{
  "index": "1952-A-7",
  "type": "COMB",
  "tag": [
    "COMB",
    "NT",
    "ANA"
  ],
  "difficulty": "",
  "question": "7. Directed lines are drawn from the center of a circle, making angles of 0 , \\( \\pm 1, \\pm 2, \\pm 3, \\ldots \\) (measured in radians from a prime direction). If these lines meet the circle in points \\( P_{0}, P_{1}, P_{-1}, P_{2}, P_{-2}, \\ldots \\), show that there is no interval on the circumference of the circle which does not contain some \\( P_{ \\pm i} \\). (You may assume that \\( \\pi \\) is irrational.)",
  "solution": "Solution. The various points \\( P_{0}, P_{ \\pm 1}, P_{ \\pm 2}, \\ldots \\) must all be distinct, else \\( \\pi \\) would be a rational number. (For \\( P_{t}=P_{l} \\) implies that \\( j-i \\) is an integral multiple of \\( 2 \\pi \\).)\nLet \\( I \\) be a given interval on the circumference, say of length \\( \\epsilon \\). Choose \\( N \\) so that \\( \\epsilon>2 \\pi / N \\). The points \\( P_{0}, P_{1}, P_{2}, \\ldots, P_{N} \\), are all different and divide the circle into \\( N \\) arcs, of which one must have length at most \\( 2 \\pi / N \\). Hence there are indices \\( m \\) and \\( m+k(k>0) \\) such that \\( P_{m} \\) and \\( P_{m+k} \\) bound an arc of length \\( \\delta<\\epsilon \\). If \\( q \\) is the greatest integer in \\( 2 \\pi / \\delta \\), then\n\\[\nP_{m}, P_{m+k}, P_{m+2 k}, \\ldots, P_{m+4 k}\n\\]\ndivide the circle into \\( q+1 \\) arcs each of length \\( \\delta \\) or less. Since \\( I \\) is longer than any of these arcs, it contains one of the points listed in (1).\n\nRemark. We needed only the points \\( P_{0}, P_{1}, P_{2}, \\ldots \\). Clearly, the same argument will apply if \\( P_{n} \\) is at \\( n \\alpha \\pi \\) radians from \\( P_{0} \\), where \\( \\alpha \\) is any irrational number.",
  "vars": [
    "P_0",
    "P_1",
    "P_-1",
    "P_2",
    "P_-2",
    "P_i",
    "P_t",
    "P_l",
    "P_m",
    "P_m+k",
    "P_m+2k",
    "P_m+4k",
    "P_n",
    "i",
    "j",
    "k",
    "m",
    "n",
    "t",
    "l",
    "q"
  ],
  "params": [
    "N",
    "I",
    "\\\\epsilon",
    "\\\\delta",
    "\\\\alpha"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "P_0": "pointzero",
        "P_1": "pointone",
        "P_-1": "pointnegone",
        "P_2": "pointtwo",
        "P_-2": "pointnegtwo",
        "P_i": "pointindex",
        "P_t": "pointtee",
        "P_l": "pointell",
        "P_m": "pointemu",
        "P_m+k": "pointemuk",
        "P_m+2k": "pointemutwok",
        "P_m+4k": "pointemufourk",
        "P_n": "pointenn",
        "i": "indexi",
        "j": "indexj",
        "k": "indexk",
        "m": "indexm",
        "n": "indexn",
        "t": "indext",
        "l": "indexl",
        "q": "indexq",
        "N": "paramnumb",
        "I": "intervalsym",
        "\\epsilon": "paramepsilon",
        "\\delta": "deltaparam",
        "\\alpha": "alphaparam"
      },
      "question": "Directed lines are drawn from the center of a circle, making angles of 0 , \\( \\pm 1, \\pm 2, \\pm 3, \\ldots \\) (measured in radians from a prime direction). If these lines meet the circle in points \\( pointzero, pointone, pointnegone, pointtwo, pointnegtwo, \\ldots \\), show that there is no interval on the circumference of the circle which does not contain some \\( pointindex \\). (You may assume that \\( \\pi \\) is irrational.)",
      "solution": "Solution. The various points \\( pointzero, pointone, pointnegone, pointtwo, pointnegtwo, \\ldots \\) must all be distinct, else \\( \\pi \\) would be a rational number. (For \\( pointtee = pointell \\) implies that \\( indexj - indexi \\) is an integral multiple of \\( 2 \\pi \\).)\nLet \\( intervalsym \\) be a given interval on the circumference, say of length \\( paramepsilon \\). Choose \\( paramnumb \\) so that \\( paramepsilon > 2 \\pi / paramnumb \\). The points \\( pointzero, pointone, pointtwo, \\ldots , P_{paramnumb} \\), are all different and divide the circle into \\( paramnumb \\) arcs, of which one must have length at most \\( 2 \\pi / paramnumb \\). Hence there are indices \\( indexm \\) and \\( indexm + indexk (indexk > 0) \\) such that \\( pointemu \\) and \\( pointemuk \\) bound an arc of length \\( deltaparam < paramepsilon \\). If \\( indexq \\) is the greatest integer in \\( 2 \\pi / deltaparam \\), then\n\\[\npointemu, pointemuk, pointemutwok, \\ldots , pointemufourk\n\\]\ndivide the circle into \\( indexq + 1 \\) arcs each of length \\( deltaparam \\) or less. Since \\( intervalsym \\) is longer than any of these arcs, it contains one of the points listed in (1).\n\nRemark. We needed only the points \\( pointzero, pointone, pointtwo, \\ldots \\). Clearly, the same argument will apply if \\( pointenn \\) is at \\( indexn\\, alphaparam \\pi \\) radians from \\( pointzero \\), where \\( alphaparam \\) is any irrational number."
