{
  "index": "1971-A-2",
  "type": "ALG",
  "tag": [
    "ALG",
    "NT"
  ],
  "difficulty": "",
  "question": "\\text { A-2. Determine all polynomials } P(x) \\text { such that } P\\left(x^{2}+1\\right)=(P(x))^{2}+1 \\text { and } P(0)=0 \\text {. }",
  "solution": "A-2 \\( \\quad P(0)=0, \\quad P(1)=[P(0)]^{2}+1=1, \\quad P(2)=[P(1)]^{2}+1=2, \\quad P(5) \\) \\( =[P(2)]^{2}+1=5, P\\left(5^{2}+1\\right)=[P(5)]^{2}+1=26 \\), etc. Thus the polynomial \\( P(x) \\) agrees with \\( x \\) for more values than the degree of \\( P(x) \\), so \\( P(x) \\equiv x \\).",
  "vars": [
    "x"
  ],
  "params": [
    "P"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "variable",
        "P": "polynom"
      },
      "question": "\\text { A-2. Determine all polynomials } polynom(variable) \\text { such that } polynom\\left(variable^{2}+1\\right)=(polynom(variable))^{2}+1 \\text { and } polynom(0)=0 \\text {. }",
      "solution": "A-2 \\( \\quad polynom(0)=0, \\quad polynom(1)=[polynom(0)]^{2}+1=1, \\quad polynom(2)=[polynom(1)]^{2}+1=2, \\quad polynom(5) \\) \\( =[polynom(2)]^{2}+1=5, polynom\\left(5^{2}+1\\right)=[polynom(5)]^{2}+1=26 \\), etc. Thus the polynomial \\( polynom(variable) \\) agrees with \\( variable \\) for more values than the degree of \\( polynom(variable) \\), so \\( polynom(variable) \\equiv variable \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "monolith",
        "P": "horizonline"
      },
      "question": "\\text { A-2. Determine all polynomials } horizonline(monolith) \\text { such that } horizonline\\left(monolith^{2}+1\\right)=(horizonline(monolith))^{2}+1 \\text { and } horizonline(0)=0 \\text {. }",
      "solution": "A-2 \\( \\quad horizonline(0)=0, \\quad horizonline(1)=[horizonline(0)]^{2}+1=1, \\quad horizonline(2)=[horizonline(1)]^{2}+1=2, \\quad horizonline(5) \\) \\( =[horizonline(2)]^{2}+1=5, horizonline\\left(5^{2}+1\\right)=[horizonline(5)]^{2}+1=26 \\), etc. Thus the polynomial \\( horizonline(monolith) \\) agrees with \\( monolith \\) for more values than the degree of \\( horizonline(monolith) \\), so \\( horizonline(monolith) \\equiv monolith \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "constantval",
        "P": "transcendfunc"
      },
      "question": "\\text { A-2. Determine all polynomials } transcendfunc(constantval) \\text { such that } transcendfunc\\left(constantval^{2}+1\\right)=(transcendfunc(constantval))^{2}+1 \\text { and } transcendfunc(0)=0 \\text {. }",
      "solution": "A-2 \\( \\quad transcendfunc(0)=0, \\quad transcendfunc(1)=[transcendfunc(0)]^{2}+1=1, \\quad transcendfunc(2)=[transcendfunc(1)]^{2}+1=2, \\quad transcendfunc(5) \\) \\( =[transcendfunc(2)]^{2}+1=5, transcendfunc\\left(5^{2}+1\\right)=[transcendfunc(5)]^{2}+1=26 \\), etc. Thus the polynomial \\( transcendfunc(constantval) \\) agrees with \\( constantval \\) for more values than the degree of \\( transcendfunc(constantval) \\), so \\( transcendfunc(constantval) \\equiv constantval \\)."
