{
  "index": "1982-A-1",
  "type": "GEO",
  "tag": [
    "GEO",
    "ALG"
  ],
  "difficulty": "",
  "question": "Problem A-1\nLet \\( V \\) be the region in the cartesian plane consisting of all points \\( (x, y) \\) satisfying the simultaneous conditions\n\\[\n|x| \\leqslant y \\leqslant|x|+3 \\text { and } y \\leqslant 4\n\\]\n\nFind the centroid \\( (\\bar{x}, \\bar{y}) \\) of \\( V \\).",
  "solution": "A-1.\nLet \\( T \\) consist of the points inside or on the triangle with vertices at \\( (0,3),(-1,4),(1,4) \\) and let \\( U \\) be the set of points inside or on the triangle with vertices at \\( (0,0),(-4,4),(4,4) \\). Then \\( T \\) and \\( V \\) overlap only on boundary points and their union is \\( U \\). The centroids of \\( T \\) and \\( U \\) are \\( (0,11 / 3) \\) and \\( (0,8 / 3) \\), respectively. The areas of \\( T, V \\), and \\( U \\) are 1,15 , and 16 , respectively. Using weighted averages with the areas as weights, one has\n\\[\n1 \\cdot 0+15 \\bar{x}=16 \\cdot 0, \\quad 1 \\cdot \\frac{11}{3}+15 \\bar{y}=16 \\cdot \\frac{8}{3} .\n\\]\n\nIt follows that \\( \\bar{x}=0, \\bar{y}=13 / 5 \\).",
  "vars": [
    "x",
    "y",
    "\\\\bar{x}",
    "\\\\bar{y}"
  ],
  "params": [
    "V",
    "T",
    "U"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "coordinatex",
        "y": "coordinatey",
        "\\bar{x}": "centroidx",
        "\\bar{y}": "centroidy",
        "V": "regionv",
        "T": "trianglet",
        "U": "triangleu"
      },
      "question": "Problem A-1\nLet \\( regionv \\) be the region in the cartesian plane consisting of all points \\( (coordinatex, coordinatey) \\) satisfying the simultaneous conditions\n\\[\n|coordinatex| \\leqslant coordinatey \\leqslant|coordinatex|+3 \\text { and } coordinatey \\leqslant 4\n\\]\n\nFind the centroid \\( (centroidx, centroidy) \\) of \\( regionv \\).",
      "solution": "A-1.\nLet \\( trianglet \\) consist of the points inside or on the triangle with vertices at \\( (0,3),(-1,4),(1,4) \\) and let \\( triangleu \\) be the set of points inside or on the triangle with vertices at \\( (0,0),(-4,4),(4,4) \\). Then \\( trianglet \\) and \\( regionv \\) overlap only on boundary points and their union is \\( triangleu \\). The centroids of \\( trianglet \\) and \\( triangleu \\) are \\( (0,11 / 3) \\) and \\( (0,8 / 3) \\), respectively. The areas of \\( trianglet, regionv \\), and \\( triangleu \\) are 1,15 , and 16 , respectively. Using weighted averages with the areas as weights, one has\n\\[\n1 \\cdot 0+15 centroidx=16 \\cdot 0, \\quad 1 \\cdot \\frac{11}{3}+15 centroidy=16 \\cdot \\frac{8}{3} .\n\\]\n\nIt follows that \\( centroidx=0, centroidy=13 / 5 \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "mapleleaf",
        "y": "monarchic",
        "\\bar{x}": "windcastle",
        "\\bar{y}": "stormcloud",
        "V": "harboring",
        "T": "lighthouse",
        "U": "snowdrifts"
      },
      "question": "Problem A-1\nLet \\( harboring \\) be the region in the cartesian plane consisting of all points \\( (mapleleaf, monarchic) \\) satisfying the simultaneous conditions\n\\[\n|mapleleaf| \\leqslant monarchic \\leqslant|mapleleaf|+3 \\text { and } monarchic \\leqslant 4\n\\]\n\nFind the centroid \\( (windcastle, stormcloud) \\) of \\( harboring \\).",
      "solution": "A-1.\nLet \\( lighthouse \\) consist of the points inside or on the triangle with vertices at \\( (0,3),(-1,4),(1,4) \\) and let \\( snowdrifts \\) be the set of points inside or on the triangle with vertices at \\( (0,0),(-4,4),(4,4) \\). Then \\( lighthouse \\) and \\( harboring \\) overlap only on boundary points and their union is \\( snowdrifts \\). The centroids of \\( lighthouse \\) and \\( snowdrifts \\) are \\( (0,11 / 3) \\) and \\( (0,8 / 3) \\), respectively. The areas of \\( lighthouse, harboring \\), and \\( snowdrifts \\) are 1,15 , and 16 , respectively. Using weighted averages with the areas as weights, one has\n\\[\n1 \\cdot 0+15 windcastle=16 \\cdot 0, \\quad 1 \\cdot \\frac{11}{3}+15 stormcloud=16 \\cdot \\frac{8}{3} .\n\\]\n\nIt follows that \\( windcastle=0, stormcloud=13 / 5 \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "constantval",
        "y": "horizontal",
        "\\bar{x}": "boundaryval",
        "\\bar{y}": "boundaryvertical",
        "V": "emptiness",
        "T": "linearity",
        "U": "singularity"
      },
      "question": "Problem:\n<<<\nProblem A-1\nLet \\( emptiness \\) be the region in the cartesian plane consisting of all points \\( (constantval, horizontal) \\) satisfying the simultaneous conditions\n\\[\n|constantval| \\leqslant horizontal \\leqslant|constantval|+3 \\text { and } horizontal \\leqslant 4\n\\]\n\nFind the centroid \\( (boundaryval, boundaryvertical) \\) of \\( emptiness \\).\n>>>\n",
