{
  "index": "1996-A-3",
  "type": "COMB",
  "tag": [
    "COMB"
  ],
  "difficulty": "",
  "question": "Suppose that each of 20 students has made a choice of anywhere from 0\nto 6 courses from a total of 6 courses offered. Prove or disprove:\nthere are 5 students and 2 courses such that all 5 have chosen both\ncourses or all 5 have chosen neither course.",
  "solution": "The claim is false. There are $\\binom{6}{3} = 20$ ways to choose 3 of the\n6 courses; have each student choose a different set of 3 courses. Then\neach pair of courses is chosen by 4 students (corresponding to the\nfour ways to complete this pair to a set of 3 courses) and is not\nchosen by 4 students (corresponding to the 3-element subsets of the\nremaining 4 courses).\n\nNote: Assuming that no two students choose the same courses,\nthe above counterexample is unique (up to permuting students).\nThis may be seen as follows: Given a group of students, suppose that\nfor any pair of courses (among the six) there are at most 4 students\ntaking both, and at most 4 taking neither.  Then there are at most\n$120=(4+4)\\binom{6}{2}$ pairs $(s,p)$, where $s$ is a student, and $p$\nis a set of two courses of which $s$ is taking either both or none.\nOn the other hand, if a student $s$ is taking $k$ courses, then he/she\noccurs in $f(k)=\\binom{k}{2}+\\binom{6-k}{2}$ such pairs $(s,p)$.  As\n$f(k)$ is minimized for $k=3$, it follows that every student occurs in\nat least $6=\\binom{3}{2}+\\binom{3}{2}$ such pairs $(s,p)$.  Hence\nthere can be at most $120/6=20$ students, with equality only if each\nstudent takes 3 courses, and for each set of two courses, there are\nexactly 4 students who take both and exactly 4 who take neither.\nSince there are only 4 ways to complete a given pair of courses to a\nset of 3, and only 4 ways to choose 3 courses not containing the given\npair, the only way for there to be 20 students (under our hypotheses)\nis if all sets of 3 courses are in fact taken.  This is the desired conclusion.\n\nHowever, Robin Chapman has pointed out that the solution is not unique\nin the problem as stated, because a given selection of courses may be\nmade by more than one student. One alternate solution is to identify\nthe 6 courses with pairs of antipodal vertices of an icosahedron, and\nhave each student pick a different face and choose the three vertices\ntouching that face. In this example, each of 10 selections is made by\na pair of students.",
  "vars": [
    "s",
    "p",
    "k",
    "f"
  ],
  "params": [],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "s": "scholar",
        "p": "pairset",
        "k": "coursenum",
        "f": "paircount"
      },
      "question": "Suppose that each of 20 students has made a choice of anywhere from 0\nto 6 courses from a total of 6 courses offered. Prove or disprove:\nthere are 5 students and 2 courses such that all 5 have chosen both\ncourses or all 5 have chosen neither course.",
      "solution": "The claim is false. There are $\\binom{6}{3} = 20$ ways to choose 3 of the\n6 courses; have each student choose a different set of 3 courses. Then\neach pair of courses is chosen by 4 students (corresponding to the\nfour ways to complete this pair to a set of 3 courses) and is not\nchosen by 4 students (corresponding to the 3-element subsets of the\nremaining 4 courses).\n\nNote: Assuming that no two students choose the same courses,\nthe above counterexample is unique (up to permuting students).\nThis may be seen as follows: Given a group of students, suppose that\nfor any pair of courses (among the six) there are at most 4 students\ntaking both, and at most 4 taking neither.  Then there are at most\n$120=(4+4)\\binom{6}{2}$ pairs $(\\scholar,\\pairset)$, where $\\scholar$ is a student, and $\\pairset$\nis a set of two courses of which $\\scholar$ is taking either both or none.\nOn the other hand, if a student $\\scholar$ is taking $\\coursenum$ courses, then he/she\noccurs in $\\paircount(\\coursenum)=\\binom{\\coursenum}{2}+\\binom{6-\\coursenum}{2}$ such pairs $(\\scholar,\\pairset)$.  As\n$\\paircount(\\coursenum)$ is minimized for $\\coursenum=3$, it follows that every student occurs in\nat least $6=\\binom{3}{2}+\\binom{3}{2}$ such pairs $(\\scholar,\\pairset)$.  Hence\nthere can be at most $120/6=20$ students, with equality only if each\nstudent takes 3 courses, and for each set of two courses, there are\nexactly 4 students who take both and exactly 4 who take neither.\nSince there are only 4 ways to complete a given pair of courses to a\nset of 3, and only 4 ways to choose 3 courses not containing the given\npair, the only way for there to be 20 students (under our hypotheses)\nis if all sets of 3 courses are in fact taken.  This is the desired conclusion.\n\nHowever, Robin Chapman has pointed out that the solution is not unique\nin the problem as stated, because a given selection of courses may be\nmade by more than one student. One alternate solution is to identify\nthe 6 courses with pairs of antipodal vertices of an icosahedron, and\nhave each student pick a different face and choose the three vertices\ntouching that face. In this example, each of 10 selections is made by\na pair of students."