    },
    "garbled_string": {
      "map": {
        "x": "bqztrnfa",
        "P": "lmxwqhtz"
      },
      "question": "\\text { A-2. Determine all polynomials } lmxwqhtz(bqztrnfa) \\text { such that } lmxwqhtz\\left(bqztrnfa^{2}+1\\right)=(lmxwqhtz(bqztrnfa))^{2}+1 \\text { and } lmxwqhtz(0)=0 \\text {. }",
      "solution": "A-2 \\( \\quad lmxwqhtz(0)=0, \\quad lmxwqhtz(1)=[lmxwqhtz(0)]^{2}+1=1, \\quad lmxwqhtz(2)=[lmxwqhtz(1)]^{2}+1=2, \\quad lmxwqhtz(5) \\) \\( =[lmxwqhtz(2)]^{2}+1=5, lmxwqhtz\\left(5^{2}+1\\right)=[lmxwqhtz(5)]^{2}+1=26 \\), etc. Thus the polynomial \\( lmxwqhtz(bqztrnfa) \\) agrees with \\( bqztrnfa \\) for more values than the degree of \\( lmxwqhtz(bqztrnfa) \\), so \\( lmxwqhtz(bqztrnfa) \\equiv bqztrnfa \\)."
    },
    "kernel_variant": {
      "question": "Let  \n f₂(x)=x²+1 and f₃(x)=x³+3.  \nDetermine all complex–coefficient polynomials P(x) that\n\n1. commute with both f₂ and f₃:  \n P∘f₂=f₂∘P and P∘f₃=f₃∘P  (for every complex x);  \n\n2. satisfy the normalising conditions  \n P(0)=0 and P(2)=2.  \n\nFind every such polynomial P(x).\n\n────────────────────────────────────────",
      "solution": "Step 1.  Structure of the centralisers of f₂ and f₃  \nA classical result of Ritt theory states that, for a non-exceptional polynomial g of degree ≥2, every polynomial that commutes with g is an iterate of g.  \nBoth f₂ and f₃ are non-exceptional (their critical points are not pre-periodic to a 2-cycle), so\n\n Cent(f₂)= { f₂ᵐ : m≥0 }  and  Cent(f₃)= { f₃ⁿ : n≥0 }. (★)\n\nStep 2.  Degree considerations  \nWrite d=deg P≥1.  \nBecause P commutes with f₂, (★) implies P=f₂ᵐ for some m≥0; hence d=2ᵐ.  \nBecause P also commutes with f₃, the same argument gives P=f₃ⁿ for some n≥0; hence d=3ⁿ.  \nTherefore  \n 2ᵐ = d = 3ⁿ.  \nThe only positive integer that is simultaneously a power of 2 and a power of 3 is 1, so\n\n d=1, i.e. P is linear.               (1)\n\nStep 3.  Write the linear candidate  \nLet P(x)=ax+b with a,b∈ℂ and a≠0.  From the normalising condition P(0)=0 we obtain b=0, so\n\n P(x)=ax.                                         (2)\n\nStep 4.  Impose the commutation with f₂  \nSubstitute (2) into P∘f₂=f₂∘P:\n\n a(x²+1)= (ax)²+1 = a²x²+1.\n\nComparing coefficients of x² and of the constant term gives\n\n a = a² and a = 1.\n\nThus a=1.\n\nStep 5.  Verification  \nP(x)=x obviously commutes with both f₂ and f₃ and fulfils P(0)=0 and P(2)=2.\n\nConclusion.  The unique polynomial satisfying all requirements is\n\n P(x)=x.\n\n────────────────────────────────────────",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.597019",
        "was_fixed": false,
        "difficulty_analysis": "• Multiple commuting relations:  The candidate must simultaneously belong to the centraliser of two different nonlinear polynomials, forcing an examination of the intersection of two non-trivial algebraic sets.  \n\n• Use of Ritt’s theory:  Identifying the full centralisers of f₂ and f₃ and invoking the uniqueness of polynomial iterates is far subtler than the one-orbit-argument that solves the original problems.\n\n• Degree arithmetic:  Equating two independent expressions for deg P introduces a number-theoretic constraint (2ᵐ=3ⁿ) whose only solution is deg P=1; this step is absent from the original problem.\n\n• Normalising conditions only after deep structure:  The values P(0)=0 and P(2)=2 are used at the very end to eliminate the linear but non-identity possibility P(x)=-x, showing that the boundary data interact with the composition theory rather than giving an immediate telescoping sequence as in the original problem.\n\nOverall, the solver must blend polynomial composition theory, group-like properties of centralisers, elementary number theory, and classical functional equations—several conceptual layers beyond the straightforward “iterate a value until exceeding the degree” technique sufficient for the original problem."
