{
  "index": "1993-A-2",
  "type": "ALG",
  "tag": [
    "ALG",
    "NT"
  ],
  "difficulty": "",
  "question": "such that $x_n^2 - x_{n-1}x_{n+1} = 1$ for $n=1,2,3,\\dots$. Prove there\nexists a real number $a$ such that $x_{n+1} = ax_n - x_{n-1}$ for all $n\n\\geq 1$.",
  "solution": "Solution 1. Since the terms are nonzero, we can define \\( a_{n}=\\left(x_{n+1}+x_{n-1}\\right) / x_{n} \\) for \\( n \\geq 1 \\). It suffices to show \\( a_{1}=a_{2}=\\cdots \\). But \\( x_{n+1}^{2}-x_{n} x_{n+2}=1=x_{n}^{2}-x_{n-1} x_{n+1} \\), so\n\\[\n\\begin{aligned}\nx_{n+1}\\left(x_{n+1}+x_{n-1}\\right) & =x_{n}\\left(x_{n+2}+x_{n}\\right) \\\\\n\\frac{x_{n+1}+x_{n-1}}{x_{n}} & =\\frac{x_{n+2}+x_{n}}{x_{n+1}} \\\\\na_{n} & =a_{n+1}\n\\end{aligned}\n\\]\nsince \\( x_{n}, x_{n+1} \\neq 0 \\). The result follows by induction.\nSolution 2. Given \\( x_{0}, x_{1}, x_{2} \\), let \\( a=\\left(x_{0}+x_{2}\\right) / x_{1} \\). From \\( x_{1}^{2}-x_{0} x_{2}=1 \\), we get \\( x_{1}^{2}-a x_{1} x_{0}+x_{0}^{2}=1 \\).\n\nNow\n\\[\n\\begin{aligned}\nx_{2}^{2}-x_{1}\\left(a x_{2}-x_{1}\\right) & =\\left(a x_{1}-x_{0}\\right)^{2}-x_{1}\\left(a\\left(a x_{1}-x_{0}\\right)-x_{1}\\right) \\\\\n& =x_{1}^{2}-a x_{1} x_{0}+x_{0}^{2}=1,\n\\end{aligned}\n\\]\nso \\( x_{3}=a x_{2}-x_{1} \\).\nBy essentially the same algebra (and a simple induction), \\( x_{n+1}=a x_{n}-x_{n-1} \\) for all \\( n \\geq 1 \\).\n\nSolution 3. Let \\( a=\\left(x_{0}+x_{2}\\right) / x_{1} \\). Consider a sequence defined by \\( y_{n+1}-a y_{n}+ \\) \\( y_{n-1}=0 \\) where \\( y_{i}=x_{i} \\) for \\( i=0,1,2 \\); we wish to show that \\( y_{i}=x_{i} \\) for all nonnegative integers \\( i \\).\n\nIf \\( a \\neq \\pm 2 \\), then by the theory of linear recursive sequences (discussed after the solution to 1988A5), the general solution is\n\\[\ny_{n}=A r^{n}+B s^{n}\n\\]\nwhere \\( A \\) and \\( B \\) are constants, and \\( r \\) and \\( s \\) are the roots of the characteristic equation \\( t^{2}-a t+1=0 \\). Then \\( r s=1 \\), and \\( (r-s)^{2}=a^{2}-4 \\) by the quadratic formula or by the identity \\( (r-s)^{2}=(r+s)^{2}-4 r s \\). Now\n\\[\n\\begin{aligned}\ny_{n}^{2}-y_{n+1} y_{n-1} & =\\left(A r^{n}+B s^{n}\\right)^{2}-\\left(A r^{n-1}+B s^{n-1}\\right)\\left(A r^{n+1}+B s^{n+1}\\right) \\\\\n& =-A B(r s)^{n-1}(r-s)^{2} \\\\\n& =-A B\\left(a^{2}-4\\right)\n\\end{aligned}\n\\]\nwhich is independent of \\( n \\). Thus \\( y_{n}^{2}-y_{n+1} y_{n-1}=y_{1}^{2}-y_{2} y_{0}=1 \\) for all \\( n \\), so \\( y_{n}=x_{n} \\) by induction on \\( n \\).\n\nIf \\( a= \\pm 2 \\), then the general solution is\n\\[\ny_{n}=(A+B n)( \\pm 1)^{n}\n\\]\n\nThen\n\\[\ny_{n}^{2}-y_{n+1} y_{n-1}=B^{2}\n\\]\nwhich is independent of \\( n \\). Hence as before, \\( y_{n}^{2}-y_{n+1} y_{n-1}=y_{1}^{2}-y_{2} y_{0}=1 \\) for all \\( n \\), so \\( y_{n}=x_{n} \\) for all \\( n \\) as well.