    },
    "descriptive_long_confusing": {
      "map": {
        "s": "kangaroo",
        "p": "goldfish",
        "k": "daffodil",
        "f": "windsock"
      },
      "question": "Suppose that each of 20 students has made a choice of anywhere from 0\nto 6 courses from a total of 6 courses offered. Prove or disprove:\nthere are 5 students and 2 courses such that all 5 have chosen both\ncourses or all 5 have chosen neither course.",
      "solution": "The claim is false. There are $\\binom{6}{3} = 20$ ways to choose 3 of the\n6 courses; have each student choose a different set of 3 courses. Then\neach pair of courses is chosen by 4 students (corresponding to the\nfour ways to complete this pair to a set of 3 courses) and is not\nchosen by 4 students (corresponding to the 3-element subsets of the\nremaining 4 courses).\n\nNote: Assuming that no two students choose the same courses,\nthe above counterexample is unique (up to permuting students).\nThis may be seen as follows: Given a group of students, suppose that\nfor any pair of courses (among the six) there are at most 4 students\ntaking both, and at most 4 taking neither.  Then there are at most\n$120=(4+4)\\binom{6}{2}$ pairs $(kangaroo,goldfish)$, where $kangaroo$ is a student, and $goldfish$\nis a set of two courses of which $kangaroo$ is taking either both or none.\nOn the other hand, if a student $kangaroo$ is taking $daffodil$ courses, then he/she\noccurs in $windsock(daffodil)=\\binom{daffodil}{2}+\\binom{6-daffodil}{2}$ such pairs $(kangaroo,goldfish)$.  As\n$windsock(daffodil)$ is minimized for $daffodil=3$, it follows that every student occurs in\nat least $6=\\binom{3}{2}+\\binom{3}{2}$ such pairs $(kangaroo,goldfish)$.  Hence\nthere can be at most $120/6=20$ students, with equality only if each\nstudent takes 3 courses, and for each set of two courses, there are\nexactly 4 students who take both and exactly 4 who take neither.\nSince there are only 4 ways to complete a given pair of courses to a\nset of 3, and only 4 ways to choose 3 courses not containing the given\npair, the only way for there to be 20 students (under our hypotheses)\nis if all sets of 3 courses are in fact taken.  This is the desired conclusion.\n\nHowever, Robin Chapman has pointed out that the solution is not unique\nin the problem as stated, because a given selection of courses may be\nmade by more than one student. One alternate solution is to identify\nthe 6 courses with pairs of antipodal vertices of an icosahedron, and\nhave each student pick a different face and choose the three vertices\ntouching that face. In this example, each of 10 selections is made by\na pair of students."