      }
    },
    "original_kernel_variant": {
      "question": "Let  \n f₂(x)=x²+1 and f₃(x)=x³+3.  \nDetermine all complex–coefficient polynomials P(x) that\n\n1. commute with both f₂ and f₃:  \n P∘f₂=f₂∘P and P∘f₃=f₃∘P  (for every complex x);  \n\n2. satisfy the normalising conditions  \n P(0)=0 and P(2)=2.  \n\nFind every such polynomial P(x).\n\n────────────────────────────────────────",
      "solution": "Step 1.  Structure of the centralisers of f₂ and f₃  \nA classical result of Ritt theory states that, for a non-exceptional polynomial g of degree ≥2, every polynomial that commutes with g is an iterate of g.  \nBoth f₂ and f₃ are non-exceptional (their critical points are not pre-periodic to a 2-cycle), so\n\n Cent(f₂)= { f₂ᵐ : m≥0 }  and  Cent(f₃)= { f₃ⁿ : n≥0 }. (★)\n\nStep 2.  Degree considerations  \nWrite d=deg P≥1.  \nBecause P commutes with f₂, (★) implies P=f₂ᵐ for some m≥0; hence d=2ᵐ.  \nBecause P also commutes with f₃, the same argument gives P=f₃ⁿ for some n≥0; hence d=3ⁿ.  \nTherefore  \n 2ᵐ = d = 3ⁿ.  \nThe only positive integer that is simultaneously a power of 2 and a power of 3 is 1, so\n\n d=1, i.e. P is linear.               (1)\n\nStep 3.  Write the linear candidate  \nLet P(x)=ax+b with a,b∈ℂ and a≠0.  From the normalising condition P(0)=0 we obtain b=0, so\n\n P(x)=ax.                                         (2)\n\nStep 4.  Impose the commutation with f₂  \nSubstitute (2) into P∘f₂=f₂∘P:\n\n a(x²+1)= (ax)²+1 = a²x²+1.\n\nComparing coefficients of x² and of the constant term gives\n\n a = a² and a = 1.\n\nThus a=1.\n\nStep 5.  Verification  \nP(x)=x obviously commutes with both f₂ and f₃ and fulfils P(0)=0 and P(2)=2.\n\nConclusion.  The unique polynomial satisfying all requirements is\n\n P(x)=x.\n\n────────────────────────────────────────",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.478336",
        "was_fixed": false,
        "difficulty_analysis": "• Multiple commuting relations:  The candidate must simultaneously belong to the centraliser of two different nonlinear polynomials, forcing an examination of the intersection of two non-trivial algebraic sets.  \n\n• Use of Ritt’s theory:  Identifying the full centralisers of f₂ and f₃ and invoking the uniqueness of polynomial iterates is far subtler than the one-orbit-argument that solves the original problems.\n\n• Degree arithmetic:  Equating two independent expressions for deg P introduces a number-theoretic constraint (2ᵐ=3ⁿ) whose only solution is deg P=1; this step is absent from the original problem.\n\n• Normalising conditions only after deep structure:  The values P(0)=0 and P(2)=2 are used at the very end to eliminate the linear but non-identity possibility P(x)=-x, showing that the boundary data interact with the composition theory rather than giving an immediate telescoping sequence as in the original problem.\n\nOverall, the solver must blend polynomial composition theory, group-like properties of centralisers, elementary number theory, and classical functional equations—several conceptual layers beyond the straightforward “iterate a value until exceeding the degree” technique sufficient for the original problem."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}