{
  "index": "1954-A-2",
  "type": "GEO",
  "tag": [
    "GEO",
    "COMB"
  ],
  "difficulty": "",
  "question": "2. Consider any five points \\( P_{1}, P_{2}, P_{3}, P_{4}, P_{5} \\) in the interior of a square \\( S \\) of side-length 1. Denote by \\( d_{i j} \\) the distance between the points \\( P_{i} \\) and \\( P_{i} \\). Prove that at least one of the distances \\( d_{i j} \\) is less than \\( \\sqrt{2} / 2 \\). Can \\( \\sqrt{2} / 2 \\) be replaced by a smaller number in this statement?",
  "solution": "Solution. Let the square be divided into four small squares, as indicated in the sketch, each of side-length \\( \\frac{1}{2} \\). Considering each of the smaller squares as closed sets, two of the five points must fall in the same small square, and these two points are at a distance less than \\( \\frac{1}{2} \\sqrt{2} \\) from each other. For a formal proof of this, note that with axes parallel to the sides of the square, both coordinate differences are less than \\( \\frac{1}{2} \\).\n\nGiven \\( \\epsilon>0 \\), choose the center and four points on the diagonals, one within \\( \\epsilon \\) of each corner. This gives five points in the interior of the square such that the minimum distance is more than \\( \\frac{1}{2} \\sqrt{2}-\\epsilon \\). Hence no number smaller than \\( \\frac{1}{2} \\sqrt{2} \\) will do.",
  "vars": [
    "P_1",
    "P_2",
    "P_3",
    "P_4",
    "P_5",
    "d_ij"
  ],
  "params": [
    "S",
    "\\\\epsilon"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "P_1": "pointone",
        "P_2": "pointtwo",
        "P_3": "pointthree",
        "P_4": "pointfour",
        "P_5": "pointfive",
        "d_ij": "interdist",
        "S": "mainsquare",
        "\\epsilon": "epsparam"
      },
      "question": "2. Consider any five points \\( pointone, pointtwo, pointthree, pointfour, pointfive \\) in the interior of a square \\( mainsquare \\) of side-length 1. Denote by \\( interdist \\) the distance between the points \\( P_{i} \\) and \\( P_{i} \\). Prove that at least one of the distances \\( interdist \\) is less than \\( \\sqrt{2} / 2 \\). Can \\( \\sqrt{2} / 2 \\) be replaced by a smaller number in this statement?",
      "solution": "Solution. Let the square be divided into four small squares, as indicated in the sketch, each of side-length \\( \\frac{1}{2} \\). Considering each of the smaller squares as closed sets, two of the five points must fall in the same small square, and these two points are at a distance less than \\( \\frac{1}{2} \\sqrt{2} \\) from each other. For a formal proof of this, note that with axes parallel to the sides of the square, both coordinate differences are less than \\( \\frac{1}{2} \\).\n\nGiven \\( epsparam>0 \\), choose the center and four points on the diagonals, one within \\( epsparam \\) of each corner. This gives five points in the interior of the square such that the minimum distance is more than \\( \\frac{1}{2} \\sqrt{2}-epsparam \\). Hence no number smaller than \\( \\frac{1}{2} \\sqrt{2} \\) will do."
