{
  "index": "1955-A-2",
  "type": "GEO",
  "tag": [
    "GEO",
    "ALG"
  ],
  "difficulty": "",
  "question": "2. \\( A_{1} A_{2} \\ldots A_{n} \\) is a regular polygon inscribed in a circle of radius \\( r \\) and center \\( O . P \\) is a point on line \\( O A_{1} \\) extended beyond \\( A_{1} \\). Show that\n\\[\n\\prod_{i=1}^{n} \\overline{P A}_{i}=\\overline{O P}^{n}-r^{n}\n\\]",
  "solution": "Solution. We may assume the polygon is in the complex plane with its center at the origin and \\( A_{1} \\) on the positive real axis. Then the other vertices are\n\\[\nr \\omega, r \\omega^{2}, \\ldots, r \\omega^{n-1}\n\\]\nwhere \\( \\omega \\) is a primitive \\( n \\)th root of unity.\nIf \\( P \\) is at the point \\( x \\), then \\( \\overline{P A_{i}}=\\left|x-r \\omega^{i-1}\\right| \\). So\n\\[\n\\begin{aligned}\n\\prod_{i=1}^{n} \\overline{P A_{i}} & =\\left|\\prod_{i=1}^{n}\\left(x-r \\omega^{i-1}\\right)\\right|=r^{n}\\left|\\prod_{i=1}^{n}\\left(\\frac{x}{r}-\\omega^{i-1}\\right)\\right|=r^{n}\\left|\\left(\\frac{x}{r}\\right)^{n}-1\\right| \\\\\n& =\\left|x^{n}-r^{n}\\right|=x^{n}-r^{n}=\\overline{O P}^{n}-r^{n} .\n\\end{aligned}\n\\]\n\nAt the third step we used the factorization\n\\[\nX^{n}-1=\\prod_{i=1}^{n}\\left(X-\\omega^{i-1}\\right)\n\\]\nwhich is valid because \\( 1, \\omega, \\omega^{2}, \\ldots, \\omega^{n-1} \\) are the zeros of \\( X^{n}-1 \\).",
  "vars": [
    "P",
    "x",
    "X",
    "i"
  ],
  "params": [
    "n",
    "r",
    "O",
    "A_1",
    "A_2",
    "A_n",
    "A_i",
    "\\\\omega"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "P": "pointpe",
        "x": "complexx",
        "X": "symbolex",
        "i": "indexvar",
        "n": "sidesnum",
        "r": "radiusln",
        "O": "centerpt",
        "A_1": "vertexon",
        "A_2": "vertextw",
        "A_n": "vertexnn",
        "A_i": "vertexii",
        "\\\\omega": "rootunit"
      },
      "question": "2. \\( vertexon vertextw \\ldots vertexnn \\) is a regular polygon inscribed in a circle of radius \\( radiusln \\) and center \\( centerpt . pointpe \\) is a point on line \\( centerpt vertexon \\) extended beyond \\( vertexon \\). Show that\n\\[\n\\prod_{indexvar=1}^{sidesnum} \\overline{pointpe vertexii}=\\overline{centerpt pointpe}^{sidesnum}-radiusln^{sidesnum}\n\\]",
      "solution": "Solution. We may assume the polygon is in the complex plane with its center at the origin and \\( vertexon \\) on the positive real axis. Then the other vertices are\n\\[\nradiusln\\ rootunit, radiusln\\ rootunit^{2}, \\ldots, radiusln\\ rootunit^{sidesnum-1}\n\\]\nwhere \\( rootunit \\) is a primitive \\( sidesnum \\)th root of unity.