{
  "index": "1979-A-4",
  "type": "COMB",
  "tag": [
    "COMB",
    "GEO"
  ],
  "difficulty": "",
  "question": "Problem A-4\nLet \\( A \\) be a set of \\( 2 n \\) points in the plane, no three of which are collinear. Suppose that \\( n \\) of them are colored red and the remaining \\( n \\) blue. Prove or disprove: there are \\( n \\) closed straight line segments, no two with a point in common, such that the endpoints of each segment are points of \\( A \\) having different colors.",
  "solution": "A-4.\nThere are a finite number (actually \\( n! \\) ) of ways of pairing each of the red points with a blue point in a 1-to-1 way. Hence, there exists a pairing for which the sum of the lengths of the segments joining paired points is minimal. We now show that for such a pairing no two of the \\( n \\) segments intersect.\nLet red points \\( R \\) and \\( R^{\\prime} \\) be paired with \\( B \\) and \\( B^{\\prime} \\), respectively, and assume that segments \\( R B \\) and \\( R^{\\prime} B^{\\prime} \\) intersect. The triangle inequality implies that the sum of the lengths of these segments exceeds the sum of the lengths of segments \\( R B^{\\prime} \\) and \\( R^{\\prime} B \\). Then interchanging \\( B \\) and \\( B^{\\prime} \\) would give us a new pairing with a smaller sum of segment lengths. This contradiction proves the existence of a pairing with nonintersecting segments.",
  "vars": [
    "A",
    "R",
    "B"
  ],
  "params": [
    "n"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "A": "pointset",
        "R": "redpoint",
        "B": "bluepoint",
        "n": "paircount"
      },
      "question": "Problem A-4\nLet \\( pointset \\) be a set of \\( 2 paircount \\) points in the plane, no three of which are collinear. Suppose that \\( paircount \\) of them are colored red and the remaining \\( paircount \\) blue. Prove or disprove: there are \\( paircount \\) closed straight line segments, no two with a point in common, such that the endpoints of each segment are points of \\( pointset \\) having different colors.",
      "solution": "A-4.\nThere are a finite number (actually \\( paircount! \\) ) of ways of pairing each of the red points with a blue point in a 1-to-1 way. Hence, there exists a pairing for which the sum of the lengths of the segments joining paired points is minimal. We now show that for such a pairing no two of the \\( paircount \\) segments intersect.\nLet red points \\( redpoint \\) and \\( redpoint^{\\prime} \\) be paired with \\( bluepoint \\) and \\( bluepoint^{\\prime} \\), respectively, and assume that segments \\( redpoint bluepoint \\) and \\( redpoint^{\\prime} bluepoint^{\\prime} \\) intersect. The triangle inequality implies that the sum of the lengths of these segments exceeds the sum of the lengths of segments \\( redpoint bluepoint^{\\prime} \\) and \\( redpoint^{\\prime} bluepoint \\). Then interchanging \\( bluepoint \\) and \\( bluepoint^{\\prime} \\) would give us a new pairing with a smaller sum of segment lengths. This contradiction proves the existence of a pairing with nonintersecting segments."
    },
    "descriptive_long_confusing": {
      "map": {
        "A": "velocity",
        "R": "momentum",
        "B": "pressure",
        "n": "longitude"
      },
      "question": "Problem A-4\nLet \\( velocity \\) be a set of \\( 2 longitude \\) points in the plane, no three of which are collinear. Suppose that \\( longitude \\) of them are colored red and the remaining \\( longitude \\) blue. Prove or disprove: there are \\( longitude \\) closed straight line segments, no two with a point in common, such that the endpoints of each segment are points of \\( velocity \\) having different colors.",
      "solution": "A-4.\nThere are a finite number (actually \\( longitude! \\) ) of ways of pairing each of the red points with a blue point in a 1-to-1 way. Hence, there exists a pairing for which the sum of the lengths of the segments joining paired points is minimal. We now show that for such a pairing no two of the \\( longitude \\) segments intersect.\nLet red points \\( momentum \\) and \\( momentum^{\\prime} \\) be paired with \\( pressure \\) and \\( pressure^{\\prime} \\), respectively, and assume that segments \\( momentum pressure \\) and \\( momentum^{\\prime} pressure^{\\prime} \\) intersect. The triangle inequality implies that the sum of the lengths of these segments exceeds the sum of the lengths of segments \\( momentum pressure^{\\prime} \\) and \\( momentum^{\\prime} pressure \\). Then interchanging \\( pressure \\) and \\( pressure^{\\prime} \\) would give us a new pairing with a smaller sum of segment lengths. This contradiction proves the existence of a pairing with nonintersecting segments."
