) and let \\( C_{1}, \\ldots, C_{t} \\) be the \\( B_{j} \\) not containing \\( x_{2} \\). As before, one can show that \\( t \\leqslant 265 \\).\n\nWe continue in this way. The 4th sequence of sets \\( D_{1}, \\ldots, D_{u} \\) will number no more than 132 . The numbers of sets in the 5 th through 10 th sequences will number no more than \\( 65,32,15,7,3 \\), and 1 , respectively. Thus we obtain the desired elements \\( x_{1}, \\ldots, x_{10} \\) unless \\( X \\) has fewer than 10 elements.",
  "vars": [
    "x_1",
    "x_2",
    "x_3",
    "x_10",
    "x_i",
    "x_t",
    "x_m"
  ],
  "params": [
    "A_1",
    "A_2",
    "A_1066",
    "A_i",
    "A_j",
    "X",
    "m",
    "n_t",
    "n_1",
    "n_2",
    "N",
    "s",
    "B_1",
    "B_s",
    "B_j",
    "Y",
    "C_1",
    "C_t",
    "D_1",
    "D_u",
    "t",
    "u",
    "j",
    "i"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x_1": "elemone",
        "x_2": "elemtwo",
        "x_3": "elemthree",
        "x_10": "elemtenth",
        "x_i": "elemindex",
        "x_t": "elemtindex",
        "x_m": "elemlast",
        "A_1": "subsetone",
        "A_2": "subsettwo",
        "A_1066": "subsetfinal",
        "A_i": "subsetindex",
        "A_j": "subsetjindex",
        "X": "wholeset",
        "m": "totalcount",
        "n_t": "containcount",
        "n_1": "containone",
        "n_2": "containtwo",
        "N": "pairtotal",
        "s": "subsetcount",
        "B_1": "subcollectionone",
        "B_s": "subcollectionlast",
        "B_j": "subcollectionj",
        "Y": "reducedset",
        "C_1": "subgroupone",
        "C_t": "subgrouplast",
        "D_1": "sequenceone",
        "D_u": "sequencelast",
        "t": "stagecount",
        "u": "finalcount",
        "j": "indexjay",
        "i": "indexeye"
      },
      "question": "Problem B-4\nLet \\( subsetone, subsettwo, \\ldots, subsetfinal \\) be subsets of a finite set \\( wholeset \\) suck that \\( \\left|subsetone\\right|>\\frac{1}{2}|wholeset| \\) for \\( 1 \\leqslant indexeye \\leqslant 1066 \\). Prove there exist ten elements \\( elemone, \\ldots, elemtenth \\) of \\( wholeset \\) such that every \\( subsetone \\) contains at least one of \\( elemone, \\ldots, elemtenth \\).\n(Here \\( |S| \\) means the number of elements in the set \\( S \\).)",
      "solution": "B-4.\nThe result we are asked to prove is clearly not true if \\( |wholeset|<10 \\). Hence we assume that \\( |wholeset| \\geqslant 10 \\) or that the \\( A \\), are distinct, which implies that \\( |wholeset| \\geqslant 10 \\).\n\nLet \\( wholeset=\\left\\{elemone, \\ldots, elemlast\\right\\} \\), with \\( totalcount=|wholeset| \\), and let containcount, be the number of indexjay such that elemtindex is in subsetjindex. Let pairtotal be the number of ordered pairs \\( (indexeye, indexjay) \\) such that elemindex is in \\( A_{,} \\). Then\n\\[\npairtotal=containone+containtwo+\\cdots+n_{totalcount}=\\left|subsetone\\right|+\\left|subsettwo\\right|+\\cdots+\\left|subsetfinal\\right|>1066(\\mathrm{~totalcount} / 2)=533 \\mathrm{~totalcount}\n\\]\n\nHence one of the \\( n_{indexeye} \\), say containone, exceeds 533 .\nLet subcollectionone, \\ldots, subcollectionlast be those sets subsetjindex not containing elemone and reducedset=\\left\\{elemtwo, elemthree, \\ldots, elemlast\\right\\}. Then subsetcount=1066- containone \\leqslant 532 and each \\( \\left|subcollectionj\\right|>|reducedset| / 2 \\). We can assume that elemtwo is in at least as many subcollectionj as any other elemindex and let subgroupone, \\ldots, subgrouplast be the subcollectionj not containing elemtwo. As before, one can show that stagecount \\leqslant 265.\n\nWe continue in this way. The 4th sequence of sets sequenceone, \\ldots, sequencelast will number no more than 132 . The numbers of sets in the 5 th through 10 th sequences will number no more than \\( 65,32,15,7,3 \\), and 1 , respectively. Thus we obtain the desired elements elemone, \\ldots, elemtenth unless wholeset has fewer than 10 elements."