    },
    "descriptive_long_confusing": {
      "map": {
        "P_0": "marigold",
        "P_1": "lighthouse",
        "P_-1": "pinecones",
        "P_2": "rattlesnake",
        "P_-2": "cinnamon",
        "P_i": "dragonfly",
        "P_t": "toothbrush",
        "P_l": "blackboard",
        "P_m": "sunflower",
        "P_m+k": "shoelaces",
        "P_m+2k": "hairbrush",
        "P_m+4k": "watermelon",
        "P_n": "screwdriver",
        "i": "pendulum",
        "j": "snowflake",
        "k": "harmonica",
        "m": "skateboard",
        "n": "buttercup",
        "t": "enchilada",
        "l": "armadillo",
        "q": "courtyard",
        "N": "paperclip",
        "I": "chandelier",
        "\\epsilon": "lemonade",
        "\\delta": "paintbrush",
        "\\alpha": "horseshoe"
      },
      "question": "7. Directed lines are drawn from the center of a circle, making angles of 0 , \\( \\pm 1, \\pm 2, \\pm 3, \\ldots \\) (measured in radians from a prime direction). If these lines meet the circle in points \\( marigold, lighthouse, pinecones, rattlesnake, cinnamon, \\ldots \\), show that there is no interval on the circumference of the circle which does not contain some \\( dragonfly \\). (You may assume that \\pi is irrational.)",
      "solution": "Solution. The various points \\( marigold, lighthouse, pinecones, rattlesnake, cinnamon, \\ldots \\) must all be distinct, else \\( \\pi \\) would be a rational number. (For \\( dragonfly=blackboard \\) implies that \\( snowflake-pendulum \\) is an integral multiple of \\( 2 \\pi \\).)\nLet \\( chandelier \\) be a given interval on the circumference, say of length \\( lemonade \\). Choose \\( paperclip \\) so that \\( lemonade>2 \\pi / paperclip \\). The points \\( marigold, lighthouse, rattlesnake, \\ldots , P_{paperclip} \\), are all different and divide the circle into \\( paperclip \\) arcs, of which one must have length at most \\( 2 \\pi / paperclip \\). Hence there are indices \\( skateboard \\) and \\( skateboard+harmonica(harmonica>0) \\) such that \\( sunflower \\) and \\( shoelaces \\) bound an arc of length \\( paintbrush<lemonade \\). If \\( courtyard \\) is the greatest integer in \\( 2 \\pi / paintbrush \\), then\n\\[\nsunflower, shoelaces, hairbrush, \\ldots , watermelon\n\\]\ndivide the circle into \\( courtyard+1 \\) arcs each of length \\( paintbrush \\) or less. Since \\( chandelier \\) is longer than any of these arcs, it contains one of the points listed in (1).\n\nRemark. We needed only the points \\( marigold, lighthouse, rattlesnake, \\ldots \\). Clearly, the same argument will apply if screwdriver is at \\( buttercup horseshoe \\pi \\) radians from \\( marigold \\), where \\( horseshoe \\) is any irrational number."