\nIf \\( pointpe \\) is at the point \\( complexx \\), then \\( \\overline{pointpe\\ vertexii}=\\left|complexx-radiusln\\ rootunit^{indexvar-1}\\right| \\). So\n\\[\n\\begin{aligned}\n\\prod_{indexvar=1}^{sidesnum} \\overline{pointpe\\ vertexii} & =\\left|\\prod_{indexvar=1}^{sidesnum}\\left(complexx-radiusln\\ rootunit^{indexvar-1}\\right)\\right|=radiusln^{sidesnum}\\left|\\prod_{indexvar=1}^{sidesnum}\\left(\\frac{complexx}{radiusln}-rootunit^{indexvar-1}\\right)\\right|=radiusln^{sidesnum}\\left|\\left(\\frac{complexx}{radiusln}\\right)^{sidesnum}-1\\right| \\\\\n& =\\left|complexx^{sidesnum}-radiusln^{sidesnum}\\right|=complexx^{sidesnum}-radiusln^{sidesnum}=\\overline{centerpt\\ pointpe}^{sidesnum}-radiusln^{sidesnum} .\n\\end{aligned}\n\\]\n\nAt the third step we used the factorization\n\\[\nsymbolex^{sidesnum}-1=\\prod_{indexvar=1}^{sidesnum}\\left(symbolex-rootunit^{indexvar-1}\\right)\n\\]\nwhich is valid because \\( 1, rootunit, rootunit^{2}, \\ldots, rootunit^{sidesnum-1} \\) are the zeros of \\( symbolex^{sidesnum}-1 \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "P": "butterfly",
        "x": "rainwater",
        "X": "lighthouse",
        "i": "seashell",
        "n": "pinecones",
        "r": "stargazer",
        "O": "shipwreck",
        "A_1": "jellyfish",
        "A_2": "snowflake",
        "A_n": "gemstone",
        "A_i": "driftwood",
        "\\\\omega": "raincloud"
      },
      "question": "2. \\( jellyfish snowflake \\ldots gemstone \\) is a regular polygon inscribed in a circle of radius \\( stargazer \\) and center \\( shipwreck . butterfly \\) is a point on line \\( shipwreck jellyfish \\) extended beyond \\( jellyfish \\). Show that\n\\[\n\\prod_{seashell=1}^{pinecones} \\overline{butterfly driftwood}=\\overline{shipwreck butterfly}^{pinecones}-stargazer^{pinecones}\n\\]",
      "solution": "Solution. We may assume the polygon is in the complex plane with its center at the origin and \\( jellyfish \\) on the positive real axis. Then the other vertices are\n\\[\nstargazer raincloud, stargazer raincloud^{2}, \\ldots, stargazer raincloud^{pinecones-1}\n\\]\nwhere \\( raincloud \\) is a primitive \\( pinecones \\)th root of unity.\nIf \\( butterfly \\) is at the point \\( rainwater \\), then \\( \\overline{butterfly driftwood}=\\left|rainwater-stargazer raincloud^{seashell-1}\\right| \\). So\n\\[\n\\begin{aligned}\n\\prod_{seashell=1}^{pinecones} \\overline{butterfly driftwood} & =\\left|\\prod_{seashell=1}^{pinecones}\\left(rainwater-stargazer raincloud^{seashell-1}\\right)\\right|=stargazer^{pinecones}\\left|\\prod_{seashell=1}^{pinecones}\\left(\\frac{rainwater}{stargazer}-raincloud^{seashell-1}\\right)\\right|=stargazer^{pinecones}\\left|\\left(\\frac{rainwater}{stargazer}\\right)^{pinecones}-1\\right| \\\\\n& =\\left|rainwater^{pinecones}-stargazer^{pinecones}\\right|=rainwater^{pinecones}-stargazer^{pinecones}=\\overline{shipwreck butterfly}^{pinecones}-stargazer^{pinecones} .\n\\end{aligned}\n\\]\n\nAt the third step we used the factorization\n\\[\nlighthouse^{pinecones}-1=\\prod_{seashell=1}^{pinecones}\\left(lighthouse-raincloud^{seashell-1}\\right)\n\\]\nwhich is valid because \\( 1, raincloud, raincloud^{2}, \\ldots, raincloud^{pinecones-1} \\) are the zeros of \\( lighthouse^{pinecones}-1 \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "P": "insidepoint",
        "x": "imaginaryvalue",
