{
  "index": "1991-A-4",
  "type": "GEO",
  "tag": [
    "GEO",
    "ANA"
  ],
  "difficulty": "",
  "question": "Does there exist an infinite sequence of closed discs $D_1, D_2, D_3,\n\\dots$ in the plane, with centers $c_1, c_2, c_3, \\dots$, respectively,\nsuch that\n\\begin{enumerate}\n    \\item the $c_i$ have no limit point in the finite plane,\n    \\item the sum of the areas of the $D_i$ is finite, and\n    \\item every line in the plane intersects at least one of the $D_i$?\n\\end{enumerate}",
  "solution": "Solution. Let \\( a_{i}=1 / i \\) for \\( i \\geq 1 \\) (or choose any other sequence of positive numbers \\( a_{i} \\) satisfying \\( \\sum_{i=1}^{\\infty} a_{i}=\\infty \\) and \\( \\left.\\sum_{i=1}^{\\infty} a_{i}^{2}<\\infty\\right) \\). For \\( n \\geq 1 \\), let \\( A_{n}=a_{1}+a_{2}+\\cdots+a_{n} \\). Let \\( U \\) be the union of the discs of radius \\( a_{n} \\) centered at \\( \\left(A_{n}, 0\\right),\\left(-A_{n}, 0\\right),\\left(0, A_{n}\\right) \\), \\( \\left(0,-A_{n}\\right) \\), for all \\( n \\geq 1 \\). Then \\( U \\) covers the two coordinate axes, and has finite total area. Every line in the plane meets at least one axis, and hence meets \\( U \\). Finally, the centers have no limit point, since every circle \\( C \\) centered at the origin encloses at most finitely many centers: if \\( C \\) has radius \\( R \\), we can choose \\( n \\) such that \\( A_{n}>R \\), and then less than \\( 4 n \\) centers lie inside \\( C \\).",
  "vars": [
    "i",
    "n"
  ],
  "params": [
    "D_i",
    "c_i",
    "a_i",
    "A_n",
    "U",
    "R"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "i": "indexer",
        "n": "counter",
        "D_i": "discfamily",
        "c_i": "centerarray",
        "a_i": "radseq",
        "A_n": "partialsum",
        "U": "unionset",
        "R": "radiuslimit"
      },
      "question": "Does there exist an infinite sequence of closed discs $discfamily_1, discfamily_2, discfamily_3, \\dots$ in the plane, with centers $centerarray_1, centerarray_2, centerarray_3, \\dots$, respectively, such that\n\\begin{enumerate}\n    \\item the $centerarray_{indexer}$ have no limit point in the finite plane,\n    \\item the sum of the areas of the $discfamily_{indexer}$ is finite, and\n    \\item every line in the plane intersects at least one of the $discfamily_{indexer}$?\n\\end{enumerate}",
      "solution": "Solution. Let \\( radseq_{indexer}=1 / indexer \\) for \\( indexer \\geq 1 \\) (or choose any other sequence of positive numbers \\( radseq_{indexer} \\) satisfying \\( \\sum_{indexer=1}^{\\infty} radseq_{indexer}=\\infty \\) and \\( \\left.\\sum_{indexer=1}^{\\infty} radseq_{indexer}^{2}<\\infty\\right) \\). For \\( counter \\geq 1 \\), let \\( partialsum_{counter}=radseq_{1}+radseq_{2}+\\cdots+radseq_{counter} \\). Let \\( unionset \\) be the union of the discs of radius \\( radseq_{counter} \\) centered at \\( \\left(partialsum_{counter}, 0\\right),\\left(-partialsum_{counter}, 0\\right),\\left(0, partialsum_{counter}\\right) \\), \\( \\left(0,-partialsum_{counter}\\right) \\), for all \\( counter \\geq 1 \\). Then \\( unionset \\) covers the two coordinate axes, and has finite total area. Every line in the plane meets at least one axis, and hence meets \\( unionset \\). Finally, the centers have no limit point, since every circle \\( C \\) centered at the origin encloses at most finitely many centers: if \\( C \\) has radius \\( radiuslimit \\), we can choose \\( counter \\) such that \\( partialsum_{counter}>radiuslimit \\), and then less than \\( 4 counter \\) centers lie inside \\( C \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "i": "lanterns",
        "n": "marigolds",
        "D_i": "blueberry",
        "c_i": "sailcloth",
        "a_i": "rainstorm",
        "A_n": "pinecones",
        "U": "goldcrest",
        "R": "kingfisher"
      },