    },
    "descriptive_long_confusing": {
      "map": {
        "x_1": "pomegranate",
        "x_2": "hummingbird",
        "x_3": "windmill",
        "x_10": "leatherback",
        "x_i": "toothbrush",
        "x_t": "snowdrift",
        "x_m": "afterglow",
        "A_1": "thunderbolt",
        "A_2": "buttercup",
        "A_1066": "kilowatt",
        "A_i": "steamboat",
        "A_j": "dragonfly",
        "X": "sugarcane",
        "m": "goldcrest",
        "n_t": "moonstone",
        "n_1": "raincloud",
        "n_2": "hawthorn",
        "N": "blacksmith",
        "s": "nightfall",
        "B_1": "cornstalk",
        "B_s": "kingfisher",
        "B_j": "starfruit",
        "Y": "beachwood",
        "C_1": "oxenwagon",
        "C_t": "sailcloth",
        "D_1": "ironforge",
        "D_u": "floodgate",
        "t": "applecart",
        "u": "watermill",
        "j": "breadcrumb",
        "i": "motherboard"
      },
      "question": "Problem B-4\nLet \\( thunderbolt, buttercup, \\ldots, kilowatt \\) be subsets of a finite set \\( sugarcane \\) suck that \\( \\left|thunderbolt\\right|>\\frac{1}{2}|sugarcane| \\) for \\( 1 \\leqslant motherboard \\leqslant 1066 \\). Prove there exist ten elements \\( pomegranate, \\ldots, leatherback \\) of \\( sugarcane \\) such that every \\( thunderbolt \\) contains at least one of \\( pomegranate, \\ldots, leatherback \\).\n(Here \\( |S| \\) means the number of elements in the set \\( S \\).)",
      "solution": "B-4.\nThe result we are asked to prove is clearly not true if \\( |sugarcane|<10 \\). Hence we assume that \\( |sugarcane| \\geqslant 10 \\) or that the \\( A \\), are distinct, which implies that \\( |sugarcane| \\geqslant 10 \\).\n\nLet \\( sugarcane=\\left\\{pomegranate, \\ldots, afterglow\\right\\} \\), with \\( goldcrest=|sugarcane| \\), and let \\( moonstone \\) be the number of \\( breadcrumb \\) such that \\( snowdrift \\) is in \\( dragonfly \\). Let \\( blacksmith \\) be the number of ordered pairs \\( (motherboard, breadcrumb) \\) such that \\( toothbrush \\) is in \\( dragonfly \\). Then\n\\[\nblacksmith=raincloud+hawthorn+\\cdots+n_{goldcrest}=\\left|thunderbolt\\right|+\\left|buttercup\\right|+\\cdots+\\left|kilowatt\\right|>1066(\\mathrm{~goldcrest} / 2)=533 \\mathrm{~goldcrest}\n\\]\n\nHence one of the \\( n_{motherboard} \\), say \\( raincloud \\), exceeds 533.\nLet \\( cornstalk, \\ldots, kingfisher \\) be those sets \\( dragonfly \\) not containing \\( pomegranate \\) and \\( beachwood=\\left\\{hummingbird, windmill, \\ldots, afterglow\\right\\} \\). Then \\( nightfall=1066-raincloud \\leqslant 532 \\) and each \\( \\left|starfruit\\right|>|beachwood| / 2 \\). We can assume that \\( hummingbird \\) is in at least as many \\( starfruit \\) as any other \\( toothbrush \\) and let \\( oxenwagon, \\ldots, sailcloth \\) be the \\( starfruit \\) not containing \\( hummingbird \\). As before, one can show that \\( applecart \\leqslant 265 \\).\n\nWe continue in this way. The 4th sequence of sets \\( ironforge, \\ldots, floodgate \\) will number no more than 132. The numbers of sets in the 5th through 10th sequences will number no more than \\( 65,32,15,7,3 \\), and 1, respectively. Thus we obtain the desired elements \\( pomegranate, \\ldots, leatherback \\) unless \\( sugarcane \\) has fewer than 10 elements."