\n\nSolution 4. For all \\( n \\),\n\\[\n\\begin{aligned}\n\\operatorname{det}\\left(\\begin{array}{cc}\nx_{n-1}+x_{n+1} & x_{n}+x_{n+2} \\\\\nx_{n} & x_{n+1}\n\\end{array}\\right) & =\\operatorname{det}\\left(\\begin{array}{cc}\nx_{n-1} & x_{n} \\\\\nx_{n} & x_{n+1}\n\\end{array}\\right)+\\operatorname{det}\\left(\\begin{array}{cc}\nx_{n+1} & x_{n+2} \\\\\nx_{n} & x_{n+1}\n\\end{array}\\right) \\\\\n& =-1+1=0\n\\end{aligned}\n\\]\n\nThus \\( \\left(x_{n-1}+x_{n+1}, x_{n}+x_{n+2}\\right)=a_{n}\\left(x_{n}, x_{n+1}\\right) \\) for some scalar \\( a_{n} \\). Hence\n\\[\n\\frac{x_{n+1}+x_{n-1}}{x_{n}}=\\frac{x_{n+2}+x_{n}}{x_{n+1}}\n\\]\nfor all \\( n \\geq 1 \\), since \\( x_{n}, x_{n+1} \\neq 0 \\), so we are done by induction.",
  "vars": [
    "x_n",
    "x_n-1",
    "x_n+1",
    "x_n+2",
    "x_0",
    "x_1",
    "x_2",
    "x_3",
    "y_n",
    "y_n+1",
    "y_n-1",
    "y_0",
    "y_1",
    "y_2",
    "a_n",
    "n",
    "i"
  ],
  "params": [
    "a",
    "A",
    "B",
    "r",
    "s",
    "t"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x_n": "seqvar",
        "x_n-1": "seqprev",
        "x_n+1": "seqnext",
        "x_n+2": "seqnexttwo",
        "x_0": "seqinitzero",
        "x_1": "seqinitone",
        "x_2": "seqinittwo",
        "x_3": "seqinitthree",
        "y_n": "altseqvar",
        "y_n+1": "altseqnext",
        "y_n-1": "altseqprev",
        "y_0": "altseqzero",
        "y_1": "altseqone",
        "y_2": "altseqtwo",
        "a_n": "coeffvar",
        "n": "indexvar",
        "i": "indexalt",
        "a": "coefconst",
        "A": "solconsta",
        "B": "solconstb",
        "r": "rootone",
        "s": "roottwo",
        "t": "charvar"
      },
      "question": "such that $seqvar^2 - seqprev seqnext = 1$ for $indexvar=1,2,3,\\dots$. Prove there exists a real number $coefconst$ such that $seqnext = coefconst seqvar - seqprev$ for all $indexvar \\geq 1$.",
      "solution": "Solution 1. Since the terms are nonzero, we can define \\( coeffvar=\\left(seqnext+seqprev\\right) / seqvar \\) for \\( indexvar \\geq 1 \\). It suffices to show \\( a_{1}=a_{2}=\\cdots \\). But \\( seqnext^{2}-seqvar seqnexttwo=1=seqvar^{2}-seqprev seqnext \\), so\n\\[\n\\begin{aligned}\nseqnext\\left(seqnext+seqprev\\right) & =seqvar\\left(seqnexttwo+seqvar\\right) \\\\\n\\frac{seqnext+seqprev}{seqvar} & =\\frac{seqnexttwo+seqvar}{seqnext} \\\\\ncoeffvar & =a_{indexvar+1}\n\\end{aligned}\n\\]\nsince \\( seqvar, seqnext \\neq 0 \\). The result follows by induction.\n\nSolution 2. Given \\( seqinitzero, seqinitone, seqinittwo \\), let \\( coefconst=\\left(seqinitzero+seqinittwo\\right) / seqinitone \\). From \\( seqinitone^{2}-seqinitzero seqinittwo=1 \\), we get \\( seqinitone^{2}-coefconst seqinitone seqinitzero+seqinitzero^{2}=1 \\).\n\nNow\n\\[\n\\begin{aligned}\nseqinittwo^{2}-seqinitone\\left(coefconst seqinittwo-seqinitone\\right) & =\\left(coefconst seqinitone-seqinitzero\\right)^{2}-seqinitone\\left(coefconst\\left(coefconst seqinitone-seqinitzero\\right)-seqinitone\\right) \\\\\n& =seqinitone^{2}-coefconst seqinitone seqinitzero+seqinitzero^{2}=1,\n\\end{aligned}\n\\]\nso \\( seqinitthree=coefconst seqinittwo-seqinitone \\).