    },
    "descriptive_long_confusing": {
      "map": {
        "P_1": "blueberry",
        "P_2": "compassrose",
        "P_3": "daybreaker",
        "P_4": "firecandle",
        "P_5": "moonshadow",
        "d_ij": "thunderstone",
        "S": "orchardlane",
        "\\epsilon": "whisperwind"
      },
      "question": "2. Consider any five points \\( blueberry, compassrose, daybreaker, firecandle, moonshadow \\) in the interior of a square \\( orchardlane \\) of side-length 1. Denote by \\( thunderstone \\) the distance between the points \\( P_{i} \\) and \\( P_{i} \\). Prove that at least one of the distances \\( thunderstone \\) is less than \\( \\sqrt{2} / 2 \\). Can \\( \\sqrt{2} / 2 \\) be replaced by a smaller number in this statement?",
      "solution": "Solution. Let the square be divided into four small squares, as indicated in the sketch, each of side-length \\( \\frac{1}{2} \\). Considering each of the smaller squares as closed sets, two of the five points must fall in the same small square, and these two points are at a distance less than \\( \\frac{1}{2} \\sqrt{2} \\) from each other. For a formal proof of this, note that with axes parallel to the sides of the square, both coordinate differences are less than \\( \\frac{1}{2} \\).\n\nGiven \\( whisperwind>0 \\), choose the center and four points on the diagonals, one within \\( whisperwind \\) of each corner. This gives five points in the interior of the square such that the minimum distance is more than \\( \\frac{1}{2} \\sqrt{2}-whisperwind \\). Hence no number smaller than \\( \\frac{1}{2} \\sqrt{2} \\) will do."
    },
    "descriptive_long_misleading": {
      "map": {
        "P_{1}": "voidpointa",
        "P_{2}": "voidpointb",
        "P_{3}": "voidpointc",
        "P_{4}": "voidpointd",
        "P_{5}": "voidpointe",
        "d_{i j}": "nearness_{i j}",
        "S": "roundshape",
        "\\epsilon": "megadelta"
      },
      "question": "2. Consider any five points \\( voidpointa, voidpointb, voidpointc, voidpointd, voidpointe \\) in the interior of a square \\( roundshape \\) of side-length 1. Denote by \\( nearness_{i j} \\) the distance between the points \\( P_{i} \\) and \\( P_{i} \\). Prove that at least one of the distances \\( nearness_{i j} \\) is less than \\( \\sqrt{2} / 2 \\). Can \\( \\sqrt{2} / 2 \\) be replaced by a smaller number in this statement?",
      "solution": "Solution. Let the square be divided into four small squares, as indicated in the sketch, each of side-length \\( \\frac{1}{2} \\). Considering each of the smaller squares as closed sets, two of the five points must fall in the same small square, and these two points are at a distance less than \\( \\frac{1}{2} \\sqrt{2} \\) from each other. For a formal proof of this, note that with axes parallel to the sides of the square, both coordinate differences are less than \\( \\frac{1}{2} \\).\n\nGiven \\( megadelta>0 \\), choose the center and four points on the diagonals, one within \\( megadelta \\) of each corner. This gives five points in the interior of the square such that the minimum distance is more than \\( \\frac{1}{2} \\sqrt{2}-megadelta \\). Hence no number smaller than \\( \\frac{1}{2} \\sqrt{2} \\) will do."
    },
    "garbled_string": {
      "map": {
        "P_1": "qzxwvtnp",
        "P_2": "hjgrksla",
        "P_3": "mfldqzke",
        "P_4": "wprjctgu",
        "P_5": "ybndsxam",
        "d_ij": "slvrcqwn",
        "S": "fthjgmln",
        "\\epsilon": "ktbsfqwr"
      },
      "question": "2. Consider any five points \\( qzxwvtnp, hjgrksla, mfldqzke, wprjctgu, ybndsxam \\) in the interior of a square \\( fthjgmln \\) of side-length 1. Denote by \\( slvrcqwn \\) the distance between the points \\( P_{i} \\) and \\( P_{i} \\). Prove that at least one of the distances \\( slvrcqwn \\) is less than \\( \\sqrt{2} / 2 \\). Can \\( \\sqrt{2} / 2 \\) be replaced by a smaller number in this statement?",
      "solution": "Solution. Let the square be divided into four small squares, as indicated in the sketch, each of side-length \\( \\frac{1}{2} \\). Considering each of the smaller squares as closed sets, two of the five points must fall in the same small square, and these two points are at a distance less than \\( \\frac{1}{2} \\sqrt{2} \\) from each other. For a formal proof of this, note that with axes parallel to the sides of the square, both coordinate differences are less than \\( \\frac{1}{2} \\).\n\nGiven \\( ktbsfqwr>0 \\), choose the center and four points on the diagonals, one within \\( ktbsfqwr \\) of each corner. This gives five points in the interior of the square such that the minimum distance is more than \\( \\frac{1}{2} \\sqrt{2}-ktbsfqwr \\). Hence no number smaller than \\( \\frac{1}{2} \\sqrt{2} \\) will do."