    },
    "descriptive_long_misleading": {
      "map": {
        "A": "emptiness",
        "R": "colorless",
        "B": "yellowish",
        "n": "infinite"
      },
      "question": "Problem A-4\nLet \\( emptiness \\) be a set of \\( 2 infinite \\) points in the plane, no three of which are collinear. Suppose that \\( infinite \\) of them are colored red and the remaining \\( infinite \\) blue. Prove or disprove: there are \\( infinite \\) closed straight line segments, no two with a point in common, such that the endpoints of each segment are points of \\( emptiness \\) having different colors.",
      "solution": "A-4.\nThere are a finite number (actually \\( infinite! \\) ) of ways of pairing each of the red points with a blue point in a 1-to-1 way. Hence, there exists a pairing for which the sum of the lengths of the segments joining paired points is minimal. We now show that for such a pairing no two of the \\( infinite \\) segments intersect.\nLet red points \\( colorless \\) and \\( colorless^{\\prime} \\) be paired with \\( yellowish \\) and \\( yellowish^{\\prime} \\), respectively, and assume that segments \\( colorless yellowish \\) and \\( colorless^{\\prime} yellowish^{\\prime} \\) intersect. The triangle inequality implies that the sum of the lengths of these segments exceeds the sum of the lengths of segments \\( colorless yellowish^{\\prime} \\) and \\( colorless^{\\prime} yellowish \\). Then interchanging \\( yellowish \\) and \\( yellowish^{\\prime} \\) would give us a new pairing with a smaller sum of segment lengths. This contradiction proves the existence of a pairing with nonintersecting segments."
    },
    "garbled_string": {
      "map": {
        "A": "qzxwvtnp",
        "R": "hjgrksla",
        "B": "mdfqpezi",
        "n": "lcvharud"
      },
      "question": "Problem A-4\nLet \\( qzxwvtnp \\) be a set of \\( 2 lcvharud \\) points in the plane, no three of which are collinear. Suppose that \\( lcvharud \\) of them are colored red and the remaining \\( lcvharud \\) blue. Prove or disprove: there are \\( lcvharud \\) closed straight line segments, no two with a point in common, such that the endpoints of each segment are points of \\( qzxwvtnp \\) having different colors.",
      "solution": "A-4.\nThere are a finite number (actually \\( lcvharud! \\) ) of ways of pairing each of the red points with a blue point in a 1-to-1 way. Hence, there exists a pairing for which the sum of the lengths of the segments joining paired points is minimal. We now show that for such a pairing no two of the \\( lcvharud \\) segments intersect.\nLet red points \\( hjgrksla \\) and \\( hjgrksla^{\\prime} \\) be paired with \\( mdfqpezi \\) and \\( mdfqpezi^{\\prime} \\), respectively, and assume that segments \\( hjgrksla mdfqpezi \\) and \\( hjgrksla^{\\prime} mdfqpezi^{\\prime} \\) intersect. The triangle inequality implies that the sum of the lengths of these segments exceeds the sum of the lengths of segments \\( hjgrksla mdfqpezi^{\\prime} \\) and \\( hjgrksla^{\\prime} mdfqpezi \\). Then interchanging \\( mdfqpezi \\) and \\( mdfqpezi^{\\prime} \\) would give us a new pairing with a smaller sum of segment lengths. This contradiction proves the existence of a pairing with nonintersecting segments."