    },
    "descriptive_long_misleading": {
      "map": {
        "s": "nonlearner",
        "p": "singleton",
        "k": "emptiness",
        "f": "dysfunction"
      },
      "question": "Suppose that each of 20 students has made a choice of anywhere from 0\nto 6 courses from a total of 6 courses offered. Prove or disprove:\nthere are 5 students and 2 courses such that all 5 have chosen both\ncourses or all 5 have chosen neither course.",
      "solution": "The claim is false. There are $\\binom{6}{3} = 20$ ways to choose 3 of the\n6 courses; have each student choose a different set of 3 courses. Then\neach pair of courses is chosen by 4 students (corresponding to the\nfour ways to complete this pair to a set of 3 courses) and is not\nchosen by 4 students (corresponding to the 3-element subsets of the\nremaining 4 courses).\n\nNote: Assuming that no two students choose the same courses,\nthe above counterexample is unique (up to permuting students).\nThis may be seen as follows: Given a group of students, suppose that\nfor any pair of courses (among the six) there are at most 4 students\ntaking both, and at most 4 taking neither.  Then there are at most\n$120=(4+4)\\binom{6}{2}$ pairs $(nonlearner,singleton)$, where $nonlearner$ is a student, and $singleton$\nis a set of two courses of which $nonlearner$ is taking either both or none.\nOn the other hand, if a student $nonlearner$ is taking $emptiness$ courses, then he/she\noccurs in $dysfunction(emptiness)=\\binom{emptiness}{2}+\\binom{6-emptiness}{2}$ such pairs $(nonlearner,singleton)$.  As\n$dysfunction(emptiness)$ is minimized for $emptiness=3$, it follows that every student occurs in\nat least $6=\\binom{3}{2}+\\binom{3}{2}$ such pairs $(nonlearner,singleton)$.  Hence\nthere can be at most $120/6=20$ students, with equality only if each\nstudent takes 3 courses, and for each set of two courses, there are\nexactly 4 students who take both and exactly 4 who take neither.\nSince there are only 4 ways to complete a given pair of courses to a\nset of 3, and only 4 ways to choose 3 courses not containing the given\npair, the only way for there to be 20 students (under our hypotheses)\nis if all sets of 3 courses are in fact taken.  This is the desired conclusion.\n\nHowever, Robin Chapman has pointed out that the solution is not unique\nin the problem as stated, because a given selection of courses may be\nmade by more than one student. One alternate solution is to identify\nthe 6 courses with pairs of antipodal vertices of an icosahedron, and\nhave each student pick a different face and choose the three vertices\ntouching that face. In this example, each of 10 selections is made by\na pair of students."
    },
    "garbled_string": {
      "map": {
        "s": "qzxwvtnp",
        "p": "hjgrksla",
        "k": "mfdlqzre",
        "f": "bwtnchsu"
      },
      "question": "Suppose that each of 20 students has made a choice of anywhere from 0\nto 6 courses from a total of 6 courses offered. Prove or disprove:\nthere are 5 students and 2 courses such that all 5 have chosen both\ncourses or all 5 have chosen neither course.",
      "solution": "The claim is false. There are $\\binom{6}{3} = 20$ ways to choose 3 of the\n6 courses; have each student choose a different set of 3 courses. Then\neach pair of courses is chosen by 4 students (corresponding to the\nfour ways to complete this pair to a set of 3 courses) and is not\nchosen by 4 students (corresponding to the 3-element subsets of the\nremaining 4 courses).\n\nNote: Assuming that no two students choose the same courses,\nthe above counterexample is unique (up to permuting students).\nThis may be seen as follows: Given a group of students, suppose that\nfor any pair of courses (among the six) there are at most 4 students\ntaking both, and at most 4 taking neither.  Then there are at most\n$120=(4+4)\\binom{6}{2}$ pairs $(qzxwvtnp,hjgrksla)$, where $qzxwvtnp$ is a student, and $hjgrksla$ is a set of two courses of which $qzxwvtnp$ is taking either both or none.\nOn the other hand, if a student $qzxwvtnp$ is taking $mfdlqzre$ courses, then he/she\noccurs in $bwtnchsu(mfdlqzre)=\\binom{mfdlqzre}{2}+\\binom{6-mfdlqzre}{2}$ such pairs $(qzxwvtnp,hjgrksla)$.  As\n$bwtnchsu(mfdlqzre)$ is minimized for $mfdlqzre=3$, it follows that every student occurs in\nat least $6=\\binom{3}{2}+\\binom{3}{2}$ such pairs $(qzxwvtnp,hjgrksla)$.  Hence\nthere can be at most $120/6=20$ students, with equality only if each\nstudent takes 3 courses, and for each set of two courses, there are\nexactly 4 students who take both and exactly 4 who take neither.\nSince there are only 4 ways to complete a given pair of courses to a\nset of 3, and only 4 ways to choose 3 courses not containing the given\npair, the only way for there to be 20 students (under our hypotheses)\nis if all sets of 3 courses are in fact taken.  This is the desired conclusion.\n\nHowever, Robin Chapman has pointed out that the solution is not unique\nin the problem as stated, because a given selection of courses may be\nmade by more than one student. One alternate solution is to identify\nthe 6 courses with pairs of antipodal vertices of an icosahedron, and\nhave each student pick a different face and choose the three vertices\ntouching that face. In this example, each of 10 selections is made by\na pair of students."