    },
    "kernel_variant": {
      "question": "Let $D=\\{(x,y)\\in\\mathbb R^{2}\bigm|x^{2}+y^{2}<1\\}$ be the open disk of radius $1$.  For any seven points $P_{1},\\dots ,P_{7}$ chosen in $D$ let $d_{ij}=|P_{i}P_{j}|$ denote the distance between $P_{i}$ and $P_{j}$.\\n\\n(a)  Prove that at least one of the $\\,d_{ij}\\,$ is strictly smaller than $1$.\\n\\n(b)  Show that the number $1$ cannot be replaced by any smaller positive constant; that is, for every $\\varepsilon>0$ there exist seven points in $D$ whose mutual distances are all greater than $1-\\varepsilon$.",
      "solution": "a)  Divide the disk by three diameters that meet at angles of 60°; this produces six congruent sectors, each with central angle π/3.  These sectors will serve as the “pigeonholes.”\n\nThe chord connecting the two extreme points of a sector subtends an angle of π/3 at the centre, so its length is\n\n2 sin(π/6)=1.\n\nBecause the seven given points lie in the open disk, any two points that fall in the same sector are actually closer than that chord; hence their distance is strictly less than 1.\n\nWith six sectors and seven points, the pigeonhole principle guarantees that two of the points must lie in the same sector, completing the proof of part (a).\n\nb)  Fix ε>0 and set r=1–ε/2 (>0).  Place a point O at the origin (the centre of the disk) and six further points Q_k (k=0,1,…,5) at polar coordinates (r, kπ/3).  All seven points lie in D because r<1.\n\nDistances involving the centre:  |OQ_k|=r=1–ε/2>1–ε.\n\nDistances between two peripheral points: if Q_k and Q_{k+m} are m sectors apart, the angle between them is mπ/3, so\n\n|Q_kQ_{k+m}|=2r sin(mπ/6)≥2r sin(π/6)=r>1–ε.\n\nThus every pair of the seven points is more than 1–ε apart.  Since ε was arbitrary, no number smaller than 1 can replace the constant in part (a), and the bound is sharp.",
      "_meta": {
        "core_steps": [
          "Bisect the unit square both horizontally and vertically, creating 4 congruent subsquares.",
          "Apply the pigeonhole principle: 5 interior points, 4 subsquares → two points fall in the same subsquare.",
          "The farthest two points that can lie in one subsquare are at most its diagonal, √2·(1/2) = √2⁄2, so that pair’s distance is < √2⁄2.",
          "Exhibit 5 points (center + four near the corners) whose mutual distances approach √2⁄2, proving the constant is sharp."
        ],
        "mutable_slots": {
          "slot_points": {
            "description": "Total number of interior points; must exceed the number of regions to force a shared region by pigeonhole.",
            "original": 5
          },
          "slot_regions": {
            "description": "How the big square is partitioned (here a 2×2 grid of congruent squares); only the count (4) and bounded diameter matter.",
            "original": "4 congruent subsquares formed by drawing both midlines"
          },
          "slot_sidelength": {
            "description": "Side length of the original square; all distance bounds scale linearly with it.",
            "original": 1
          },
          "slot_bound": {
            "description": "Numerical upper bound on the distance, equal to the diameter of one region; changes with the two previous slots.",
            "original": "√2⁄2"
          },
          "slot_epsilon": {
            "description": "Positive margin used in the sharpness construction; any small ε>0 suffices.",
            "original": "ε > 0 (arbitrary small)"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}