{
  "index": "1990-B-1",
  "type": "ANA",
  "tag": [
    "ANA",
    "ALG"
  ],
  "difficulty": "",
  "question": "on the real line such that for all $x$,\n\\[\n(f(x))^2 = \\int_0^x [(f(t))^2 + (f'(t))^2]\\,dt + 1990.\n\\]",
  "solution": "Solution. For a given \\( f \\), the functions on the left- and right-hand sides are equal if and only if their values at 0 are equal, i.e., \\( f(0)^{2}=1990 \\), and their derivatives are equal for all \\( x \\), i.e.,\n\\[\n2 f(x) f^{\\prime}(x)=(f(x))^{2}+\\left(f^{\\prime}(x)\\right)^{2} \\quad \\text { for all } x\n\\]\n\nThe latter condition is equivalent to each of the following: \\( \\left(f(x)-f^{\\prime}(x)\\right)^{2}=0 \\), \\( f^{\\prime}(x)=f(x), f(x)=C e^{x} \\) for some constant \\( C \\). Combining this condition with \\( f(0)^{2}=1990 \\) yields \\( C= \\pm \\sqrt{1990} \\), so the desired functions are \\( f(x)= \\pm \\sqrt{1990} e^{x} \\).",
  "vars": [
    "f",
    "t",
    "x"
  ],
  "params": [
    "C"
  ],
  "sci_consts": [
    "e"
  ],
  "variants": {
    "descriptive_long": {
      "map": {
        "f": "function",
        "t": "timevar",
        "x": "variable",
        "C": "parameter"
      },
      "question": "on the real line such that for all $variable$, \n\\[\n(function(variable))^2 = \\int_0^{variable} [(function(timevar))^2 + (function'(timevar))^2]\\,dtimevar + 1990.\n\\]",
      "solution": "Solution. For a given \\( function \\), the functions on the left- and right-hand sides are equal if and only if their values at 0 are equal, i.e., \\( function(0)^{2}=1990 \\), and their derivatives are equal for all \\( variable \\), i.e.,\n\\[\n2 function(variable) function^{\\prime}(variable)=(function(variable))^{2}+\\left(function^{\\prime}(variable)\\right)^{2} \\quad \\text { for all } variable\n\\]\n\nThe latter condition is equivalent to each of the following: \\( \\left(function(variable)-function^{\\prime}(variable)\\right)^{2}=0 \\), \\( function^{\\prime}(variable)=function(variable), function(variable)=parameter e^{variable} \\) for some constant \\( parameter \\). Combining this condition with \\( function(0)^{2}=1990 \\) yields \\( parameter= \\pm \\sqrt{1990} \\), so the desired functions are \\( function(variable)= \\pm \\sqrt{1990} e^{variable} \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "f": "lanternfish",
        "t": "dandelion",
        "x": "marshmallow",
        "C": "hurricane"
      },
      "question": "on the real line such that for all $marshmallow$,\n\\[\n(lanternfish(marshmallow))^2 = \\int_0^{marshmallow} [(lanternfish(dandelion))^2 + (lanternfish'(dandelion))^2]\\,d dandelion + 1990.\n\\]",
      "solution": "Solution. For a given \\( lanternfish \\), the functions on the left- and right-hand sides are equal if and only if their values at 0 are equal, i.e., \\( lanternfish(0)^{2}=1990 \\), and their derivatives are equal for all \\( marshmallow \\), i.e.,\n\\[\n2\\, lanternfish(marshmallow)\\, lanternfish^{\\prime}(marshmallow)=(lanternfish(marshmallow))^{2}+\\left(lanternfish^{\\prime}(marshmallow)\\right)^{2} \\quad \\text { for all } marshmallow\n\\]\n\nThe latter condition is equivalent to each of the following: \\( \\left(lanternfish(marshmallow)-lanternfish^{\\prime}(marshmallow)\\right)^{2}=0 \\), \\( lanternfish^{\\prime}(marshmallow)=lanternfish(marshmallow), lanternfish(marshmallow)=hurricane e^{marshmallow} \\) for some constant \\( hurricane \\). Combining this condition with \\( lanternfish(0)^{2}=1990 \\) yields \\( hurricane= \\pm \\sqrt{1990} \\), so the desired functions are \\( lanternfish(marshmallow)= \\pm \\sqrt{1990} e^{marshmallow} \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "f": "staticval",
        "t": "spacecoord",
        "x": "momentvar",
        "C": "changeable"
      },
      "question": "on the real line such that for all $momentvar$,\n\\[\n(staticval(momentvar))^2 = \\int_0^{momentvar} [(staticval(spacecoord))^2 + (staticval'(spacecoord))^2]\\,dspacecoord + 1990.\n\\]",
      "solution": "Solution. For a given \\( staticval \\), the functions on the left- and right-hand sides are equal if and only if their values at 0 are equal, i.e., \\( staticval(0)^{2}=1990 \\), and their derivatives are equal for all \\( momentvar \\), i.e.,\n\\[\n2\\,staticval(momentvar)\\,staticval^{\\prime}(momentvar)=(staticval(momentvar))^{2}+\\left(staticval^{\\prime}(momentvar)\\right)^{2} \\quad \\text { for all } momentvar\n\\]\n\nThe latter condition is equivalent to each of the following: \\( \\left(staticval(momentvar)-staticval^{\\prime}(momentvar)\\right)^{2}=0 \\), \\( staticval^{\\prime}(momentvar)=staticval(momentvar),\\ staticval(momentvar)=changeable e^{momentvar} \\) for some constant \\( changeable \\). Combining this condition with \\( staticval(0)^{2}=1990 \\) yields \\( changeable= \\pm \\sqrt{1990} \\), so the desired functions are \\( staticval(momentvar)= \\pm \\sqrt{1990} e^{momentvar} \\)."