    },
    "kernel_variant": {
      "question": "Nine graduate seminars are numbered $1,2,\\dots ,9$.  \nEvery student may register for an arbitrary (possibly empty) subset of the nine seminars.  \n\n(a)  Show that among any $73$ students one can always find a set of $7$ students and three distinct seminars such that  \n\n • either all $7$ of those students registered for all three of the chosen seminars,  \n\n • or all $7$ registered for none of the three seminars.  \n\n(Equivalently: every $0/1$–matrix with more than $72$ rows and $9$ columns always contains a monochromatic $7\\times 3$ sub-matrix.)  \n\n(b)  Prove that if in a $0/1$–matrix with $9$ columns no colour (neither $0$ nor $1$) occurs at least $7$ times in any $3$ columns simultaneously, then the matrix has at most $72$ rows.  (Thus $72$ is the largest number of rows *compatible* with the numerical constraints derived in part (a).  Whether a $72$–row matrix satisfying the restriction exists is left open.)",
      "solution": "Throughout we identify every student with the $0/1$–vector  \n$x=(x_{1},\\dots ,x_{9})\\in\\{0,1\\}^{9}$ where $x_{i}=1$ iff the student took seminar $i$.  \nThe *weight* of $x$ is $|x|=\\sum_{i=1}^{9}x_{i}$.\n\n-----------------------------------------------------------------------------\n\n1.  How many monochromatic triples does one student generate?  \n\nFor $k=0,1,\\dots ,9$ set  \n\\[\ng(k)=\\binom{k}{3}+\\binom{9-k}{3}.\n\\]\nIf $|x|=k$ then exactly $\\binom{k}{3}$ triples of seminars are taken completely by the student and $\\binom{9-k}{3}$ triples are avoided completely.  Hence the student is monochromatic on exactly $g(k)$ triples.  A direct evaluation gives  \n\n\\[\n\\begin{array}{c|cccccccccc}\nk & 0&1&2&3&4&5&6&7&8&9\\\\\\hline\ng(k)&84&56&35&21&14&14&21&35&56&84\n\\end{array}\n\\]\n\nIn particular  \n\\[\ng(k)\\ge 14\\quad\\text{for every }k,\\qquad\\text{and}\\qquad g(k)=14\\Longleftrightarrow k\\in\\{4,5\\}.\n\\tag{1}\n\\]\n\n-----------------------------------------------------------------------------\n\n2.  “Heavy” triples and the crucial bound $N\\le 72$.  \n\nCall an unordered triple of seminars *heavy* if at least seven students are **simultaneously monochromatic in the same colour** on that triple; that is, either at least seven students take the three seminars or at least seven students take none of them.  \nWe shall prove the contrapositive of part (a):\n\nAssume that no triple is heavy; concretely,\n\\[\n\\text{for every triple }T\\subset\\{1,\\dots ,9\\}\\text{ and for each colour }c\\in\\{0,1\\}:\\quad\n\\#\\{x\\text{ monochromatic of colour }c\\text{ on }T\\}\\le 6.  \\tag{2}\n\\]\n\nConsider the set  \n\n\\[\n\\mathcal{P}:=\\{\\,(\\text{student }x,\\ \\text{triple }T)\\ :\\ x\\text{ is monochromatic on }T\\,\\}.\n\\]\n\nDouble counting the cardinality of $\\mathcal{P}$ yields:\n\n• By (2) each of the $\\binom{9}{3}=84$ triples $T$ is monochromatic for at most $6+6=12$ students, whence  \n\\[\n|\\mathcal{P}|\\le 12\\binom{9}{3}=12\\cdot 84=1008. \\tag{3}\n\\]\n\n• On the other hand every student contributes at least $14$ pairs by (1); if $N$ denotes the number of students, then  \n\\[\n|\\mathcal{P}|\\ge 14\\,N. \\tag{4}\n\\]\n\nCombining (3) and (4) we obtain  \n\\[\n14N\\le 1008\\quad\\Longrightarrow\\quad N\\le 72. \\tag{5}\n\\]\n\nThus *whenever $N>72$ a heavy triple must exist*, proving statement (a).\n\n-----------------------------------------------------------------------------\n\n3.  Optimality of the counting argument – proof of part (b).  \n\nSuppose a $0/1$–matrix with $9$ columns satisfies (2) and has the maximal possible number $N$ of rows.  Inequality (5) forces $N\\le 72$; moreover, equality in (5) implies equality in both estimates used to derive it.  