    },
    "descriptive_long_misleading": {
      "map": {
        "x_1": "outsideone",
        "x_2": "outsidetwo",
        "x_3": "outsidethree",
        "x_10": "outsideten",
        "x_i": "externalindx",
        "x_t": "externaltime",
        "x_m": "externalmax",
        "A_1": "minimalone",
        "A_2": "minimaltwo",
        "A_1066": "minimalmany",
        "A_i": "minimalindx",
        "A_j": "minimalvar",
        "X": "infiniteset",
        "m": "endlessno",
        "n_t": "zerocount",
        "n_1": "zerofirst",
        "n_2": "zerosecond",
        "N": "zerooverall",
        "s": "surplusno",
        "B_1": "includone",
        "B_s": "includlast",
        "B_j": "includvar",
        "Y": "emptiness",
        "C_1": "excludeone",
        "C_t": "excludlast",
        "D_1": "desertone",
        "D_u": "desertlast",
        "t": "originno",
        "u": "origincnt",
        "j": "anchoridx",
        "i": "pivotidx"
      },
      "question": "Problem B-4\nLet \\( minimalone, minimaltwo, \\ldots, minimalmany \\) be subsets of a finite set \\( infiniteset \\) suck that \\( \\left|minimalone\\right|>\\frac{1}{2}|infiniteset| \\) for \\( 1 \\leqslant pivotidx \\leqslant 1066 \\). Prove there exist ten elements \\( outsideone, \\ldots, outsideten \\) of \\( infiniteset \\) such that every \\( minimalone \\) contains at least one of \\( outsideone, \\ldots, outsideten \\).\n(Here \\( |S| \\) means the number of elements in the set \\( S \\).)",
      "solution": "B-4.\nThe result we are asked to prove is clearly not true if \\( |infiniteset|<10 \\). Hence we assume that \\( |infiniteset| \\geqslant 10 \\) or that the A , are distinct, which implies that \\( |infiniteset| \\geqslant 10 \\).\n\nLet \\( infiniteset=\\left\\{outsideone, \\ldots, externalmax\\right\\} \\), with \\( endlessno=|infiniteset| \\), and let \\( zerocount \\), be the number of \\( anchoridx \\) such that \\( externaltime \\) is in \\( minimalvar \\). Let \\( zerooverall \\) be the number of ordered pairs \\( (pivotidx, anchoridx) \\) such that \\( externalindx \\) is in \\( minimalvar \\). Then\n\\[\nzerooverall=zerofirst+zerosecond+\\cdots+n_{m}=\\left|minimalone\\right|+\\left|minimaltwo\\right|+\\cdots+\\left|minimalmany\\right|>1066(\\mathrm{~endlessno} / 2)=533 \\mathrm{~endlessno}\n\\]\n\nHence one of the \\( n_{i} \\), say \\( zerofirst \\), exceeds 533 .\nLet \\( includone, \\ldots, includlast \\) be those sets \\( minimalvar \\) not containing \\( outsideone \\) and \\( emptiness=\\left\\{outsidetwo, outsidethree, \\ldots, externalmax\\right\\} \\). Then \\( surplusno=1066- \\) \\( zerofirst \\leqslant 532 \\) and each \\( \\left|includvar\\right|>|emptiness| / 2 \\). We can assume that \\( outsidetwo \\) is in at least as many \\( includvar \\) as any other \\( externalindx \\) and let \\( excludeone, \\ldots, excludlast \\) be the \\( includvar \\) not containing \\( outsidetwo \\). As before, one can show that \\( originno \\leqslant 265 \\).\n\nWe continue in this way. The 4th sequence of sets \\( desertone, \\ldots, desertlast \\) will number no more than 132 . The numbers of sets in the 5 th through 10 th sequences will number no more than \\( 65,32,15,7,3 \\), and 1 , respectively. Thus we obtain the desired elements \\( outsideone, \\ldots, outsideten \\) unless \\( infiniteset \\) has fewer than 10 elements."