      "solution": "Solution:\n<<<\nA-1.\nLet \\( linearity \\) consist of the points inside or on the triangle with vertices at \\( (0,3),(-1,4),(1,4) \\) and let \\( singularity \\) be the set of points inside or on the triangle with vertices at \\( (0,0),(-4,4),(4,4) \\). Then \\( linearity \\) and \\( emptiness \\) overlap only on boundary points and their union is \\( singularity \\). The centroids of \\( linearity \\) and \\( singularity \\) are \\( (0,11 / 3) \\) and \\( (0,8 / 3) \\), respectively. The areas of \\( linearity, emptiness \\), and \\( singularity \\) are 1,15 , and 16 , respectively. Using weighted averages with the areas as weights, one has\n\\[\n1 \\cdot 0+15 boundaryval=16 \\cdot 0, \\quad 1 \\cdot \\frac{11}{3}+15 boundaryvertical=16 \\cdot \\frac{8}{3} .\n\\]\n\nIt follows that \\( boundaryval=0, boundaryvertical=13 / 5 \\).\n>>>\n"
    },
    "garbled_string": {
      "map": {
        "x": "qzxwvtnp",
        "y": "hjgrksla",
        "\\bar{x}": "\\bar{mbczlqsj}",
        "\\bar{y}": "\\bar{nxfjegop}",
        "V": "pqlkdrtm",
        "T": "akmsidpq",
        "U": "jvrcehzw"
      },
      "question": "Let \\( pqlkdrtm \\) be the region in the cartesian plane consisting of all points \\( (qzxwvtnp, hjgrksla) \\) satisfying the simultaneous conditions\n\\[\n|qzxwvtnp| \\leqslant hjgrksla \\leqslant|qzxwvtnp|+3 \\text { and } hjgrksla \\leqslant 4\n\\]\n\nFind the centroid \\( (\\bar{mbczlqsj}, \\bar{nxfjegop}) \\) of \\( pqlkdrtm \\).",
      "solution": "A-1.\nLet \\( akmsidpq \\) consist of the points inside or on the triangle with vertices at \\( (0,3),(-1,4),(1,4) \\) and let \\( jvrcehzw \\) be the set of points inside or on the triangle with vertices at \\( (0,0),(-4,4),(4,4) \\). Then \\( akmsidpq \\) and \\( pqlkdrtm \\) overlap only on boundary points and their union is \\( jvrcehzw \\). The centroids of \\( akmsidpq \\) and \\( jvrcehzw \\) are \\( (0,11 / 3) \\) and \\( (0,8 / 3) \\), respectively. The areas of \\( akmsidpq, pqlkdrtm \\), and \\( jvrcehzw \\) are 1,15 , and 16 , respectively. Using weighted averages with the areas as weights, one has\n\\[\n1 \\cdot 0+15 \\bar{mbczlqsj}=16 \\cdot 0, \\quad 1 \\cdot \\frac{11}{3}+15 \\bar{nxfjegop}=16 \\cdot \\frac{8}{3} .\n\\]\n\nIt follows that \\( \\bar{mbczlqsj}=0, \\bar{nxfjegop}=13 / 5 \\)."
    },
    "kernel_variant": {
      "question": "Let\n\\[\nV=\\{(x,y)\\in\\mathbb R^{2}\\mid |x|\\le y\\le |x|+5\\text{ and }y\\le 9\\}.\n\\]\nFind the coordinates \\((\\bar x,\\bar y)\\) of the centroid of the planar region $V$.",
      "solution": "1.  Symmetry.\n   The defining inequalities are unchanged when x is replaced by −x, so V is symmetric about the y–axis.  Hence x̄=0.\n\n2.  View V as a difference of two triangles.\n   The lines y=|x| and y=9 intersect at (−9,9) and (9,9), while y=|x|+5 and y=9 meet at (−4,9) and (4,9).  Define\n     U = Δ((0,0),(−9,9),(9,9)),\n     T = Δ((0,5),(−4,9),(4,9)).\n   One checks that T and V share only boundary points and that U=T∪V.  Thus V=U∖T.\n\n3.  Areas of U and T.\n   Both triangles have horizontal bases and vertical heights:\n     [U]=½·18·9=81,   [T]=½·8·4=16.\n   Hence [V]=81−16=65.\n\n4.  Centroids of U and T.\n   The centroid of a triangle is the arithmetic mean of its vertices:\n     C_U=(0,6),   C_T=(0,23/3).\n\n5.  Area–weighted addition.\n   Since U = T⊔V (up to boundary),\n     [U]C_U = [T]C_T + [V](x̄,ȳ).\n   From the x–coordinate:  81·0 =16·0+65·x̄ ⇒ x̄=0.\n   From the y–coordinate:\n     81·6 =16·(23/3)+65·ȳ\n     ⇒486=368/3+65ȳ\n     ⇒65ȳ=1090/3\n     ⇒ȳ=218/39.\n\nTherefore the centroid of V is\n   (x̄,ȳ) = (0, 218/39).",
      "_meta": {
        "core_steps": [
          "Note symmetry about the y-axis to deduce \\bar{x}=0",
          "View V as the difference U \\ T where U and T are two triangles that share only boundary",
          "Find the areas of triangles U and T with the usual ½·base·height formula",
          "Find the centroids of U and T by averaging their three vertex coordinates",
          "Use the area-weighted (additive) centroid formula to obtain \\bar{y}"
        ],
        "mutable_slots": {
          "slot1": {
            "description": "vertical offset between the two slanted boundaries |x| ≤ y ≤ |x| + c",
            "original": "3"
          },
          "slot2": {
            "description": "height of the horizontal cap y ≤ h",
            "original": "4"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "calculation"
}