      "question": "Does there exist an infinite sequence of closed discs $blueberry_1, blueberry_2, blueberry_3, \\dots$ in the plane, with centers $sailcloth_1, sailcloth_2, sailcloth_3, \\dots$, respectively, such that\n\\begin{enumerate}\n    \\item the $sailcloth_{lanterns}$ have no limit point in the finite plane,\n    \\item the sum of the areas of the $blueberry_{lanterns}$ is finite, and\n    \\item every line in the plane intersects at least one of the $blueberry_{lanterns}$?\n\\end{enumerate}",
      "solution": "Solution. Let \\( rainstorm_{lanterns}=1 / lanterns \\) for \\( lanterns \\geq 1 \\) (or choose any other sequence of positive numbers \\( rainstorm_{lanterns} \\) satisfying \\( \\sum_{lanterns=1}^{\\infty} rainstorm_{lanterns}=\\infty \\) and \\( \\left.\\sum_{lanterns=1}^{\\infty} rainstorm_{lanterns}^{2}<\\infty\\right) \\). For \\( marigolds \\geq 1 \\), let \\( pinecones_{marigolds}=rainstorm_{1}+rainstorm_{2}+\\cdots+rainstorm_{marigolds} \\). Let \\( goldcrest \\) be the union of the discs of radius \\( rainstorm_{marigolds} \\) centered at \\( \\left(pinecones_{marigolds}, 0\\right),\\left(-pinecones_{marigolds}, 0\\right),\\left(0, pinecones_{marigolds}\\right), \\left(0,-pinecones_{marigolds}\\right) \\), for all \\( marigolds \\geq 1 \\). Then \\( goldcrest \\) covers the two coordinate axes, and has finite total area. Every line in the plane meets at least one axis, and hence meets \\( goldcrest \\). Finally, the centers have no limit point, since every circle \\( C \\) centered at the origin encloses at most finitely many centers: if \\( C \\) has radius \\( kingfisher \\), we can choose \\( marigolds \\) such that \\( pinecones_{marigolds}>kingfisher \\), and then less than \\( 4 marigolds \\) centers lie inside \\( C \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "i": "endpointvar",
        "n": "fewcount",
        "D_i": "linedomain",
        "c_i": "borderpoint",
        "a_i": "negativescalar",
        "A_n": "subtractionval",
        "U": "intersection",
        "R": "linearity"
      },
      "question": "Does there exist an infinite sequence of closed discs $linedomain_1, linedomain_2, linedomain_3,\n\\dots$ in the plane, with centers $borderpoint_1, borderpoint_2, borderpoint_3, \\dots$, respectively,\nsuch that\n\\begin{enumerate}\n    \\item the $borderpoint_{endpointvar}$ have no limit point in the finite plane,\n    \\item the sum of the areas of the $linedomain_{endpointvar}$ is finite, and\n    \\item every line in the plane intersects at least one of the $linedomain_{endpointvar}$?\n\\end{enumerate}",
      "solution": "Solution. Let \\( negativescalar_{endpointvar}=1 / endpointvar \\) for \\( endpointvar \\geq 1 \\) (or choose any other sequence of positive numbers \\( negativescalar_{endpointvar} \\) satisfying \\( \\sum_{endpointvar=1}^{\\infty} negativescalar_{endpointvar}=\\infty \\) and \\( \\left.\\sum_{endpointvar=1}^{\\infty} negativescalar_{endpointvar}^{2}<\\infty\\right) \\). For \\( fewcount \\geq 1 \\), let \\( subtractionval_{fewcount}=negativescalar_{1}+negativescalar_{2}+\\cdots+negativescalar_{fewcount} \\). Let \\( intersection \\) be the union of the discs of radius \\( negativescalar_{fewcount} \\) centered at \\( \\left(subtractionval_{fewcount}, 0\\right),\\left(-subtractionval_{fewcount}, 0\\right),\\left(0, subtractionval_{fewcount}\\right) \\), \\( \\left(0,-subtractionval_{fewcount}\\right) \\), for all \\( fewcount \\geq 1 \\). Then \\( intersection \\) covers the two coordinate axes, and has finite total area. Every line in the plane meets at least one axis, and hence meets \\( intersection \\). Finally, the centers have no limit point, since every circle \\( C \\) centered at the origin encloses at most finitely many centers: if \\( C \\) has radius \\( linearity \\), we can choose \\( fewcount \\) such that \\( subtractionval_{fewcount}>linearity \\), and then less than \\( 4 fewcount \\) centers lie inside \\( C \\)."