        "X": "constantvalue",
        "i": "aggregate",
        "n": "singularity",
        "r": "diameterlength",
        "O": "edgepoint",
        "A_1": "lastvertex",
        "A_2": "middlevertex",
        "A_n": "absentvertex",
        "A_i": "sidepoint",
        "\\omega": "realnumber"
      },
      "question": "2. \\( lastvertex middlevertex \\ldots absentvertex \\) is a regular polygon inscribed in a circle of radius \\( diameterlength \\) and center \\( edgepoint . insidepoint \\) is a point on line \\( edgepoint lastvertex \\) extended beyond \\( lastvertex \\). Show that\n\\[\n\\prod_{aggregate=1}^{singularity} \\overline{insidepoint sidepoint}= \\overline{edgepoint insidepoint}^{singularity}-diameterlength^{singularity}\n\\]",
      "solution": "Solution. We may assume the polygon is in the complex plane with its center at the origin and \\( lastvertex \\) on the positive real axis. Then the other vertices are\n\\[\ndiameterlength\\, realnumber,\\; diameterlength\\, realnumber^{2}, \\ldots, diameterlength\\, realnumber^{singularity-1}\n\\]\nwhere \\( realnumber \\) is a primitive \\( singularity \\)th root of unity.\nIf \\( insidepoint \\) is at the point \\( imaginaryvalue \\), then \\( \\overline{insidepoint sidepoint}=\\left|imaginaryvalue-diameterlength\\, realnumber^{aggregate-1}\\right| \\). So\n\\[\n\\begin{aligned}\n\\prod_{aggregate=1}^{singularity} \\overline{insidepoint sidepoint} & =\\left|\\prod_{aggregate=1}^{singularity}\\left(imaginaryvalue-diameterlength\\, realnumber^{aggregate-1}\\right)\\right|=diameterlength^{singularity}\\left|\\prod_{aggregate=1}^{singularity}\\left(\\frac{imaginaryvalue}{diameterlength}-realnumber^{aggregate-1}\\right)\\right| \\\\\n& =diameterlength^{singularity}\\left|\\left(\\frac{imaginaryvalue}{diameterlength}\\right)^{singularity}-1\\right| \\\\\n& =\\left|imaginaryvalue^{singularity}-diameterlength^{singularity}\\right|=imaginaryvalue^{singularity}-diameterlength^{singularity}=\\overline{edgepoint insidepoint}^{singularity}-diameterlength^{singularity} .\n\\end{aligned}\n\\]\n\nAt the third step we used the factorization\n\\[\nconstantvalue^{singularity}-1=\\prod_{aggregate=1}^{singularity}\\left(constantvalue-realnumber^{aggregate-1}\\right)\n\\]\nwhich is valid because \\( 1, realnumber, realnumber^{2}, \\ldots, realnumber^{singularity-1} \\) are the zeros of \\( constantvalue^{singularity}-1 \\)."
    },
    "garbled_string": {
      "map": {
        "P": "qzxwvtnp",
        "x": "hjgrksla",
        "X": "mldkfjwe",
        "i": "zasdghjk",
        "n": "lqwertyu",
        "r": "zmxncbva",
        "O": "plokmijn",
        "A_1": "asdfghjk",
        "A_2": "qweruiop",
        "A_n": "zxcvlkjh",
        "A_i": "poiulkjh",
        "\\omega": "kjhgfdsa"
      },
      "question": "2. \\( asdfghjk qweruiop \\ldots zxcvlkjh \\) is a regular polygon inscribed in a circle of radius \\( zmxncbva \\) and center \\( plokmijn . qzxwvtnp \\) is a point on line \\( plokmijn asdfghjk \\) extended beyond \\( asdfghjk \\). Show that\n\\[\n\\prod_{zasdghjk=1}^{lqwertyu} \\overline{qzxwvtnp A}_{zasdghjk}=\\overline{plokmijn qzxwvtnp}^{lqwertyu}-zmxncbva^{lqwertyu}\n\\]",
