{
  "index": "1989-A-3",
  "type": "COMB",
  "tag": [
    "COMB",
    "ALG",
    "ANA"
  ],
  "difficulty": "",
  "question": "Prove that if\n\\[\n11z^{10}+10iz^9+10iz-11=0,\n\\]\nthen $|z|=1.$ (Here $z$ is a complex number and $i^2=-1$.)",
  "solution": "Solution 1. Let \\( g(z)=11 z^{10}+10 i z^{9}+10 i z-11 \\). Let \\( p(w)=-w^{-5} g(i w)= \\) \\( 11 w^{5}+10 w^{4}+10 w^{-4}+11 w^{-5} \\). As \\( \\theta \\) increases from 0 to \\( 2 \\pi \\), the real-valued function \\( p\\left(e^{i \\theta}\\right)=22 \\cos (5 \\theta)+20 \\cos (4 \\theta) \\) changes sign at least 10 times, since at \\( \\theta=2 \\pi k / 10 \\) for \\( k=0,1, \\ldots, 9 \\) its value is \\( 22(-1)^{k}+20 \\cos (4 \\theta) \\), which has the sign of \\( (-1)^{k} \\). By the Intermediate Value Theorem [Spv, Ch. 7, Theorem 5], \\( p\\left(e^{i \\theta}\\right) \\) has at least one zero between \\( \\theta=2 \\pi k / 10 \\) and \\( \\theta=2 \\pi i(k+1) / 10 \\) for \\( k=0, \\ldots, 9 \\); this makes at least 10 zeros. Thus \\( g(z) \\) also has at least 10 zeros on the circle \\( |z|=1 \\), and these are all the zeros since \\( g(z) \\) is of degree 10 .\n\nSolution 2. The equation can be rewritten as \\( z^{9}=\\frac{11-10 i z}{11 z+10 i} \\). If \\( z=a+b i \\), then\n\\[\n|z|^{9}=\\left|\\frac{11-10 i z}{11 z+10 i}\\right|=\\frac{\\sqrt{11^{2}+220 b+10^{2}\\left(a^{2}+b^{2}\\right)}}{\\sqrt{11^{2}\\left(a^{2}+b^{2}\\right)+220 b+10^{2}}} .\n\\]\n\nLet \\( f(a, b) \\) and \\( g(a, b) \\) denote the numerator and denominator of the right-hand side. If \\( |z|>1 \\), then \\( a^{2}+b^{2}>1 \\), so \\( g(a, b)>f(a, b) \\), making \\( \\left|z^{9}\\right|<1 \\), a contradiction. If \\( |z|<1 \\), then \\( a^{2}+b^{2}<1 \\), so \\( g(a, b)<f(a, b) \\), making \\( \\left|z^{9}\\right|>1 \\), again a contradiction. Hence \\( |z|=1 \\).\n\nSolution 3. Rouché's Theorem [Ah, p. 153] states that if \\( f \\) and \\( g \\) are analytic functions on an open set containing a closed disc, and if \\( |g(z)-f(z)|<|f(z)| \\) everywhere on the boundary of the disc, then \\( f \\) and \\( g \\) have the same number of zeros inside the disc. Let \\( f(z)=10 i z-11 \\) and \\( g(z)=11 z^{10}+10 i z^{9}+10 i z-11 \\), and consider the discs \\( |z| \\leq \\alpha \\) with \\( \\alpha \\in(0,1) \\). Then\n\\[\n|g(z)-f(z)|=\\left|11 z^{10}+10 i z^{9}\\right|=|z|^{9}|11 z+10 i|<|10 i z-11|=|f(z)|\n\\]\nif \\( |z|<1 \\), by the same calculation as in Solution 2 . But \\( f \\) has its only zero at \\( 11 /(10 i) \\), outside \\( |z|=1 \\), so \\( g \\) has no zeros with \\( |z|<\\alpha \\) for any \\( \\alpha \\in(0,1) \\), and hence no zeros with \\( |z|<1 \\). Finally, \\( g(-1 / z)=-z^{-10} g(z) \\), so the nonvanishing of \\( g \\) for \\( |z|<1 \\) implies the nonvanishing of \\( g \\) for \\( |z|>1 \\). Therefore, if \\( g(z)=0 \\), then \\( |z|=1 \\).",
  "vars": [
    "z",
    "w",
    "\\\\theta",
    "a",
    "b"
  ],
  "params": [
    "g",
    "p",
    "k",
    "f",
    "\\\\alpha"
  ],
  "sci_consts": [
    "i",
    "e"
  ],
  "variants": {
    "descriptive_long": {
      "map": {
        "z": "complexvar",
        "w": "auxiliaryvar",