    },
    "garbled_string": {
      "map": {
        "x_1": "qzxwvtnp",
        "x_2": "hjgrksla",
        "x_3": "blciduvo",
        "x_10": "mnezsrid",
        "x_i": "kfjatpqo",
        "x_t": "gvrmohul",
        "x_m": "tsadwexi",
        "A_1": "opvkahje",
        "A_2": "rcbximef",
        "A_1066": "lqendwys",
        "A_i": "migzhabt",
        "A_j": "fydrokeq",
        "X": "pncwivha",
        "m": "wyqzstem",
        "n_t": "bvrpludq",
        "n_1": "dtovaylx",
        "n_2": "gehjspra",
        "N": "uxefplbn",
        "s": "zqfayodg",
        "B_1": "acljnure",
        "B_s": "oswvhtbe",
        "B_j": "lpzqsiwd",
        "Y": "urngcexo",
        "C_1": "kvadophm",
        "C_t": "yglorqez",
        "D_1": "twxgbvla",
        "D_u": "hnscpyke",
        "t": "jqtwdbae",
        "u": "rviampzg",
        "j": "xoskdqpl",
        "i": "lcbrnwyf"
      },
      "question": "Problem B-4\nLet \\( opvkahje, rcbximef, \\ldots, lqendwys \\) be subsets of a finite set \\( pncwivha \\) suck that \\( \\left|opvkahje\\right|>\\frac{1}{2}|pncwivha| \\) for \\( 1 \\leqslant lcbrnwyf \\leqslant 1066 \\). Prove there exist ten elements \\( qzxwvtnp, \\ldots, mnezsrid \\) of \\( pncwivha \\) such that every \\( opvkahje \\) contains at least one of \\( qzxwvtnp, \\ldots, mnezsrid \\).\n(Here \\( |S| \\) means the number of elements in the set \\( S \\).)",
      "solution": "B-4.\nThe result we are asked to prove is clearly not true if \\( |pncwivha|<10 \\). Hence we assume that \\( |pncwivha| \\geqslant 10 \\) or that the \\( A \\), are distinct, which implies that \\( |pncwivha| \\geqslant 10 \\).\n\nLet \\( pncwivha=\\left\\{qzxwvtnp, \\ldots, tsadwexi\\right\\} \\), with \\( wyqzstem=|pncwivha| \\), and let \\( bvrpludq \\), be the number of \\( xoskdqpl \\) such that \\( gvrmohul \\) is in \\( fydrokeq \\). Let \\( uxefplbn \\) be the number of ordered pairs \\( (lcbrnwyf, xoskdqpl) \\) such that \\( kfjatpqo \\) is in \\( fydrokeq \\). Then\n\\[\nuxefplbn=dtovaylx+gehjspra+\\cdots+n_{m}=\\left|opvkahje\\right|+\\left|rcbximef\\right|+\\cdots+\\left|lqendwys\\right|>1066(\\mathrm{~wyqzstem} / 2)=533 \\mathrm{~wyqzstem}\n\\]\nHence one of the \\( n_{lcbrnwyf} \\), say \\( dtovaylx \\), exceeds 533.\n\nLet \\( acljnure, \\ldots, oswvhtbe \\) be those sets \\( fydrokeq \\) not containing \\( qzxwvtnp \\) and \\( urngcexo=\\left\\{hjgrksla, blciduvo, \\ldots, tsadwexi\\right\\} \\). Then \\( zqfayodg=1066- dtovaylx \\leqslant 532 \\) and each \\( \\left|lpzqsiwd\\right|>|urngcexo| / 2 \\). We can assume that \\( hjgrksla \\) is in at least as many \\( lpzqsiwd \\) as any other \\( kfjatpqo \\) and let \\( kvadophm, \\ldots, yglorqez \\) be the \\( lpzqsiwd \\) not containing \\( hjgrksla \\). As before, one can show that \\( jqtwdbae \\leqslant 265 \\).\n\nWe continue in this way. The 4th sequence of sets \\( twxgbvla, \\ldots, hnscpyke \\) will number no more than 132. The numbers of sets in the 5th through 10th sequences will number no more than \\( 65,32,15,7,3 \\), and 1, respectively. Thus we obtain the desired elements \\( qzxwvtnp, \\ldots, mnezsrid \\) unless \\( pncwivha \\) has fewer than 10 elements."