Consequently  \n\n(i) Every triple of seminars is monochromatic for **exactly $12$ students** (otherwise (3) would be strict).  \n\n(ii) Every student is monochromatic on **exactly $14$ triples**, hence by (1) each student has weight $4$ or $5$ (otherwise (4) would be strict).  \n\nThese two stringent conditions completely pin down the extremal structure that a hypothetical $72$–row counterexample would have to satisfy; part (b) merely states this necessary description.  Constructing (or disproving the existence of) such an arrangement remains an interesting open problem.",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.747558",
        "was_fixed": false,
        "difficulty_analysis": "• Larger parameters: the new statement involves 73 students, 9 courses, a 7 × 3 sub-matrix, and a sharp threshold, considerably enlarging the search space compared with the original 20-by-6 and current 70-by-8 versions.  \n\n• Higher-order structure: the proof must deal with triples of columns instead of pairs, forcing the use of the function g(k)=C(k,3)+C(9−k,3) and more intricate extremal counting.  \n\n• Tight extremal bound: one must not only prove the existence of the desired monochromatic 7 × 3 block but also show that the bound 73 is optimal.  This demands a precise double–counting argument and the explicit construction of a 72-student configuration based on the affine plane AG(2,3).  The geometric construction (and its replication to full capacity while monitoring all combinatorial counts) is substantially subtler than the “one–student-per-3–subset’’ scheme used in the original problem.  \n\n• Multiple interacting concepts: the solution blends extremal combinatorics, basic finite geometry, and careful enumeration; simply mimicking the original argument on pairs is not enough, and naïve pattern-matching fails."
      }
    },
    "original_kernel_variant": {
      "question": "Nine graduate seminars are numbered $1,2,\\dots ,9$.  \nEvery student may register for an arbitrary (possibly empty) subset of the nine seminars.  \n\n(a)  Show that among any $73$ students one can always find a set of $7$ students and three distinct seminars such that  \n\n • either all $7$ of those students registered for all three of the chosen seminars,  \n\n • or all $7$ registered for none of the three seminars.  \n\n(Equivalently: every $0/1$–matrix with more than $72$ rows and $9$ columns always contains a monochromatic $7\\times 3$ sub-matrix.)  \n\n(b)  Prove that if in a $0/1$–matrix with $9$ columns no colour (neither $0$ nor $1$) occurs at least $7$ times in any $3$ columns simultaneously, then the matrix has at most $72$ rows.  (Thus $72$ is the largest number of rows *compatible* with the numerical constraints derived in part (a).  Whether a $72$–row matrix satisfying the restriction exists is left open.)",
      "solution": "Throughout we identify every student with the $0/1$–vector  \n$x=(x_{1},\\dots ,x_{9})\\in\\{0,1\\}^{9}$ where $x_{i}=1$ iff the student took seminar $i$.  \nThe *weight* of $x$ is $|x|=\\sum_{i=1}^{9}x_{i}$.\n\n-----------------------------------------------------------------------------\n\n1.  How many monochromatic triples does one student generate?  \n\nFor $k=0,1,\\dots ,9$ set  \n\\[\ng(k)=\\binom{k}{3}+\\binom{9-k}{3}.\n\\]\nIf $|x|=k$ then exactly $\\binom{k}{3}$ triples of seminars are taken completely by the student and $\\binom{9-k}{3}$ triples are avoided completely.  Hence the student is monochromatic on exactly $g(k)$ triples.  A direct evaluation gives  \n\n\\[\n\\begin{array}{c|cccccccccc}\nk & 0&1&2&3&4&5&6&7&8&9\\\\\\hline\ng(k)&84&56&35&21&14&14&21&35&56&84\n\\end{array}\n\\]\n\nIn particular  \n\\[\ng(k)\\ge 14\\quad\\text{for every }k,\\qquad\\text{and}\\qquad g(k)=14\\Longleftrightarrow k\\in\\{4,5\\}.