        "\\theta": "rotationangle",
        "a": "realcoordinate",
        "b": "imaginarycoord",
        "g": "polyfunctiong",
        "p": "polyfunctionp",
        "k": "indexinteger",
        "f": "analyticfunc",
        "\\alpha": "discsradius"
      },
      "question": "Prove that if\n\\[\n11complexvar^{10}+10i complexvar^9+10i complexvar-11=0,\n\\],\nthen $|complexvar|=1.$ (Here $complexvar$ is a complex number and $i^2=-1$.)",
      "solution": "Solution 1. Let \\( polyfunctiong(complexvar)=11 complexvar^{10}+10 i complexvar^{9}+10 i complexvar-11 \\). Let \\( polyfunctionp(auxiliaryvar)=-auxiliaryvar^{-5} polyfunctiong(i auxiliaryvar)= \\) \\( 11 auxiliaryvar^{5}+10 auxiliaryvar^{4}+10 auxiliaryvar^{-4}+11 auxiliaryvar^{-5} \\). As \\( rotationangle \\) increases from 0 to \\( 2 \\pi \\), the real-valued function \\( polyfunctionp\\left(e^{i rotationangle}\\right)=22 \\cos (5 rotationangle)+20 \\cos (4 rotationangle) \\) changes sign at least 10 times, since at \\( rotationangle=2 \\pi indexinteger / 10 \\) for \\( indexinteger=0,1, \\ldots, 9 \\) its value is \\( 22(-1)^{indexinteger}+20 \\cos (4 rotationangle) \\), which has the sign of \\( (-1)^{indexinteger} \\). By the Intermediate Value Theorem [Spv, Ch. 7, Theorem 5], \\( polyfunctionp\\left(e^{i rotationangle}\\right) \\) has at least one zero between \\( rotationangle=2 \\pi indexinteger / 10 \\) and \\( rotationangle=2 \\pi i(indexinteger+1) / 10 \\) for \\( indexinteger=0, \\ldots, 9 \\); this makes at least 10 zeros. Thus \\( polyfunctiong(complexvar) \\) also has at least 10 zeros on the circle \\( |complexvar|=1 \\), and these are all the zeros since \\( polyfunctiong(complexvar) \\) is of degree 10.\n\nSolution 2. The equation can be rewritten as \\( complexvar^{9}=\\frac{11-10 i complexvar}{11 complexvar+10 i} \\). If \\( complexvar=realcoordinate+imaginarycoord i \\), then\n\\[\n|complexvar|^{9}=\\left|\\frac{11-10 i complexvar}{11 complexvar+10 i}\\right|=\\frac{\\sqrt{11^{2}+220 imaginarycoord+10^{2}\\left(realcoordinate^{2}+imaginarycoord^{2}\\right)}}{\\sqrt{11^{2}\\left(realcoordinate^{2}+imaginarycoord^{2}\\right)+220 imaginarycoord+10^{2}}} .\n\\]\n\nLet \\( analyticfunc(realcoordinate, imaginarycoord) \\) and \\( polyfunctiong(realcoordinate, imaginarycoord) \\) denote the numerator and denominator of the right-hand side. If \\( |complexvar|>1 \\), then \\( realcoordinate^{2}+imaginarycoord^{2}>1 \\), so \\( polyfunctiong(realcoordinate, imaginarycoord)>analyticfunc(realcoordinate, imaginarycoord) \\), making \\( |complexvar^{9}|<1 \\), a contradiction. If \\( |complexvar|<1 \\), then \\( realcoordinate^{2}+imaginarycoord^{2}<1 \\), so \\( polyfunctiong(realcoordinate, imaginarycoord)<analyticfunc(realcoordinate, imaginarycoord) \\), making \\( |complexvar^{9}|>1 \\), again a contradiction. Hence \\( |complexvar|=1 \\).