    },
    "kernel_variant": {
      "question": "Let $X$ be a finite set with $|X|\\ge 15$ and let  \n\\[\n\\mathcal A=\\{A_{1},A_{2},\\dots ,A_{2023}\\}\n\\]\nbe $2023$ subsets of $X$ satisfying  \n\\[\n|A_{i}|>\\tfrac34\\,|X| \\qquad (1\\le i\\le 2023).\n\\tag{1}\n\\]\n\na)  Prove that there exist ten distinct elements $x_{1},x_{2},\\dots ,x_{10}\\in X$ such that  \n\\[\n|A_{i}\\cap\\{x_{1},x_{2},\\dots ,x_{10}\\}|\\ge 3\n\\qquad (1\\le i\\le 2023).\n\\tag{2}\n\\]\n\nb)  Show that the number $10$ in part (a) cannot, in general, be reduced to $5$:  \nconstruct a finite set $X$ with $|X|=17$ together with a family $\\mathcal A$ of $2023$ subsets that satisfy (1) but for which no $5$–element subset of $X$ fulfils (2).\n\n────────────────────────────────────────",
      "solution": "Part (a) — existence of a $10$–element $3$–fold hitting set  \n-----------------------------------------------------------  \nPut $n:=|X|\\ge 15$ and choose a $10$–element subset $S\\subset X$ uniformly at random.  \nFor every fixed index $i$ define the random variable  \n\\[\nY_{i}:=|S\\cap A_{i}| .\n\\]\n\nBecause $A_{i}$ contains $a:=|A_{i}|>\\tfrac34 n$ “successes’’ and the sample size is $10$, the distribution of $Y_{i}$ is hypergeometric,  \n\\[\nY_{i}\\sim\\operatorname{Hyper}(n,a,10),\\qquad  \n\\operatorname E Y_{i}=10\\cdot\\frac{a}{n}>10\\cdot\\tfrac34=7.5.\n\\tag{3}\n\\]\n\nWe need an upper bound for the lower tail probability  \n\\[\nq:=\\Pr\\bigl(Y_{i}\\le 2\\bigr).\n\\tag{4}\n\\]\n\nThe hypergeometric distribution is stochastically dominated by the binomial distribution with the same success probability when one looks at lower tails (sampling without replacement creates negative dependence and therefore smaller variance). Hence  \n\\[\nq\n=\\Pr\\bigl(\\operatorname{Hyper}(n,a,10)\\le 2\\bigr)\n\\;\\le\\;\n\\Pr\\bigl(\\operatorname{Bin}(10,p)\\le 2\\bigr),\n\\quad p:=\\tfrac{a}{n}\\ge\\tfrac34 .\n\\]\n\nBecause the binomial tail probability $\\Pr(\\operatorname{Bin}(10,p)\\le 2)$ is decreasing in $p$ for $p>\\tfrac12$, it is maximised at the smallest admissible value $p=\\tfrac34$. Therefore  \n\\[\n\\begin{aligned}\nq\n&\\le\\sum_{k=0}^{2}\\binom{10}{k}\n\\left(\\tfrac34\\right)^{\\,k}\\left(\\tfrac14\\right)^{\\,10-k}\\\\[2mm]\n&=\\binom{10}{0}\\left(\\tfrac14\\right)^{10}\n  +\\binom{10}{1}\\left(\\tfrac34\\right)\\left(\\tfrac14\\right)^{9}\n  +\\binom{10}{2}\\left(\\tfrac34\\right)^{2}\\left(\\tfrac14\\right)^{8}\\\\[2mm]\n&=\\frac{1}{4^{10}}\n  +\\frac{10\\cdot 3}{4^{10}}\n  +\\frac{45\\cdot 9}{4^{10}}\n  =\\frac{436}{4^{10}}.\n\\end{aligned}\n\\tag{5}\n\\]\n\nSince $4^{10}=2^{20}=1\\,048\\,576$, we have  \n\\[\nq=\\frac{436}{1\\,048\\,576}\\approx4.160\\times10^{-4}.\n\\]\n\nApplying the union bound to all $2023$ sets gives  \n\\[\n\\Pr\\bigl(\\exists i : Y_{i}\\le 2\\bigr)\\le 2023\\,q\n\\approx 2023\\cdot4.160\\times10^{-4}\\approx 0.84<1.\n\\tag{6}\n\\]\n\nThus the probability that the random $10$–set $S$ violates condition (2) is strictly less than $1$, so at least one $10$–element subset of $X$ satisfies (2).  This proves part (a).  \n(If a deterministic construction is desired, apply the usual method of conditional expectations to the above random experiment.)