{
  "index": "1959-B-1",
  "type": "COMB",
  "tag": [
    "COMB",
    "GEO"
  ],
  "difficulty": "",
  "question": "1. Let each of \\( m \\) distinct points on the positive part of the \\( X \\)-axis be joined to \\( n \\) distinct points on the positive part of the \\( \\boldsymbol{Y} \\)-axis. Obtain a formula for the number of intersection points of these segments (exclusive of endpoints), assuming that no three of the segments are concurrent.",
  "solution": "Solution. Each pair of points on the \\( X \\)-axis together with each pair of points on the \\( Y \\)-axis determine a convex quadrilateral whose diagonals meet somewhere in the first quadrant. Conversely, each intersection point arises in this way. Since no three segments are concurrent, except at the endpoints, each intersection point arises uniquely. There are, therefore,\n\\[\n\\binom{m}{2}\\binom{n}{2}=m n(m-1)(n-1) / 4\n\\]\npoints of intersection. (Compare Problem A.M. 4 of the Fifteenth Competition.)",
  "vars": [
    "m",
    "n"
  ],
  "params": [
    "X",
    "Y"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "m": "xpointcount",
        "n": "ypointcount",
        "X": "xaxisline",
        "Y": "yaxisline"
      },
      "question": "1. Let each of \\( xpointcount \\) distinct points on the positive part of the \\( xaxisline \\)-axis be joined to \\( ypointcount \\) distinct points on the positive part of the \\( \\boldsymbol{yaxisline} \\)-axis. Obtain a formula for the number of intersection points of these segments (exclusive of endpoints), assuming that no three of the segments are concurrent.",
      "solution": "Solution. Each pair of points on the \\( xaxisline \\)-axis together with each pair of points on the \\( yaxisline \\)-axis determine a convex quadrilateral whose diagonals meet somewhere in the first quadrant. Conversely, each intersection point arises in this way. Since no three segments are concurrent, except at the endpoints, each intersection point arises uniquely. There are, therefore,\n\\[\n\\binom{xpointcount}{2}\\binom{ypointcount}{2}=xpointcount\\, ypointcount(xpointcount-1)(ypointcount-1) / 4\n\\]\npoints of intersection. (Compare Problem A.M. 4 of the Fifteenth Competition.)"
    },
    "descriptive_long_confusing": {
      "map": {
        "m": "sunflower",
        "n": "seahorse",
        "X": "walnuttree",
        "Y": "blueberry"
      },
      "question": "1. Let each of \\( sunflower \\) distinct points on the positive part of the \\( walnuttree \\)-axis be joined to \\( seahorse \\) distinct points on the positive part of the \\( \\boldsymbol{blueberry} \\)-axis. Obtain a formula for the number of intersection points of these segments (exclusive of endpoints), assuming that no three of the segments are concurrent.",
      "solution": "Solution. Each pair of points on the \\( walnuttree \\)-axis together with each pair of points on the \\( blueberry \\)-axis determine a convex quadrilateral whose diagonals meet somewhere in the first quadrant. Conversely, each intersection point arises in this way. Since no three segments are concurrent, except at the endpoints, each intersection point arises uniquely. There are, therefore,\n\\[\n\\binom{sunflower}{2}\\binom{seahorse}{2}=sunflower\\,seahorse(sunflower-1)(seahorse-1) / 4\n\\]\npoints of intersection. (Compare Problem A.M. 4 of the Fifteenth Competition.)"
    },
    "descriptive_long_misleading": {
      "map": {
        "m": "emptiness",
        "n": "voidness",
        "X": "vertical",
        "Y": "horizontal"
      },
      "question": "1. Let each of \\( emptiness \\) distinct points on the positive part of the \\( vertical \\)-axis be joined to \\( voidness \\) distinct points on the positive part of the \\( \\boldsymbol{horizontal} \\)-axis. Obtain a formula for the number of intersection points of these segments (exclusive of endpoints), assuming that no three of the segments are concurrent.",
      "solution": "Solution. Each pair of points on the \\( vertical \\)-axis together with each pair of points on the \\( horizontal \\)-axis determine a convex quadrilateral whose diagonals meet somewhere in the first quadrant. Conversely, each intersection point arises in this way. Since no three segments are concurrent, except at the endpoints, each intersection point arises uniquely. There are, therefore,\n\\[\n\\binom{emptiness}{2}\\binom{voidness}{2}=emptiness voidness(emptiness-1)(voidness-1) / 4\n\\]\npoints of intersection. (Compare Problem A.M. 4 of the Fifteenth Competition.)"
    },
    "garbled_string": {
      "map": {
        "m": "qzxwvtnp",
        "n": "hjgrksla",
        "X": "fcdnxwzp",
        "Y": "gjlmvktr"
      },
      "question": "1. Let each of \\( qzxwvtnp \\) distinct points on the positive part of the \\( fcdnxwzp \\)-axis be joined to \\( hjgrksla \\) distinct points on the positive part of the \\( \\boldsymbol{gjlmvktr} \\)-axis. Obtain a formula for the number of intersection points of these segments (exclusive of endpoints), assuming that no three of the segments are concurrent.",
      "solution": "Solution. Each pair of points on the \\( fcdnxwzp \\)-axis together with each pair of points on the \\( gjlmvktr \\)-axis determine a convex quadrilateral whose diagonals meet somewhere in the first quadrant. Conversely, each intersection point arises in this way. Since no three segments are concurrent, except at the endpoints, each intersection point arises uniquely. There are, therefore,\n\\[\n\\binom{qzxwvtnp}{2}\\binom{hjgrksla}{2}=qzxwvtnp hjgrksla(qzxwvtnp-1)(hjgrksla-1) / 4\n\\]\npoints of intersection. (Compare Problem A.M. 4 of the Fifteenth Competition.)"