\n\nSolution 3. Rouché's Theorem [Ah, p. 153] states that if \\( analyticfunc \\) and \\( polyfunctiong \\) are analytic functions on an open set containing a closed disc, and if \\( |polyfunctiong(complexvar)-analyticfunc(complexvar)|<|analyticfunc(complexvar)| \\) everywhere on the boundary of the disc, then \\( analyticfunc \\) and \\( polyfunctiong \\) have the same number of zeros inside the disc. Let \\( analyticfunc(complexvar)=10 i complexvar-11 \\) and \\( polyfunctiong(complexvar)=11 complexvar^{10}+10 i complexvar^{9}+10 i complexvar-11 \\), and consider the discs \\( |complexvar| \\leq discsradius \\) with \\( discsradius \\in(0,1) \\). Then\n\\[\n|polyfunctiong(complexvar)-analyticfunc(complexvar)|=\\left|11 complexvar^{10}+10 i complexvar^{9}\\right|=|complexvar|^{9}|11 complexvar+10 i|<|10 i complexvar-11|=|analyticfunc(complexvar)|\n\\]\nif \\( |complexvar|<1 \\), by the same calculation as in Solution 2. But \\( analyticfunc \\) has its only zero at \\( 11 /(10 i) \\), outside \\( |complexvar|=1 \\), so \\( polyfunctiong \\) has no zeros with \\( |complexvar|<discsradius \\) for any \\( discsradius \\in(0,1) \\), and hence no zeros with \\( |complexvar|<1 \\). Finally, \\( polyfunctiong(-1 / complexvar)=-complexvar^{-10} polyfunctiong(complexvar) \\), so the nonvanishing of \\( polyfunctiong \\) for \\( |complexvar|<1 \\) implies the nonvanishing of \\( polyfunctiong \\) for \\( |complexvar|>1 \\). Therefore, if \\( polyfunctiong(complexvar)=0 \\), then \\( |complexvar|=1 \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "z": "marigolds",
        "w": "toffeejar",
        "\\theta": "lighthouse",
        "a": "sandglass",
        "b": "cuckoobird",
        "g": "raincloud",
        "p": "driftwood",
        "k": "pebblestone",
        "f": "moonluster",
        "\\alpha": "bluewhale"
      },
      "question": "Prove that if\n\\[\n11marigolds^{10}+10 i marigolds^{9}+10 i marigolds-11=0,\n\\]\nthen $|marigolds|=1.$ (Here $marigolds$ is a complex number and $i^2=-1$.)",
      "solution": "Solution 1. Let \\( raincloud(marigolds)=11 marigolds^{10}+10 i marigolds^{9}+10 i marigolds-11 \\). Let \\( driftwood(toffeejar)=-toffeejar^{-5} raincloud(i toffeejar)= \\) \\( 11 toffeejar^{5}+10 toffeejar^{4}+10 toffeejar^{-4}+11 toffeejar^{-5} \\). As \\( lighthouse \\) increases from 0 to \\( 2 \\pi \\), the real-valued function \\( driftwood\\left(e^{i lighthouse}\\right)=22 \\cos (5 lighthouse)+20 \\cos (4 lighthouse) \\) changes sign at least 10 times, since at \\( lighthouse=2 \\pi pebblestone / 10 \\) for \\( pebblestone=0,1, \\ldots, 9 \\) its value is \\( 22(-1)^{pebblestone}+20 \\cos (4 lighthouse) \\), which has the sign of \\( (-1)^{pebblestone} \\). By the Intermediate Value Theorem [Spv, Ch. 7, Theorem 5], \\( driftwood\\left(e^{i lighthouse}\\right) \\) has at least one zero between \\( lighthouse=2 \\pi pebblestone / 10 \\) and \\( lighthouse=2 \\pi i(pebblestone+1) / 10 \\) for \\( pebblestone=0, \\ldots, 9 \\); this makes at least 10 zeros. Thus \\( raincloud(marigolds) \\) also has at least 10 zeros on the circle \\( |marigolds|=1 \\), and these are all the zeros since \\( raincloud(marigolds) \\) is of degree 10.\n\nSolution 2. The equation can be rewritten as \\( marigolds^{9}=\\frac{11-10 i marigolds}{11 marigolds+10 i} \\). If \\( marigolds=sandglass+cuckoobird i \\), then\n\\[\n|marigolds|^{9}=\\left|\\frac{11-10 i marigolds}{11 marigolds+10 i}\\right|=\\frac{\\sqrt{11^{2}+220 cuckoobird+10^{2}\\left(sandglass^{2}+cuckoobird^{2}\\right)}}{\\sqrt{11^{2}\\left(sandglass^{2}+cuckoobird^{2}\\right)+220 cuckoobird+10^{2}}} .