    },
    "descriptive_long_misleading": {
      "map": {
        "P_0": "extendedarea",
        "P_1": "massivefield",
        "P_-1": "limitlesszone",
        "P_2": "sprawlingland",
        "P_-2": "boundlessplot",
        "P_i": "infiniteplace",
        "P_t": "vastterritory",
        "P_l": "giantregion",
        "P_m": "broadsector",
        "P_m+k": "enlargedsector",
        "P_m+2k": "amplifiedsector",
        "P_m+4k": "intensifiedsector",
        "P_n": "colossaltract",
        "i": "finishline",
        "j": "terminalnode",
        "k": "haltingmark",
        "m": "concludingid",
        "n": "ceasingtag",
        "t": "stoptoken",
        "l": "limitflag",
        "q": "irrationalno",
        "N": "minuscule",
        "I": "unboundedspace",
        "\\epsilon": "grandomega",
        "\\delta": "vastlambda",
        "\\alpha": "terminusa"
      },
      "question": "Directed lines are drawn from the center of a circle, making angles of 0 , \\( \\pm 1, \\pm 2, \\pm 3, \\ldots \\) (measured in radians from a prime direction). If these lines meet the circle in points \\( extendedarea, massivefield, limitlesszone, sprawlingland, boundlessplot, \\ldots \\), show that there is no interval on the circumference of the circle which does not contain some \\( infiniteplace_{ \\pm finishline} \\). (You may assume that \\( \\pi \\) is irrational.)",
      "solution": "Solution. The various points \\( extendedarea, massivefield, limitlesszone, sprawlingland, boundlessplot, \\ldots \\) must all be distinct, else \\( \\pi \\) would be a rational number. (For \\( vastterritory=giantregion \\) implies that \\( terminalnode-finishline \\) is an integral multiple of \\( 2 \\pi \\).)\nLet unboundedspace be a given interval on the circumference, say of length grandomega. Choose minuscule so that grandomega>2 \\pi / minuscule. The points \\( extendedarea, massivefield, sprawlingland, \\ldots, P_{minuscule} \\) are all different and divide the circle into minuscule arcs, of which one must have length at most \\( 2 \\pi / minuscule \\). Hence there are indices concludingid and concludingid+haltingmark(haltingmark>0) such that broadsector and enlargedsector bound an arc of length vastlambda<grandomega. If irrationalno is the greatest integer in \\( 2 \\pi / vastlambda \\), then\n\\[\nbroadsector, enlargedsector, amplifiedsector, \\ldots, intensifiedsector\n\\]\ndivide the circle into irrationalno+1 arcs each of length vastlambda or less. Since unboundedspace is longer than any of these arcs, it contains one of the points listed in (1).\n\nRemark. We needed only the points extendedarea, massivefield, sprawlingland, \\ldots. Clearly, the same argument will apply if colossaltract is at ceasingtag terminusa \\pi radians from extendedarea, where terminusa is any irrational number."
    },
    "garbled_string": {
      "map": {
        "P_0": "qzxwvtnp",
        "P_1": "hjgrksla",
        "P_-1": "vbnmlqwe",
        "P_2": "tyredfgh",
        "P_-2": "poklazxc",
        "P_i": "sdfghjkl",
        "P_t": "xcvbnmas",
        "P_l": "qwertyui",
        "P_m": "asdfzxcv",
        "P_m+k": "ghjklqwe",
        "P_m+2k": "zxcvtyui",
        "P_m+4k": "bnmpoiuy",
        "P_n": "lkjhgfds",
        "i": "mnbvcxza",
        "j": "poiuytre",
        "k": "lkjhgfdq",
        "m": "plmoknij",
        "n": "ujmnhygt",
        "t": "rfvcdexs",
        "l": "tgbnhyuj",
        "q": "edcvfrbg",
        "N": "yhnujmik",
        "I": "ikmjnhyb",
        "\\epsilon": "olpzaqws",
        "\\delta": "xswecdfr",
        "\\alpha": "zaqwsxed"
      },
      "question": "7. Directed lines are drawn from the center of a circle, making angles of 0 , \\( \\pm 1, \\pm 2, \\pm 3, \\ldots \\) (measured in radians from a prime direction). If these lines meet the circle in points \\( qzxwvtnp, hjgrksla, vbnmlqwe, tyredfgh, poklazxc, \\ldots \\), show that there is no interval on the circumference of the circle which does not contain some \\( P_{ \\pm mnbvcxza} \\). (You may assume that \\( \\pi \\) is irrational.)",
      "solution": "Solution. The various points \\( qzxwvtnp, P_{ \\pm 1}, P_{ \\pm 2}, \\ldots \\) must all be distinct, else \\( \\pi \\) would be a rational number. (For \\( xcvbnmas=qwertyui \\) implies that \\( poiuytre-mnbvcxza \\) is an integral multiple of \\( 2 \\pi \\).)\nLet \\( ikmjnhyb \\) be a given interval on the circumference, say of length \\( olpzaqws \\). Choose \\( yhnujmik \\) so that \\( olpzaqws>2 \\pi / yhnujmik \\). The points \\( qzxwvtnp, hjgrksla, tyredfgh, \\ldots, P_{yhnujmik} \\), are all different and divide the circle into \\( yhnujmik \\) arcs, of which one must have length at most \\( 2 \\pi / yhnujmik \\). Hence there are indices \\( plmoknij \\) and \\( plmoknij+lkjhgfdq(lkjhgfdq>0) \\) such that \\( asdfzxcv \\) and \\( ghjklqwe \\) bound an arc of length \\( xswecdfr<olpzaqws \\). If \\( edcvfrbg \\) is the greatest integer in \\( 2 \\pi / xswecdfr \\), then\n\\[\nasdfzxcv, ghjklqwe, zxcvtyui, \\ldots, bnmpoiuy\n\\]\ndivide the circle into \\( edcvfrbg+1 \\) arcs each of length \\( xswecdfr \\) or less. Since \\( ikmjnhyb \\) is longer than any of these arcs, it contains one of the points listed in (1).\n\nRemark. We needed only the points \\( qzxwvtnp, hjgrksla, tyredfgh, \\ldots \\). Clearly, the same argument will apply if \\( lkjhgfds \\) is at \\( ujmnhygt zaqwsxed \\pi \\) radians from \\( qzxwvtnp \\), where \\( zaqwsxed \\) is any irrational number."