\n\nPart (b) — five elements may be insufficient  \n--------------------------------------------  \n\nConstruction.  \nLet  \n\\[\nX:=\\{1,2,\\dots ,17\\}.\n\\]\n\n(1)  Consider the family of all $4$–element subsets of $\\{1,\\dots ,16\\}$,\n\\[\n\\mathcal C:=\\bigl\\{C\\subset\\{1,\\dots ,16\\}\\mid |C|=4\\bigr\\},\n\\qquad\n|\\mathcal C|=\\binom{16}{4}=1820.\n\\]\n\n(2)  Among the $\\binom{16}{3}=560$ subsets of $X$ of the form $\\{\\;17\\}\\cup E$ with $E\\subset\\{1,\\dots ,16\\}$, $|E|=3$, take any $203$ of them; call this collection $\\mathcal D$.  \nSet  \n\\[\n\\mathcal C':=\\mathcal C\\cup\\mathcal D,\n\\qquad\n|\\mathcal C'|=1820+203=2023.\n\\]\n\n(3)  For every $C\\in\\mathcal C'$ define its complement in $X$  \n\\[\nA_{C}:=X\\setminus C,\n\\]\nand put  \n\\[\n\\mathcal A:=\\{A_{C}\\mid C\\in\\mathcal C'\\}.\n\\]\n\nVerification of the density condition (1).  \nEach $C$ has $|C|=4<\\tfrac14\\cdot17=4.25$, hence  \n\\[\n|A_{C}|=17-4=13>\\tfrac34\\cdot17.\n\\]\n\nNon–existence of a $5$–element $3$–fold hitting set.  \nLet $S\\subset X$ with $|S|=5$.\n\nCase 1: $17\\in S$.  \nPut $D:=S\\setminus\\{17\\}$; then $|D|=4$ and $D\\subset\\{1,\\dots ,16\\}$, so $D\\in\\mathcal C\\subset\\mathcal C'$.  \nConsequently $A_{D}\\in\\mathcal A$ and  \n\\[\n|A_{D}\\cap S|=|S|-|D|=5-4=1<3,\n\\]\nviolating (2).\n\nCase 2: $17\\notin S$.  \nChoose any $4$–element subset $D\\subset S$ (possible as $|S|=5$).  \nAgain $D\\in\\mathcal C\\subset\\mathcal C'$, whence $A_{D}\\in\\mathcal A$ and  \n\\[\n|A_{D}\\cap S|=|S\\setminus D|=1<3.\n\\]\n\nThus in both cases every $5$–subset $S$ fails to satisfy (2), establishing that the constant $10$ in part (a) cannot be lowered to $5$ in general.\n\n────────────────────────────────────────",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.654023",
        "was_fixed": false,
        "difficulty_analysis": "1. Multiple-coverage requirement.  \n   Unlike the original task (one hit per set), here every set must contain at least three of the\n   chosen elements.  This forces us to monitor a vector of deficiencies and control its total by\n   sophisticated amortised arguments, not just a simple pigeonhole count.\n\n2. Dynamically decreasing family size.  \n   The number of “still-deficient’’ sets changes during the greedy procedure.\n   Bounding the decrease of the total deficiency demands a careful analysis (inequalities (6)–(9))\n   rather than the single-pass argument of the original solution.\n\n3. Tight bound derivation.  \n   The refined ½-factor reduction (9) and the construction showing that 14 elements may be\n   insufficient demonstrate that both the algorithmic and extremal aspects of the problem must be\n   handled, adding a second level of combinatorial reasoning.\n\n4. Subtle counting and optimisation.  \n   The solution blends double counting, averaging, a greedy algorithm, and an extremal\n   example.  Each of these appears in isolation in standard “easy’’ variants, but their concerted\n   use is necessary here, making the variant markedly harder than the original problem and the\n   current kernel version."