\n\\]\nLet \\( moonluster(sandglass, cuckoobird) \\) and \\( raincloud(sandglass, cuckoobird) \\) denote the numerator and denominator of the right-hand side. If \\( |marigolds|>1 \\), then \\( sandglass^{2}+cuckoobird^{2}>1 \\), so \\( raincloud(sandglass, cuckoobird)>moonluster(sandglass, cuckoobird) \\), making \\( \\left|marigolds^{9}\\right|<1 \\), a contradiction. If \\( |marigolds|<1 \\), then \\( sandglass^{2}+cuckoobird^{2}<1 \\), so \\( raincloud(sandglass, cuckoobird)<moonluster(sandglass, cuckoobird) \\), making \\( \\left|marigolds^{9}\\right|>1 \\), again a contradiction. Hence \\( |marigolds|=1 \\).\n\nSolution 3. Rouché's Theorem [Ah, p. 153] states that if \\( moonluster \\) and \\( raincloud \\) are analytic functions on an open set containing a closed disc, and if \\( |raincloud(marigolds)-moonluster(marigolds)|<|moonluster(marigolds)| \\) everywhere on the boundary of the disc, then \\( moonluster \\) and \\( raincloud \\) have the same number of zeros inside the disc. Let \\( moonluster(marigolds)=10 i marigolds-11 \\) and \\( raincloud(marigolds)=11 marigolds^{10}+10 i marigolds^{9}+10 i marigolds-11 \\), and consider the discs \\( |marigolds| \\leq bluewhale \\) with \\( bluewhale \\in(0,1) \\). Then\n\\[\n|raincloud(marigolds)-moonluster(marigolds)|=\\left|11 marigolds^{10}+10 i marigolds^{9}\\right|=|marigolds|^{9}|11 marigolds+10 i|<|10 i marigolds-11|=|moonluster(marigolds)|\n\\]\nif \\( |marigolds|<1 \\), by the same calculation as in Solution 2. But \\( moonluster \\) has its only zero at \\( 11 /(10 i) \\), outside \\( |marigolds|=1 \\), so \\( raincloud \\) has no zeros with \\( |marigolds|<bluewhale \\) for any \\( bluewhale \\in(0,1) \\), and hence no zeros with \\( |marigolds|<1 \\). Finally, \\( raincloud(-1 / marigolds)=-marigolds^{-10} raincloud(marigolds) \\), so the nonvanishing of \\( raincloud \\) for \\( |marigolds|<1 \\) implies the nonvanishing of \\( raincloud \\) for \\( |marigolds|>1 \\). Therefore, if \\( raincloud(marigolds)=0 \\), then \\( |marigolds|=1 \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "z": "knownvalue",
        "w": "steadystate",
        "\\theta": "distance",
        "a": "totality",
        "b": "tangible",
        "g": "malfunction",
        "p": "chaosform",
        "k": "continuum",
        "f": "breakage",
        "\\alpha": "limitless"
      },
      "question": "Prove that if\n\\[\n11 knownvalue^{10}+10 i knownvalue^{9}+10 i knownvalue-11=0,\n\\]\nthen $|knownvalue|=1.$ (Here knownvalue is a complex number and $i^2=-1$.)",
      "solution": "Solution 1. Let \\( malfunction(knownvalue)=11 knownvalue^{10}+10 i knownvalue^{9}+10 i knownvalue-11 \\). Let \\( chaosform(steadystate)=-steadystate^{-5} malfunction(i steadystate)= \\) \\( 11 steadystate^{5}+10 steadystate^{4}+10 steadystate^{-4}+11 steadystate^{-5} \\). As \\( distance \\) increases from 0 to \\( 2 \\pi \\), the real-valued function \\( chaosform\\left(e^{i distance}\\right)=22 \\cos (5 distance)+20 \\cos (4 distance) \\) changes sign