\nBy essentially the same algebra (and a simple induction), \\( seqnext=coefconst seqvar-seqprev \\) for all \\( indexvar \\geq 1 \\).\n\nSolution 3. Let \\( coefconst=\\left(seqinitzero+seqinittwo\\right) / seqinitone \\). Consider a sequence defined by \\( altseqnext-coefconst altseqvar+altseqprev=0 \\) where \\( y_{indexalt}=x_{indexalt} \\) for \\( indexalt=0,1,2 \\); we wish to show that \\( y_{indexalt}=x_{indexalt} \\) for all nonnegative integers \\( indexalt \\).\n\nIf \\( coefconst \\neq \\pm 2 \\), then by the theory of linear recursive sequences (discussed after the solution to 1988A5), the general solution is\n\\[\naltseqvar=solconsta rootone^{indexvar}+solconstb roottwo^{indexvar}\n\\]\nwhere \\( solconsta \\) and \\( solconstb \\) are constants, and \\( rootone \\) and \\( roottwo \\) are the roots of the characteristic equation \\( charvar^{2}-coefconst charvar+1=0 \\). Then \\( rootone roottwo=1 \\), and \\( (rootone-roottwo)^{2}=coefconst^{2}-4 \\) by the quadratic formula or by the identity \\( (rootone-roottwo)^{2}=(rootone+roottwo)^{2}-4 rootone roottwo \\). Now\n\\[\n\\begin{aligned}\naltseqvar^{2}-altseqnext altseqprev & =\\left(solconsta rootone^{indexvar}+solconstb roottwo^{indexvar}\\right)^{2}-\\left(solconsta rootone^{indexvar-1}+solconstb roottwo^{indexvar-1}\\right)\\left(solconsta rootone^{indexvar+1}+solconstb roottwo^{indexvar+1}\\right) \\\\\n& =-solconsta solconstb(rootone roottwo)^{indexvar-1}(rootone-roottwo)^{2} \\\\\n& =-solconsta solconstb\\left(coefconst^{2}-4\\right)\n\\end{aligned}\n\\]\nwhich is independent of \\( indexvar \\). Thus \\( altseqvar^{2}-altseqnext altseqprev=altseqone^{2}-altseqtwo altseqzero=1 \\) for all \\( indexvar \\), so \\( altseqvar=seqvar \\) by induction on \\( indexvar \\).\n\nIf \\( coefconst= \\pm 2 \\), then the general solution is\n\\[\naltseqvar=(solconsta+solconstb indexvar)( \\pm 1)^{indexvar}\n\\]\n\nThen\n\\[\naltseqvar^{2}-altseqnext altseqprev=solconstb^{2}\n\\]\nwhich is independent of \\( indexvar \\). Hence as before, \\( altseqvar^{2}-altseqnext altseqprev=altseqone^{2}-altseqtwo altseqzero=1 \\) for all \\( indexvar \\), so \\( altseqvar=seqvar \\) for all \\( indexvar \\) as well.\n\nSolution 4. For all \\( indexvar \\),\n\\[\n\\begin{aligned}\n\\operatorname{det}\\left(\\begin{array}{cc}\nseqprev+seqnext & seqvar+seqnexttwo \\\\\nseqvar & seqnext\n\\end{array}\\right) & =\\operatorname{det}\\left(\\begin{array}{cc}\nseqprev & seqvar \\\\\nseqvar & seqnext\n\\end{array}\\right)+\\operatorname{det}\\left(\\begin{array}{cc}\nseqnext & seqnexttwo \\\\\nseqvar & seqnext\n\\end{array}\\right) \\\\\n& =-1+1=0\n\\end{aligned}\n\\]\n\nThus \\( \\left(seqprev+seqnext, seqvar+seqnexttwo\\right)=a_{indexvar}\\left(seqvar, seqnext\\right) \\) for some scalar \\( a_{indexvar} \\). Hence\n\\[\n\\frac{seqnext+seqprev}{seqvar}=\\frac{seqnexttwo+seqvar}{seqnext}\n\\]\nfor all \\( indexvar \\geq 1 \\), since \\( seqvar, seqnext \\neq 0 \\), so we are done by induction."