    },
    "garbled_string": {
      "map": {
        "f": "qzxwvtnp",
        "t": "hjgrksla",
        "x": "mbcdefghi",
        "C": "plmnkqrst"
      },
      "question": "Problem:\n<<<\non the real line such that for all $mbcdefghi$,\n\\[\n(qzxwvtnp(mbcdefghi))^2 = \\int_0^{mbcdefghi} [(qzxwvtnp(hjgrksla))^2 + (qzxwvtnp'(hjgrksla))^2]\\,d hjgrksla + 1990.\n\\]\n>>>",
      "solution": "Solution:\n<<<\nSolution. For a given \\( qzxwvtnp \\), the functions on the left- and right-hand sides are equal if and only if their values at 0 are equal, i.e., \\( qzxwvtnp(0)^{2}=1990 \\), and their derivatives are equal for all \\( mbcdefghi \\), i.e.,\n\\[\n2 qzxwvtnp(mbcdefghi) qzxwvtnp^{\\prime}(mbcdefghi)=(qzxwvtnp(mbcdefghi))^{2}+\\left(qzxwvtnp^{\\prime}(mbcdefghi)\\right)^{2} \\quad \\text { for all } mbcdefghi\n\\]\n\nThe latter condition is equivalent to each of the following: \\( \\left(qzxwvtnp(mbcdefghi)-qzxwvtnp^{\\prime}(mbcdefghi)\\right)^{2}=0 \\), \\( qzxwvtnp^{\\prime}(mbcdefghi)=qzxwvtnp(mbcdefghi), qzxwvtnp(mbcdefghi)=plmnkqrst e^{mbcdefghi} \\) for some constant \\( plmnkqrst \\). Combining this condition with \\( qzxwvtnp(0)^{2}=1990 \\) yields \\( plmnkqrst= \\pm \\sqrt{1990} \\), so the desired functions are \\( qzxwvtnp(mbcdefghi)= \\pm \\sqrt{1990} e^{mbcdefghi} \\).\n>>>"
    },
    "kernel_variant": {
      "question": "Fix an integer n ≥ 1 and a real exponent p > 1.  Find all continuously-differentiable maps\n\n  F : ℝ → ℝⁿ  \n\nsatisfying the non-local balance law  \n  ‖F(x)‖ᵖ = 2024 + ∫_{-π}^{x} (‖F(t)‖ᵖ + ‖F′(t)‖ᵖ) dt   (†)  \nfor every real x, where ‖·‖ is the Euclidean norm.  \nFor which exponents p does a non-trivial solution exist, and what are all such F?\n\n——————————————————————————————————",
      "solution": "2. ENHANCED SOLUTION (≈ 120 words, original style preserved)  \nDifferentiate (†).  By the Fundamental Theorem of Calculus,\n\n  p‖F‖^{p-2}⟨F,F′⟩ = ‖F‖ᵖ + ‖F′‖ᵖ. (∗)\n\nCauchy gives |⟨F,F′⟩| ≤ ‖F‖‖F′‖, so in (∗) equality of the two Hölder steps must occur; hence F′ is always a non-negative multiple of F.  Write F′ = λF with λ ≥ 0.  Substituting in (∗) yields the scalar equation  \n\n  pλ = 1 + λᵖ. (♣)\n\nFor 1 < p < 2 equation (♣) has no positive root, so (†) has no solution.  \nFor p = 2 it gives λ = 1.  \nFor p > 2 it has exactly two positive roots (one in (0,1), one in (1,∞)).  \nWith any admissible λ, integrate F′ = λF to obtain F(x) = e^{λ(x+π)}C where C is constant.  \nPut x = –π in (†):  ‖C‖ᵖe^{-λpπ} = 2024, i.e. ‖C‖ = 2024^{1/p}e^{λπ}.  \n\nThus (†) has solutions iff p ≥ 2, and they are precisely  \n\n  F(x) = e^{λ(x+π)}C,  λ>0 solving pλ = 1 + λᵖ,  ‖C‖ = 2024^{1/p}e^{λπ}.\n\n——————————————————————————————————",
      "_replacement_note": {
        "replaced_at": "2025-07-05T22:17:12.065388",
        "reason": "Original kernel variant was too easy compared to the original problem"
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}