at least 10 times, since at \\( distance=2 \\pi continuum / 10 \\) for \\( continuum=0,1, \\ldots, 9 \\) its value is \\( 22(-1)^{continuum}+20 \\cos (4 distance) \\), which has the sign of \\( (-1)^{continuum} \\). By the Intermediate Value Theorem [Spv, Ch. 7, Theorem 5], \\( chaosform\\left(e^{i distance}\\right) \\) has at least one zero between \\( distance=2 \\pi continuum / 10 \\) and \\( distance=2 \\pi i(continuum+1) / 10 \\) for \\( continuum=0, \\ldots, 9 \\); this makes at least 10 zeros. Thus \\( malfunction(knownvalue) \\) also has at least 10 zeros on the circle \\( |knownvalue|=1 \\), and these are all the zeros since \\( malfunction(knownvalue) \\) is of degree 10.\n\nSolution 2. The equation can be rewritten as \\( knownvalue^{9}=\\frac{11-10 i knownvalue}{11 knownvalue+10 i} \\). If \\( knownvalue=totality+tangible i \\), then\n\\[\n|knownvalue|^{9}=\\left|\\frac{11-10 i knownvalue}{11 knownvalue+10 i}\\right|=\\frac{\\sqrt{11^{2}+220 tangible+10^{2}\\left(totality^{2}+tangible^{2}\\right)}}{\\sqrt{11^{2}\\left(totality^{2}+tangible^{2}\\right)+220 tangible+10^{2}}} .\n\\]\nLet \\( breakage(totality, tangible) \\) and \\( malfunction(totality, tangible) \\) denote the numerator and denominator of the right-hand side. If \\( |knownvalue|>1 \\), then \\( totality^{2}+tangible^{2}>1 \\), so \\( malfunction(totality, tangible)>breakage(totality, tangible) \\), making \\( \\left|knownvalue^{9}\\right|<1 \\), a contradiction. If \\( |knownvalue|<1 \\), then \\( totality^{2}+tangible^{2}<1 \\), so \\( malfunction(totality, tangible)<breakage(totality, tangible) \\), making \\( \\left|knownvalue^{9}\\right|>1 \\), again a contradiction. Hence \\( |knownvalue|=1 \\).\n\nSolution 3. Rouché's Theorem [Ah, p. 153] states that if \\( breakage \\) and \\( malfunction \\) are analytic functions on an open set containing a closed disc, and if \\( |malfunction(knownvalue)-breakage(knownvalue)|<|breakage(knownvalue)| \\) everywhere on the boundary of the disc, then \\( breakage \\) and \\( malfunction \\) have the same number of zeros inside the disc. Let \\( breakage(knownvalue)=10 i knownvalue-11 \\) and \\( malfunction(knownvalue)=11 knownvalue^{10}+10 i knownvalue^{9}+10 i knownvalue-11 \\), and consider the discs \\( |knownvalue| \\leq limitless \\) with \\( limitless \\in(0,1) \\). Then\n\\[\n|malfunction(knownvalue)-breakage(knownvalue)|=\\left|11 knownvalue^{10}+10 i knownvalue^{9}\\right|=|knownvalue|^{9}|11 knownvalue+10 i|<|10 i knownvalue-11|=|breakage(knownvalue)|\n\\]\nif \\( |knownvalue|<1 \\), by the same calculation as in Solution 2. But \\( breakage \\) has its only zero at \\( 11 /(10 i) \\), outside \\( |knownvalue|=1 \\), so \\( malfunction \\) has no zeros with \\( |knownvalue|<limitless \\) for any \\( limitless \\in(0,1) \\), and hence no zeros with \\( |knownvalue|<1 \\). Finally, \\( malfunction(-1 / knownvalue)=-knownvalue^{-10} malfunction(knownvalue) \\), so the nonvanishing of \\( malfunction \\) for \\( |knownvalue|<1 \\) implies the nonvanishing of \\( malfunction \\) for \\( |knownvalue|>1 \\). Therefore, if \\( malfunction(knownvalue)=0 \\), then \\( |knownvalue|=1 \\)."