    },
    "garbled_string": {
      "map": {
        "i": "qzxwvtnp",
        "n": "hjgrksla",
        "D_i": "vbxqtrsm",
        "c_i": "ylqmpost",
        "a_i": "gzprxwvu",
        "A_n": "lswdoekj",
        "U": "oqierdka",
        "R": "nvxjsklm"
      },
      "question": "Does there exist an infinite sequence of closed discs $vbxqtrsm_1, vbxqtrsm_2, vbxqtrsm_3,\n\\dots$ in the plane, with centers $ylqmpost_1, ylqmpost_2, ylqmpost_3, \\dots$, respectively,\nsuch that\n\\begin{enumerate}\n    \\item the $\\ylqmpost$ have no limit point in the finite plane,\n    \\item the sum of the areas of the $\\vbxqtrsm$ is finite, and\n    \\item every line in the plane intersects at least one of the $\\vbxqtrsm$?\n\\end{enumerate}",
      "solution": "Solution. Let \\( gzprxwvu_{qzxwvtnp}=1 / qzxwvtnp \\) for \\( qzxwvtnp \\geq 1 \\) (or choose any other sequence of positive numbers \\( gzprxwvu_{qzxwvtnp} \\) satisfying \\( \\sum_{qzxwvtnp=1}^{\\infty} gzprxwvu_{qzxwvtnp}=\\infty \\) and \\( \\left.\\sum_{qzxwvtnp=1}^{\\infty} gzprxwvu_{qzxwvtnp}^{2}<\\infty\\right) \\). For \\( hjgrksla \\geq 1 \\), let \\( lswdoekj_{hjgrksla}=gzprxwvu_{1}+gzprxwvu_{2}+\\cdots+gzprxwvu_{hjgrksla} \\). Let \\( oqierdka \\) be the union of the discs of radius \\( gzprxwvu_{hjgrksla} \\) centered at \\( \\left(lswdoekj_{hjgrksla}, 0\\right),\\left(-lswdoekj_{hjgrksla}, 0\\right),\\left(0, lswdoekj_{hjgrksla}\\right) \\), \\( \\left(0,-lswdoekj_{hjgrksla}\\right) \\), for all \\( hjgrksla \\geq 1 \\). Then \\( oqierdka \\) covers the two coordinate axes, and has finite total area. Every line in the plane meets at least one axis, and hence meets \\( oqierdka \\). Finally, the centers have no limit point, since every circle \\( C \\) centered at the origin encloses at most finitely many centers: if \\( C \\) has radius \\( nvxjsklm \\), we can choose \\( hjgrksla \\) such that \\( lswdoekj_{hjgrksla}>nvxjsklm \\), and then less than \\( 4 hjgrksla \\) centers lie inside \\( C \\)."