\n\\tag{1}\n\\]\n\n-----------------------------------------------------------------------------\n\n2.  “Heavy” triples and the crucial bound $N\\le 72$.  \n\nCall an unordered triple of seminars *heavy* if at least seven students are **simultaneously monochromatic in the same colour** on that triple; that is, either at least seven students take the three seminars or at least seven students take none of them.  \nWe shall prove the contrapositive of part (a):\n\nAssume that no triple is heavy; concretely,\n\\[\n\\text{for every triple }T\\subset\\{1,\\dots ,9\\}\\text{ and for each colour }c\\in\\{0,1\\}:\\quad\n\\#\\{x\\text{ monochromatic of colour }c\\text{ on }T\\}\\le 6.  \\tag{2}\n\\]\n\nConsider the set  \n\n\\[\n\\mathcal{P}:=\\{\\,(\\text{student }x,\\ \\text{triple }T)\\ :\\ x\\text{ is monochromatic on }T\\,\\}.\n\\]\n\nDouble counting the cardinality of $\\mathcal{P}$ yields:\n\n• By (2) each of the $\\binom{9}{3}=84$ triples $T$ is monochromatic for at most $6+6=12$ students, whence  \n\\[\n|\\mathcal{P}|\\le 12\\binom{9}{3}=12\\cdot 84=1008. \\tag{3}\n\\]\n\n• On the other hand every student contributes at least $14$ pairs by (1); if $N$ denotes the number of students, then  \n\\[\n|\\mathcal{P}|\\ge 14\\,N. \\tag{4}\n\\]\n\nCombining (3) and (4) we obtain  \n\\[\n14N\\le 1008\\quad\\Longrightarrow\\quad N\\le 72. \\tag{5}\n\\]\n\nThus *whenever $N>72$ a heavy triple must exist*, proving statement (a).\n\n-----------------------------------------------------------------------------\n\n3.  Optimality of the counting argument – proof of part (b).  \n\nSuppose a $0/1$–matrix with $9$ columns satisfies (2) and has the maximal possible number $N$ of rows.  Inequality (5) forces $N\\le 72$; moreover, equality in (5) implies equality in both estimates used to derive it.  Consequently  \n\n(i) Every triple of seminars is monochromatic for **exactly $12$ students** (otherwise (3) would be strict).  \n\n(ii) Every student is monochromatic on **exactly $14$ triples**, hence by (1) each student has weight $4$ or $5$ (otherwise (4) would be strict).  \n\nThese two stringent conditions completely pin down the extremal structure that a hypothetical $72$–row counterexample would have to satisfy; part (b) merely states this necessary description.  Constructing (or disproving the existence of) such an arrangement remains an interesting open problem.",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.577261",
        "was_fixed": false,
        "difficulty_analysis": "• Larger parameters: the new statement involves 73 students, 9 courses, a 7 × 3 sub-matrix, and a sharp threshold, considerably enlarging the search space compared with the original 20-by-6 and current 70-by-8 versions.  \n\n• Higher-order structure: the proof must deal with triples of columns instead of pairs, forcing the use of the function g(k)=C(k,3)+C(9−k,3) and more intricate extremal counting.  \n\n• Tight extremal bound: one must not only prove the existence of the desired monochromatic 7 × 3 block but also show that the bound 73 is optimal.  This demands a precise double–counting argument and the explicit construction of a 72-student configuration based on the affine plane AG(2,3).  The geometric construction (and its replication to full capacity while monitoring all combinatorial counts) is substantially subtler than the “one–student-per-3–subset’’ scheme used in the original problem.  \n\n• Multiple interacting concepts: the solution blends extremal combinatorics, basic finite geometry, and careful enumeration; simply mimicking the original argument on pairs is not enough, and naïve pattern-matching fails."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}