    },
    "garbled_string": {
      "map": {
        "z": "qzxwvtnp",
        "w": "hjgrksla",
        "\\theta": "mnplxqrs",
        "a": "vjkdpsle",
        "b": "tznchmra",
        "g": "clfdgobr",
        "p": "wfrqxbnl",
        "k": "ybrtspmq",
        "f": "prndxkjo",
        "\\alpha": "sdlvcneq"
      },
      "question": "Prove that if\n\\[\n11qzxwvtnp^{10}+10iqzxwvtnp^9+10iqzxwvtnp-11=0,\n\\],\nthen $|qzxwvtnp|=1.$ (Here $qzxwvtnp$ is a complex number and $i^2=-1$.)",
      "solution": "Solution 1. Let \\( clfdgobr(qzxwvtnp)=11 qzxwvtnp^{10}+10 i qzxwvtnp^{9}+10 i qzxwvtnp-11 \\). Let \\( wfrqxbnl(hjgrksla)=-hjgrksla^{-5} clfdgobr(i hjgrksla)= \\) \\( 11 hjgrksla^{5}+10 hjgrksla^{4}+10 hjgrksla^{-4}+11 hjgrksla^{-5} \\). As \\( mnplxqrs \\) increases from 0 to \\( 2 \\pi \\), the real-valued function \\( wfrqxbnl\\left(e^{i mnplxqrs}\\right)=22 \\cos (5 mnplxqrs)+20 \\cos (4 mnplxqrs) \\) changes sign at least 10 times, since at \\( mnplxqrs=2 \\pi ybrtspmq / 10 \\) for \\( ybrtspmq=0,1, \\ldots, 9 \\) its value is \\( 22(-1)^{ybrtspmq}+20 \\cos (4 mnplxqrs) \\), which has the sign of \\( (-1)^{ybrtspmq} \\). By the Intermediate Value Theorem [Spv, Ch. 7, Theorem 5], \\( wfrqxbnl\\left(e^{i mnplxqrs}\\right) \\) has at least one zero between \\( mnplxqrs=2 \\pi ybrtspmq / 10 \\) and \\( mnplxqrs=2 \\pi i(ybrtspmq+1) / 10 \\) for \\( ybrtspmq=0, \\ldots, 9 \\); this makes at least 10 zeros. Thus \\( clfdgobr(qzxwvtnp) \\) also has at least 10 zeros on the circle \\( |qzxwvtnp|=1 \\), and these are all the zeros since \\( clfdgobr(qzxwvtnp) \\) is of degree 10.\n\nSolution 2. The equation can be rewritten as \\( qzxwvtnp^{9}=\\frac{11-10 i qzxwvtnp}{11 qzxwvtnp+10 i} \\). If \\( qzxwvtnp=vjkdpsle+tznchmra i \\), then\n\\[\n|qzxwvtnp|^{9}=\\left|\\frac{11-10 i qzxwvtnp}{11 qzxwvtnp+10 i}\\right|=\\frac{\\sqrt{11^{2}+220 tznchmra+10^{2}\\left(vjkdpsle^{2}+tznchmra^{2}\\right)}}{\\sqrt{11^{2}\\left(vjkdpsle^{2}+tznchmra^{2}\\right)+220 tznchmra+10^{2}}} .\n\\]\nLet \\( prndxkjo(vjkdpsle, tznchmra) \\) and \\( clfdgobr(vjkdpsle, tznchmra) \\) denote the numerator and denominator of the right-hand side. If \\( |qzxwvtnp|>1 \\), then \\( vjkdpsle^{2}+tznchmra^{2}>1 \\), so \\( clfdgobr(vjkdpsle, tznchmra)>prndxkjo(vjkdpsle, tznchmra) \\), making \\( \\left|qzxwvtnp^{9}\\right|<1 \\), a contradiction. If \\( |qzxwvtnp|<1 \\), then \\( vjkdpsle^{2}+tznchmra^{2}<1 \\), so \\( clfdgobr(vjkdpsle, tznchmra)<prndxkjo(vjkdpsle, tznchmra) \\), making \\( \\left|qzxwvtnp^{9}\\right|>1 \\), again a contradiction. Hence \\( |qzxwvtnp|=1 \\).\n\nSolution 3. Rouché's Theorem [Ah, p. 153] states that if \\( prndxkjo \\) and \\( clfdgobr \\) are analytic functions on an open set containing a closed disc, and if \\( |clfdgobr(qzxwvtnp)-prndxkjo(qzxwvtnp)|<|prndxkjo(qzxwvtnp)| \\) everywhere on the boundary of the disc, then \\( prndxkjo \\) and \\( clfdgobr \\) have the same number of zeros inside the disc. Let \\( prndxkjo(qzxwvtnp)=10 i qzxwvtnp-11 \\) and \\( clfdgobr(qzxwvtnp)=11 qzxwvtnp^{10}+10 i qzxwvtnp^{9}+10 i qzxwvtnp-11 \\), and consider the discs \\( |qzxwvtnp| \\leq sdlvcneq \\) with \\( sdlvcneq \\in(0,1) \\). Then\n\\[\n|clfdgobr(qzxwvtnp)-prndxkjo(qzxwvtnp)|=\\left|11 qzxwvtnp^{10}+10 i qzxwvtnp^{9}\\right|=|qzxwvtnp|^{9}|11 qzxwvtnp+10 i|<|10 i qzxwvtnp-11|=|prndxkjo(qzxwvtnp)|\n\\]\nif \\( |qzxwvtnp|<1 \\), by the same calculation as in Solution 2. But \\( prndxkjo \\) has its only zero at \\( 11 /(10 i) \\), outside \\( |qzxwvtnp|=1 \\), so \\( clfdgobr \\) has no zeros with \\( |qzxwvtnp|<sdlvcneq \\) for any \\( sdlvcneq \\in(0,1) \\), and hence no zeros with \\( |qzxwvtnp|<1 \\). Finally, \\( clfdgobr(-1 / qzxwvtnp)=-qzxwvtnp^{-10} clfdgobr(qzxwvtnp) \\), so the nonvanishing of \\( clfdgobr \\) for \\( |qzxwvtnp|<1 \\) implies the nonvanishing of \\( clfdgobr \\) for \\( |qzxwvtnp|>1 \\). Therefore, if \\( clfdgobr(qzxwvtnp)=0 \\), then \\( |qzxwvtnp|=1 \\)."
    },
    "kernel_variant": {
      "question": "Prove that every zero of  \n  17 z¹⁸ + 10 i z¹⁷ + 10 i z – 17 = 0  \nlies on the unit circle |z| = 1.  In addition, determine how many of the zeros satisfy Re z > 0.\n\n─────────────────────────────────────────────",
      "solution": "Step 1.  No roots inside the unit disc.  \nWrite g(z)=17 z¹⁸+10 i z¹⁷+10 i z–17 and split  \n f(z)=10 i z–17, h(z)=g(z)–f(z)=17 z¹⁸+10 i z¹⁷.  \n\nPut z=r e^{iθ} with 0<r<1.  Then  \n |h(z)|=r¹⁷|17 z+10 i|, |f(z)|=|10 i z–17|.  \n\nA direct calculation gives  \n (|h|/|f|)²  \n  = r³⁴ · (289 r²+100+340 r sin θ)/(289+100 r²+340 r sin θ)  \n  ≤ r³⁴ < 1.  \n\nHence |h(z)|<|f(z)| on the circle |z|=r.  By Rouché’s theorem g and f have the same number of zeros in |z|<r, namely none, because f vanishes only at 17/(10 i), which lies outside |z|=1.  Letting r→1⁻ shows that g possesses no zero with |z|<1.\n\nStep 2.  No roots outside the unit disc.  \nObserve the reciprocity symmetry  \n g(–1/z)=17(–1/z)¹⁸+10 i(–1/z)¹⁷+10 i(–1/z)–17  \n  =–z^{–18}g(z).  \n\nThus, if g(ζ)=0 and |ζ|>1, then g(–1/ζ)=0 with |–1/ζ|<1—contradicting Step 1.  Therefore every zero of g satisfies |z|=1.\n\nStep 3.  Counting the roots on the right half of the circle.  \nSet w=i z and consider the function  \n p(w)=–w^{–9}g(i w)  \n  =17 w⁹+10 w⁸+10 w^{–8}+17 w^{–9}.  \n\nFor w = e^{iθ} we obtain  \n p(e^{iθ})=34 cos(9θ)+20 cos(8θ), a purely real quantity.  \n\nBecause 34>|20|, the sign of p(e^{iθ}) is the sign of cos(9θ)= (–1)^k at the points θ=kπ/9 (k=0,…,17).  Consequently p changes sign across every consecutive pair of these 18 points, so by the Intermediate Value Theorem it has at least one zero in each of the 18 arcs between them.  As deg g=18, these are all the zeros of g; every zero is simple and sits in a distinct arc of length π/9 on |z|=1.\n\nThose arcs with θ in (–π/2, π/2) ∪ (3π/2, 5π/2) correspond to Re z>0.  Precisely nine of the eighteen equal arcs lie in this set, whence\n\n Number of zeros of g with Re z>0 = 9.\n\n─────────────────────────────────────────────",
      "_replacement_note": {
        "replaced_at": "2025-07-05T22:17:12.085584",
        "reason": "Original kernel variant was too easy compared to the original problem"
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}