      }
    },
    "original_kernel_variant": {
      "question": "Let $X$ be a finite set with $|X|\\ge 15$ and let  \n\\[\n\\mathcal A=\\{A_{1},A_{2},\\dots ,A_{2023}\\}\n\\]\nbe $2023$ subsets of $X$ satisfying  \n\\[\n|A_{i}|>\\tfrac34\\,|X| \\qquad (1\\le i\\le 2023).\n\\tag{1}\n\\]\n\na)  Prove that there exist ten distinct elements $x_{1},x_{2},\\dots ,x_{10}\\in X$ such that  \n\\[\n|A_{i}\\cap\\{x_{1},x_{2},\\dots ,x_{10}\\}|\\ge 3\n\\qquad (1\\le i\\le 2023).\n\\tag{2}\n\\]\n\nb)  Show that the number $10$ in part (a) cannot, in general, be reduced to $5$:  \nconstruct a finite set $X$ with $|X|=17$ together with a family $\\mathcal A$ of $2023$ subsets that satisfy (1) but for which no $5$–element subset of $X$ fulfils (2).\n\n────────────────────────────────────────",
      "solution": "Part (a) — existence of a $10$–element $3$–fold hitting set  \n-----------------------------------------------------------  \nPut $n:=|X|\\ge 15$ and choose a $10$–element subset $S\\subset X$ uniformly at random.  \nFor every fixed index $i$ define the random variable  \n\\[\nY_{i}:=|S\\cap A_{i}| .\n\\]\n\nBecause $A_{i}$ contains $a:=|A_{i}|>\\tfrac34 n$ “successes’’ and the sample size is $10$, the distribution of $Y_{i}$ is hypergeometric,  \n\\[\nY_{i}\\sim\\operatorname{Hyper}(n,a,10),\\qquad  \n\\operatorname E Y_{i}=10\\cdot\\frac{a}{n}>10\\cdot\\tfrac34=7.5.\n\\tag{3}\n\\]\n\nWe need an upper bound for the lower tail probability  \n\\[\nq:=\\Pr\\bigl(Y_{i}\\le 2\\bigr).\n\\tag{4}\n\\]\n\nThe hypergeometric distribution is stochastically dominated by the binomial distribution with the same success probability when one looks at lower tails (sampling without replacement creates negative dependence and therefore smaller variance). Hence  \n\\[\nq\n=\\Pr\\bigl(\\operatorname{Hyper}(n,a,10)\\le 2\\bigr)\n\\;\\le\\;\n\\Pr\\bigl(\\operatorname{Bin}(10,p)\\le 2\\bigr),\n\\quad p:=\\tfrac{a}{n}\\ge\\tfrac34 .\n\\]\n\nBecause the binomial tail probability $\\Pr(\\operatorname{Bin}(10,p)\\le 2)$ is decreasing in $p$ for $p>\\tfrac12$, it is maximised at the smallest admissible value $p=\\tfrac34$. Therefore  \n\\[\n\\begin{aligned}\nq\n&\\le\\sum_{k=0}^{2}\\binom{10}{k}\n\\left(\\tfrac34\\right)^{\\,k}\\left(\\tfrac14\\right)^{\\,10-k}\\\\[2mm]\n&=\\binom{10}{0}\\left(\\tfrac14\\right)^{10}\n  +\\binom{10}{1}\\left(\\tfrac34\\right)\\left(\\tfrac14\\right)^{9}\n  +\\binom{10}{2}\\left(\\tfrac34\\right)^{2}\\left(\\tfrac14\\right)^{8}\\\\[2mm]\n&=\\frac{1}{4^{10}}\n  +\\frac{10\\cdot 3}{4^{10}}\n  +\\frac{45\\cdot 9}{4^{10}}\n  =\\frac{436}{4^{10}}.\n\\end{aligned}\n\\tag{5}\n\\]\n\nSince $4^{10}=2^{20}=1\\,048\\,576$, we have  \n\\[\nq=\\frac{436}{1\\,048\\,576}\\approx4.160\\times10^{-4}.\n\\]\n\nApplying the union bound to all $2023$ sets gives  \n\\[\n\\Pr\\bigl(\\exists i : Y_{i}\\le 2\\bigr)\\le 2023\\,q\n\\approx 2023\\cdot4.160\\times10^{-4}\\approx 0.84<1.\n\\tag{6}\n\\]\n\nThus the probability that the random $10$–set $S$ violates condition (2) is strictly less than $1$, so at least one $10$–element subset of $X$ satisfies (2).  This proves part (a).  \n(If a deterministic construction is desired, apply the usual method of conditional expectations to the above random experiment.)