    },
    "kernel_variant": {
      "question": "Let \\(L_1\\) be the line \\(y=x\\) and let \\(L_2\\) be the line \\(y=-x+1\\).  Do there exist infinitely many closed discs \\(D_1,D_2,\\dots\\) in the plane, with centres \\(c_1,c_2,\\dots\\), such that\n\\begin{enumerate}\n\\item[(i)] the set \\(\\{c_i\\}\\) has no accumulation point in the finite plane;\n\\item[(ii)] \\(\\displaystyle\\sum_{i=1}^{\\infty}\\operatorname{area}(D_i) < \\infty;\\)\n\\item[(iii)] every straight line in the plane meets at least one of the discs \\(D_i\\)?\n\\end{enumerate}",
      "solution": "Yes.  We follow the same “axis‐covering’’ idea, but along the two perpendicular lines L₁:y=x and L₂:y=−x+1, which meet at P=(½,½).\n\n1. Choice of radii.  Let\n  aₙ=1/n^{2/3},    n≥1.\nThen ∑aₙ=∞ (since 2/3<1) while ∑aₙ²=∑n^{-4/3}<∞.  Set\n  rₙ=2aₙ=2/n^{2/3}.\n\n2. Placement of centres.  For each n define\n  Sₙ=a₁+…+aₙ,\nand place four centres at distance Sₙ from P along the two directions of L₁ and the two directions of L₂.  Concretely, if\n  v₁=(1/√2,1/√2),  v₂=(1/√2,−1/√2),\nthen the four centres are\n  P±Sₙv₁,  P±Sₙv₂.\nDenote the corresponding four discs by D_{n,1},…,D_{n,4}, each of radius rₙ.\n\n3. Covering each ray.  Along each of the four half‐lines from P the centres lie at distances S₁<S₂<… from P.  Note:\n  (i)  r₁=2a₁≥a₁=S₁, so D₁ covers P itself,\n  (ii)  the gap between successive centres is S_{n+1}−Sₙ=a_{n+1},\n  (iii)  r_{n+1}=2a_{n+1}≥a_{n+1}, so D_{n+1} contains the centre of Dₙ.\nEquivalently, Dₙ covers [Sₙ−rₙ,Sₙ+rₙ], and\n  S_{n+1}−r_{n+1}=Sₙ+a_{n+1}−2a_{n+1}=Sₙ−a_{n+1}≤Sₙ+rₙ,\nso these intervals overlap and their union is [0,∞).  Thus each ray is completely covered.\n\n4. Every line meets a disc.  Any straight line in the plane meets at least one of the two nonparallel lines L₁ or L₂.  Since each of L₁ and L₂ is fully covered by our discs, every line meets at least one Dᵢ.\n\n5. Total area finite.  In generation n there are 4 discs, each of area πrₙ²=4πaₙ², so the total area is\n  ∑_{n=1}^∞4·(4πaₙ²)=16π∑_{n=1}^∞aₙ²<∞.\n\n6. No finite accumulation of centres.  Since Sₙ→∞, only finitely many centres lie in any bounded region.  Hence the cᵢ have no limit point in the finite plane.\n\nThus (i)–(iii) are satisfied by our construction.",
      "_meta": {
        "core_steps": [
          "Pick positive radii (a_i) with ∑a_i = ∞ but ∑a_i² < ∞",
          "Set A_n = a_1 + … + a_n and center discs of radius a_n at (±A_n,0) and (0,±A_n) along two perpendicular lines",
          "Because A_{n+1} − a_{n+1} = A_n, those discs overlap consecutively and cover both entire axes",
          "Every geometric line meets at least one of the two axes, hence meets some disc",
          "Total area is π·4·∑a_i² (finite) and A_n → ∞, so the centers have no finite limit point"
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Choice of the radii sequence; only requirements are ∑a_i = ∞ and ∑a_i² < ∞",
            "original": "a_i = 1/i"
          },
          "slot2": {
            "description": "The two perpendicular lines used (their orientation and point of intersection)",
            "original": "the x- and y-axes through the origin"
          },
          "slot3": {
            "description": "Constant factor k ≥ 1 relating disc radius to successive center spacing (using radius k·a_i still works)",
            "original": "k = 1 (radius exactly a_i)"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}