    },
    "kernel_variant": {
      "question": "Let k ≥ 1 be an integer and let 𝒫 be a set of 2k points in the Euclidean plane, no three of which are collinear. Exactly k of the points are coloured gold and the remaining k silver. Prove that one can select k pairwise disjoint straight-line segments whose union of endpoints equals 𝒫 and whose every segment joins a gold point with a silver point (so that each point of 𝒫 is an endpoint of exactly one of the selected segments).",
      "solution": "Label the gold points G1, …, Gk and the silver points S1, …, Sk.\n\n1.  A length function on matchings.\n   A matching is a bijection σ : {1,…,k} → {1,…,k}; it pairs Gi with Sσ(i).\n   For every matching define\n            L(σ) = Σ_{i=1}^{k} |Gi Sσ(i)|,                           (1)\n   where |PQ| denotes the Euclidean length of segment PQ.\n   Because only k! matchings exist, there is a matching σ0 for which the\n   sum L(σ) is minimal.  Fix such a σ0 once and for all.\n\n2.  The minimal-length segments do not cross.\n   Suppose, to obtain a contradiction, that two of the segments GiSσ0(i)\n   and GjSσ0(j) (i ≠ j) intersect in an interior point X.  Relabel so that\n   i = 1 and j = 2, and abbreviate S1 := Sσ0(1),  S2 := Sσ0(2).\n\n   Because X is interior to G1S1 and to G2S2,\n          |G1S1| = |G1X| + |XS1|,                     (2a)\n          |G2S2| = |G2X| + |XS2|.                     (2b)\n\n   For the ‘cross’ pairs the triangle inequality yields\n          |G1S2| ≤ |G1X| + |XS2|,                    (3a)\n          |G2S1| ≤ |G2X| + |XS1|.                    (3b)\n\n   We now show that the two inequalities (3a) and (3b) cannot \n   simultaneously be equalities.\n\n   •  Equality in (3a) holds exactly when X lies on the segment G1S2 with\n      the order G1–X–S2.  Likewise, equality in (3b) holds exactly when\n      X lies on the segment G2S1 with the order G2–X–S1.\n   •  Assume for contradiction that both equalities do hold.  Then the\n      four original points satisfy\n           G1, X, S2 are collinear,         G2, X, S1 are collinear.   (4)\n      Because G1S1 passes through X, collinearity (4) shows that S2 also\n      lies on the straight line G1S1.  Hence the three distinct original\n      points G1, S1, S2 are collinear, contradicting the hypothesis that\n      no three of the 2k points are collinear.  Therefore at least one\n      of (3a),(3b) must be a strict inequality.\n\n   Adding (2a) and (2b) and regrouping the four summands we obtain\n          |G1S1| + |G2S2| = (|G1X| + |XS2|) + (|G2X| + |XS1|).        (5)\n   Because at least one of (3a) or (3b) is strict, the right–hand side of\n   (5) is strictly greater than |G1S2| + |G2S1|.  Thus\n          |G1S1| + |G2S2| > |G1S2| + |G2S1|.                          (6)\n\n   Define a new matching σ′ by exchanging the two silver partners:\n          σ′(1) = σ0(2),   σ′(2) = σ0(1),   and σ′(t) = σ0(t) (t > 2).\n   Inequality (6) translates exactly into L(σ′) < L(σ0), contradicting\n   the minimality of σ0.  Hence no two of the segments GiSσ0(i) intersect.\n\n3.  Completion.\n   The k segments GiSσ0(i) are pairwise disjoint, each joins one gold\n   point with one silver point, and every point of 𝒫 is an endpoint of\n   exactly one segment.  Therefore the desired set of k non-intersecting\n   gold–silver segments exists.\n\nThis completes the proof.",
      "_meta": {
        "core_steps": [
          "Consider all bijective red–blue matchings and choose one with minimal total segment length.",
          "Assume two segments of this minimal matching cross.",
          "Swap the blue (or red) endpoints; triangle inequality shortens total length, contradicting minimality.",
          "Therefore no crossings occur; the n non-intersecting bichromatic segments exist."
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Exact equality of the two color classes; any common size m works.",
            "original": "n red points and n blue points (total 2n)."
          },
          "slot2": {
            "description": "The specific color names; could be any two labels or types.",
            "original": "“red” and “blue”."
          },
          "slot3": {
            "description": "The explicit numerical count of bijections, used only to assert finiteness.",
            "original": "n! possible pairings."
          },
          "slot4": {
            "description": "The adjective “closed” attached to the segments; open or just ‘line segment’ suffices.",
            "original": "“closed straight line segments”."
          },
          "slot5": {
            "description": "Euclidean metric; any distance obeying the triangle inequality keeps the argument intact.",
            "original": "ordinary Euclidean plane distance."
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "proof",
  "iteratively_fixed": true
}