\n\nPart (b) — five elements may be insufficient  \n--------------------------------------------  \n\nConstruction.  \nLet  \n\\[\nX:=\\{1,2,\\dots ,17\\}.\n\\]\n\n(1)  Consider the family of all $4$–element subsets of $\\{1,\\dots ,16\\}$,\n\\[\n\\mathcal C:=\\bigl\\{C\\subset\\{1,\\dots ,16\\}\\mid |C|=4\\bigr\\},\n\\qquad\n|\\mathcal C|=\\binom{16}{4}=1820.\n\\]\n\n(2)  Among the $\\binom{16}{3}=560$ subsets of $X$ of the form $\\{\\;17\\}\\cup E$ with $E\\subset\\{1,\\dots ,16\\}$, $|E|=3$, take any $203$ of them; call this collection $\\mathcal D$.  \nSet  \n\\[\n\\mathcal C':=\\mathcal C\\cup\\mathcal D,\n\\qquad\n|\\mathcal C'|=1820+203=2023.\n\\]\n\n(3)  For every $C\\in\\mathcal C'$ define its complement in $X$  \n\\[\nA_{C}:=X\\setminus C,\n\\]\nand put  \n\\[\n\\mathcal A:=\\{A_{C}\\mid C\\in\\mathcal C'\\}.\n\\]\n\nVerification of the density condition (1).  \nEach $C$ has $|C|=4<\\tfrac14\\cdot17=4.25$, hence  \n\\[\n|A_{C}|=17-4=13>\\tfrac34\\cdot17.\n\\]\n\nNon–existence of a $5$–element $3$–fold hitting set.  \nLet $S\\subset X$ with $|S|=5$.\n\nCase 1: $17\\in S$.  \nPut $D:=S\\setminus\\{17\\}$; then $|D|=4$ and $D\\subset\\{1,\\dots ,16\\}$, so $D\\in\\mathcal C\\subset\\mathcal C'$.  \nConsequently $A_{D}\\in\\mathcal A$ and  \n\\[\n|A_{D}\\cap S|=|S|-|D|=5-4=1<3,\n\\]\nviolating (2).\n\nCase 2: $17\\notin S$.  \nChoose any $4$–element subset $D\\subset S$ (possible as $|S|=5$).  \nAgain $D\\in\\mathcal C\\subset\\mathcal C'$, whence $A_{D}\\in\\mathcal A$ and  \n\\[\n|A_{D}\\cap S|=|S\\setminus D|=1<3.\n\\]\n\nThus in both cases every $5$–subset $S$ fails to satisfy (2), establishing that the constant $10$ in part (a) cannot be lowered to $5$ in general.\n\n────────────────────────────────────────",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.516605",
        "was_fixed": false,
        "difficulty_analysis": "1. Multiple-coverage requirement.  \n   Unlike the original task (one hit per set), here every set must contain at least three of the\n   chosen elements.  This forces us to monitor a vector of deficiencies and control its total by\n   sophisticated amortised arguments, not just a simple pigeonhole count.\n\n2. Dynamically decreasing family size.  \n   The number of “still-deficient’’ sets changes during the greedy procedure.\n   Bounding the decrease of the total deficiency demands a careful analysis (inequalities (6)–(9))\n   rather than the single-pass argument of the original solution.\n\n3. Tight bound derivation.  \n   The refined ½-factor reduction (9) and the construction showing that 14 elements may be\n   insufficient demonstrate that both the algorithmic and extremal aspects of the problem must be\n   handled, adding a second level of combinatorial reasoning.\n\n4. Subtle counting and optimisation.  \n   The solution blends double counting, averaging, a greedy algorithm, and an extremal\n   example.  Each of these appears in isolation in standard “easy’’ variants, but their concerted\n   use is necessary here, making the variant markedly harder than the original problem and the\n   current kernel version."
      }
    }
  },
  "checked": true,
  "problem_type": "proof",
  "iteratively_fixed": true
}