      "solution": "Solution. We may assume the polygon is in the complex plane with its center at the origin and \\( asdfghjk \\) on the positive real axis. Then the other vertices are\n\\[\nzmxncbva kjhgfdsa, zmxncbva kjhgfdsa^{2}, \\ldots, zmxncbva kjhgfdsa^{lqwertyu-1}\n\\]\nwhere \\( kjhgfdsa \\) is a primitive \\( lqwertyu \\)th root of unity.\nIf \\( qzxwvtnp \\) is at the point \\( hjgrksla \\), then \\( \\overline{qzxwvtnp poiulkjh}=\\left|hjgrksla-zmxncbva kjhgfdsa^{zasdghjk-1}\\right| \\). So\n\\[\n\\begin{aligned}\n\\prod_{zasdghjk=1}^{lqwertyu} \\overline{qzxwvtnp poiulkjh} & =\\left|\\prod_{zasdghjk=1}^{lqwertyu}\\left(hjgrksla-zmxncbva kjhgfdsa^{zasdghjk-1}\\right)\\right|=zmxncbva^{lqwertyu}\\left|\\prod_{zasdghjk=1}^{lqwertyu}\\left(\\frac{hjgrksla}{zmxncbva}-kjhgfdsa^{zasdghjk-1}\\right)\\right|=zmxncbva^{lqwertyu}\\left|\\left(\\frac{hjgrksla}{zmxncbva}\\right)^{lqwertyu}-1\\right| \\\\\n& =\\left|hjgrksla^{lqwertyu}-zmxncbva^{lqwertyu}\\right|=hjgrksla^{lqwertyu}-zmxncbva^{lqwertyu}=\\overline{plokmijn qzxwvtnp}^{lqwertyu}-zmxncbva^{lqwertyu} .\n\\end{aligned}\n\\]\n\nAt the third step we used the factorization\n\\[\nmldkfjwe^{lqwertyu}-1=\\prod_{zasdghjk=1}^{lqwertyu}\\left(mldkfjwe-kjhgfdsa^{zasdghjk-1}\\right)\n\\]\nwhich is valid because \\( 1, kjhgfdsa, kjhgfdsa^{2}, \\ldots, kjhgfdsa^{lqwertyu-1} \\) are the zeros of \\( mldkfjwe^{lqwertyu}-1 \\)."
    },
    "kernel_variant": {
      "question": "Let n ≥ 3.  Two concentric regular n-gons are inscribed in the same plane circle with center O and radii r < R.  Their vertices are  \nA₁,A₂,…,Aₙ  (radius r) and B₁,B₂,…,Bₙ (radius R),  \nwith A₁ and B₁ lying on the same ray OA₁.  A point P is chosen on that ray so that r < OP < R.  Prove that  \n  ∏_{i=1}^{n}  PA_i  / ∏_{i=1}^{n}  PB_i  = (Rⁿ − OPⁿ)/(OPⁿ − rⁿ).",
      "solution": "Step 1.  Place O at the origin of the complex plane and put A₁ on the positive real axis.  With ω = e^{2πi/n},  \n A_k = r ω^{k−1}, B_k = R ω^{k−1} (1 ≤ k ≤ n).\n\nStep 2.  Let P have coordinate x, where r < x < R (x is real).  Then  \n PA_k = |x − r ω^{k−1}|, PB_k = |x − R ω^{k−1}|,  \nso  \n ∏ PA_k = |∏(x − r ω^{k−1})|, ∏ PB_k = |∏(x − R ω^{k−1})|.   (★)\n\nStep 3.  Pull r or R out of every factor in (★) and use  \n Xⁿ − 1 = ∏(X − ω^{k−1}).  \nThus  \n ∏ PA_k = |xⁿ − rⁿ|, ∏ PB_k = |xⁿ − Rⁿ|.   (†)\n\nStep 4.  Because x is real with r < x < R, the quantities in (†) are positive, so the absolute-value signs may be removed:  \n ∏_{k=1}^{n} PA_k / ∏_{k=1}^{n} PB_k = (xⁿ − rⁿ)/(Rⁿ − xⁿ).  \nFinally substitute x = OP to obtain  \n ∏_{i=1}^{n} PA_i / ∏_{i=1}^{n} PB_i = (Rⁿ − OPⁿ)/(OPⁿ − rⁿ),  \nas required.",
      "_replacement_note": {
        "replaced_at": "2025-07-05T22:17:12.151150",
        "reason": "Original kernel variant was too easy compared to the original problem"
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}