    },
    "kernel_variant": {
      "question": "Let $\\alpha=\\sqrt5$.  For every integer $n\\ge0$ draw the ray from the centre $O$ of a circle that makes an angle $n\\alpha$ (measured in radians) with a fixed prime direction.  Let this ray meet the circle again in the point $Q_n$.  Prove that for every arc $I$ of the circle having positive length $\\varepsilon$, the arc $I$ contains at least one point of the set $\\{Q_n\\}_{n\\ge0}$. (You may assume that $\\alpha/\\pi$ is irrational.)",
      "solution": "Fix any open arc I of length ε>0 on the circle and show I contains some Qₙ.  We are given α=√5 and that α/π is irrational.\n\n1.  (All Qₙ are distinct.)  If Q_i=Q_j with 0≤i<j then the directed angle from Q_i to Q_j is (j−i)α, which must equal an integer multiple of 2π.  Hence (j−i)α=2πm and α/π=m/(j−i) would be rational, contradiction.\n\n2.  (Find a small gap δ<ε.)  Choose an integer M>2π/ε.  Then the M+1 points Q₀,Q₁,…,Q_M split the circle into M+1 arcs of total length 2π, so one arc has length δ≤2π/(M+1)<ε.  Label its endpoints Q_m and Q_{m+k}, with 1≤k≤M.\n\n3.  (Step‐k rotations all give gaps ≤δ.)  For each integer t≥0, Q_{m+tk} is obtained from Q_m by rotating through t·kα, so the minor arc from Q_{m+tk} to Q_{m+(t+1)k} has length ≤δ.\n\n4.  (No arc of length ε can avoid these points.)  Set r=⌈2π/δ⌉.  Consider the r+1 points\n      Q_m, Q_{m+k}, Q_{m+2k}, …, Q_{m+r k}.\nConsecutive ones are joined by “step arcs” of length ≤δ, and the last point Q_{m+r k} returns to Q_m by a minor arc of length (sum of r step‐arcs −2π), which lies in [0,δ).  Thus these r+1 points split the circle into exactly r+1 minor arcs, each of length ≤δ<ε.\n\nAny open arc of length ε>δ must contain at least one of these r+1 points.  In particular, our given arc I contains one of Q_m,Q_{m+k},…,Q_{m+r k}.  Thus I meets the set {Qₙ}.\n\nSince I was an arbitrary arc of positive length, the points Qₙ (n≥0) are dense on the circle.  ■",
      "_meta": {
        "core_steps": [
          "Use π’s irrationality to show all points P_i are distinct (no angular coincidences).",
          "Apply the pigeon-hole principle to the first N+1 points (with N>2π/ε) to find two consecutive points bounding an arc δ<ε.",
          "Let k be the index-difference of that pair; the arithmetic progression m, m+k, m+2k, … gives points spaced by ≤δ around the circle.",
          "Because q:=⌊2π/δ⌋ ensures q+1 such arcs cover the full circumference, every length-ε interval contains at least one P_i."
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Angular step between successive directions; needs only to be an angle whose ratio with π is irrational.",
            "original": "1 radian"
          },
          "slot2": {
            "description": "Symmetric inclusion of negative indices; argument actually uses only non-negative indices.",
            "original": "Both P_{+i} and P_{−i}"
          },
          "slot3": {
            "description": "Quantifier for choosing many initial points; any N with 2π/N < ε (equivalently N > 2π/ε) works.",
            "original": "N chosen so that ε > 2π / N"
          },
          "slot4": {
            "description": "Exact truncation of the progression; ‘q = ⌊2π/δ⌋’ can be replaced by any q with (q+1)δ ≥ 2π.",
            "original": "q is the greatest integer ≤ 2π / δ"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}