    },
    "kernel_variant": {
      "question": "Let ℓ₁ be the line y = x and ℓ₂ the line y = −x.  Fix a radius R > 0 and let 𝔻 = { (x, y) | x² + y² < R² } be the open disc centred at the origin.\n\nWrite\n  ℓ₁⁺ = { (t, t) : 0 < t < R/√2 }   (the part of ℓ₁ that lies in the first quadrant),\n  ℓ₂⁺ = { (−t, t) : 0 < t < R/√2 }  (the part of ℓ₂ that lies in the second quadrant).\n\nChoose p ≥ 2 distinct points A₁ , … , A_p on ℓ₁⁺ and q ≥ 2 distinct points B₁ , … , B_q on ℓ₂⁺ (all of them belonging to 𝔻).  Join every A-point to every B-point with a straight-line segment and assume that, apart from their endpoints, no three of the pq segments are concurrent.\n\nHow many interior intersection points—that is, points that lie strictly inside 𝔻 and are not endpoints—are produced by the drawing?  Give your answer in closed form in terms of p and q.",
      "solution": "Label the chosen points along each ray by increasing distance from the origin:\n       A₁ , A₂ , … , A_p   on ℓ₁⁺      (so |OA₁| < |OA₂| < … < |OA_p|),\n       B₁ , B₂ , … , B_q   on ℓ₂⁺      (so |OB₁| < |OB₂| < … < |OB_q|).\nEvery drawn segment is of the form A_i B_j with 1 ≤ i ≤ p and 1 ≤ j ≤ q.\n\nStep 1  (From index pairs to intersection points).\nTake any two indices i₁ < i₂ and any two indices j₁ < j₂.  Because A_{i₁}, A_{i₂} lie on the same ray ℓ₁⁺ and B_{j₁}, B_{j₂} lie on the same ray ℓ₂⁺, the four points\n        A_{i₁},  B_{j₂},  A_{i₂},  B_{j₁}\noccur alternately around the origin and therefore are the vertices of a convex quadrilateral.  Its two diagonals are the segments A_{i₁}B_{j₂} and A_{i₂}B_{j₁}; these two segments meet in exactly one point, call it P.  All four vertices lie inside the disc 𝔻 and 𝔻 is convex, so their intersection point P also lies in 𝔻 and is not an endpoint.  Hence every choice of an unordered pair of A–indices and an unordered pair of B–indices gives one interior intersection point.\n\nStep 2  (From intersection points back to index pairs).\nConversely, let P be any interior intersection point.  Exactly two of the pq segments meet at P, say A_{i₁}B_{j₂} and A_{i₂}B_{j₁}, with i₁ ≠ i₂ and j₁ ≠ j₂.  Because of the ‘no three concurrent’ assumption, the indices i₁,i₂ and j₁,j₂ are uniquely determined, and P is precisely the intersection of the diagonals of the convex quadrilateral with vertices A_{i₁}, A_{i₂}, B_{j₁}, B_{j₂} chosen as in Step 1.\n\nStep 3  (Counting).\nThus there is a bijection between\n   – unordered pairs {i₁,i₂} with 1 ≤ i₁ < i₂ ≤ p, and\n   – unordered pairs {j₁,j₂} with 1 ≤ j₁ < j₂ ≤ q,\nand the set of interior intersection points.  Hence\n        # intersections = C(p,2) · C(q,2)\n                       = [p(p−1)/2] · [q(q−1)/2]\n                       = p q (p−1)(q−1) / 4.\n\nAnswer:  p q (p − 1)(q − 1) ⁄ 4.",
      "_meta": {
        "core_steps": [
          "Pick any 2 points on the first line and any 2 points on the second line; they form a convex quadrilateral.",
          "The two drawn segments that act as its diagonals necessarily intersect (exactly once).",
          "Because no three segments concur (away from endpoints), that intersection corresponds to one and only one such quadruple of endpoints.",
          "Thus intersections ↔ (2-subsets of the first set) × (2-subsets of the second set) bijectively.",
          "Count them: C(m,2)·C(n,2) = m n (m−1)(n−1)/4."
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Which two distinct lines the two point-sets lie on (only their distinctness matters).",
            "original": "the X-axis and the Y-axis"
          },
          "slot2": {
            "description": "Use of the positive rays versus the entire lines; any ordering along each line suffices.",
            "original": "“positive part” of each axis"
          },
          "slot3": {
            "description": "Specific quadrant/wedge where the intersections fall; location is irrelevant.",
            "original": "the ‘first quadrant’"
          },
          "slot4": {
            "description": "Choice of symbols for the two population sizes.",
            "original": "m and n"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "proof",
  "iteratively_fixed": true
}