    },
    "descriptive_long_confusing": {
      "map": {
        "x_n": "marigold",
        "x_n-1": "gingercat",
        "x_n+1": "candlewax",
        "x_n+2": "riverbank",
        "x_0": "birdcage",
        "x_1": "toothpick",
        "x_2": "snowflake",
        "x_3": "whistleup",
        "y_n": "paintdrop",
        "y_n+1": "moonlight",
        "y_n-1": "evercloud",
        "y_0": "sandstorm",
        "y_1": "driftwood",
        "y_2": "oatmealjar",
        "a_n": "pennywise",
        "n": "lanterns",
        "i": "quillrose",
        "a": "sailboat",
        "A": "lemonseed",
        "B": "sugarcube",
        "r": "pathfinder",
        "s": "mossyrock",
        "t": "zephyrant"
      },
      "question": "such that $marigold^{2} - gingercat candlewax = 1$ for $lanterns=1,2,3,\\dots$. Prove there\nexists a real number $sailboat$ such that $candlewax = sailboat marigold - gingercat$ for all $lanterns\n\\geq 1$.",
      "solution": "Solution 1. Since the terms are nonzero, we can define \\( pennywise=\\left(candlewax+gingercat\\right) / marigold \\) for \\( lanterns \\geq 1 \\). It suffices to show \\( pennywise=pennywise=\\cdots \\). But \\( candlewax^{2}-marigold riverbank=1=marigold^{2}-gingercat candlewax \\), so\n\\[\n\\begin{aligned}\ncandlewax\\left(candlewax+gingercat\\right) & =marigold\\left(riverbank+marigold\\right) \\\\\n\\frac{candlewax+gingercat}{marigold} & =\\frac{riverbank+marigold}{candlewax} \\\\\npennywise & =pennywise\n\\end{aligned}\n\\]\nsince \\( marigold, candlewax \\neq 0 \\). The result follows by induction.\n\nSolution 2. Given \\( birdcage, toothpick, snowflake \\), let \\( sailboat=\\left(birdcage+snowflake\\right) / toothpick \\). From \\( toothpick^{2}-birdcage snowflake=1 \\), we get \\( toothpick^{2}-sailboat toothpick birdcage+birdcage^{2}=1 \\).\n\nNow\n\\[\n\\begin{aligned}\nsnowflake^{2}-toothpick\\left(sailboat snowflake-toothpick\\right) & =\\left(sailboat toothpick-birdcage\\right)^{2}-toothpick\\left(sailboat\\left(sailboat toothpick-birdcage\\right)-toothpick\\right) \\\\\n& =toothpick^{2}-sailboat toothpick birdcage+birdcage^{2}=1,\n\\end{aligned}\n\\]\nso \\( whistleup=sailboat snowflake-toothpick \\).\nBy essentially the same algebra (and a simple induction), \\( candlewax=sailboat marigold-gingercat \\) for all \\( lanterns \\geq 1 \\).\n\nSolution 3. Let \\( sailboat=\\left(birdcage+snowflake\\right) / toothpick \\). Consider a sequence defined by \\( moonlight - sailboat paintdrop + evercloud=0 \\) where \\( y_{quillrose}=x_{quillrose} \\) for \\( quillrose=0,1,2 \\); we wish to show that \\( y_{quillrose}=x_{quillrose} \\) for all nonnegative integers \\( quillrose \\).\n\nIf \\( sailboat \\neq \\pm 2 \\), then by the theory of linear recursive sequences (discussed after the solution to 1988A5), the general solution is\n\\[\npaintdrop = lemonseed\\, pathfinder^{lanterns}+sugarcube\\, mossyrock^{lanterns}\n\\]\nwhere \\( lemonseed \\) and \\( sugarcube \\) are constants, and \\( pathfinder \\) and \\( mossyrock \\) are the roots of the characteristic equation \\( zephyrant^{2}-sailboat\\, zephyrant+1=0 \\). Then \\( pathfinder\\, mossyrock=1 \\), and \\( (pathfinder-mossyrock)^{2}=sailboat^{2}-4 \\) by the quadratic formula or by the identity \\( (pathfinder-mossyrock)^{2}=(pathfinder+mossyrock)^{2}-4\\, pathfinder\\, mossyrock \\). Now\n\\[\n\\begin{aligned}\npaintdrop^{2}-moonlight\\, evercloud & =\\left(lemonseed\\, pathfinder^{lanterns}+sugarcube\\, mossyrock^{lanterns}\\right)^{2}-\\left(lemonseed\\, pathfinder^{lanterns-1}+sugarcube\\, mossyrock^{lanterns-1}\\right)\\left(lemonseed\\, pathfinder^{lanterns+1}+sugarcube\\, mossyrock^{lanterns+1}\\right) \\\\\n& =-lemonseed\\, sugarcube(pathfinder\\, mossyrock)^{lanterns-1}(pathfinder-mossyrock)^{2} \\\\\n& =-lemonseed\\, sugarcube\\left(sailboat^{2}-4\\right)\n\\end{aligned}\n\\]\nwhich is independent of \\( lanterns \\). Thus \\( paintdrop^{2}-moonlight\\, evercloud=driftwood^{2}-oatmealjar\\, sandstorm=1 \\) for all \\( lanterns \\), so \\( paintdrop=marigold \\) by induction on \\( lanterns \\).\n\nIf \\( sailboat= \\pm 2 \\), then the general solution is\n\\[\npaintdrop=(lemonseed+sugarcube\\, lanterns)( \\pm 1)^{lanterns}\n\\]\n\nThen\n\\[\npaintdrop^{2}-moonlight\\, evercloud=sugarcube^{2}\n\\]\nwhich is independent of \\( lanterns \\). Hence as before, \\( paintdrop^{2}-moonlight\\, evercloud=driftwood^{2}-oatmealjar\\, sandstorm=1 \\) for all \\( lanterns \\), so \\( paintdrop=marigold \\) for all \\( lanterns \\) as well.\n\nSolution 4. For all \\( lanterns \\),\n\\[\n\\begin{aligned}\n\\operatorname{det}\\left(\\begin{array}{cc}\ngingercat+candlewax & marigold+riverbank \\\\\nmarigold & candlewax\n\\end{array}\\right) & =\\operatorname{det}\\left(\\begin{array}{cc}\ngingercat & marigold \\\\\nmarigold & candlewax\n\\end{array}\\right)+\\operatorname{det}\\left(\\begin{array}{cc}\ncandlewax & riverbank \\\\\nmarigold & candlewax\n\\end{array}\\right) \\\\\n& =-1+1=0\n\\end{aligned}\n\\]\n\nThus \\( \\left(gingercat+candlewax, marigold+riverbank\\right)=pennywise\\left(marigold, candlewax\\right) \\) for some scalar \\( pennywise \\). Hence\n\\[\n\\frac{candlewax+gingercat}{marigold}=\\frac{riverbank+marigold}{candlewax}\n\\]\nfor all \\( lanterns \\geq 1 \\), since \\( marigold, candlewax \\neq 0 \\), so we are done by induction."
    },
    "descriptive_long_misleading": {
      "map": {
        "x_n": "constantseq",
        "x_n-1": "constantseqprev",
        "x_n+1": "constantseqnext",
        "x_n+2": "constantseqfuture",
        "x_0": "constantseqzero",
        "x_1": "constantseqone",
        "x_2": "constantseqtwo",
        "x_3": "constantseqthree",
        "y_n": "stationarystream",
        "y_n+1": "stationarystreamnext",
        "y_n-1": "stationarystreamprev",
        "y_0": "stationarystreamzero",
        "y_1": "stationarystreamone",
        "y_2": "stationarystreamtwo",
        "a_n": "steadydrift",
        "n": "timeless",
        "i": "motionless",
        "a": "variable",
        "A": "flexible",
        "B": "shifting",
        "r": "branchless",
        "s": "stemless",
        "t": "staticval"
      },
      "question": "such that $constantseq^2 - constantseqprev constantseqnext = 1$ for $timeless=1,2,3,\\dots$. Prove there\nexists a real number $variable$ such that $constantseqnext = variable constantseq - constantseqprev$ for all $timeless\n\\geq 1$.",
      "solution": "Solution 1. Since the terms are nonzero, we can define \\( steadydrift_{timeless}=\\left(constantseqnext+constantseqprev\\right) / constantseq \\) for \\( timeless \\geq 1 \\). It suffices to show \\( steadydrift_{1}=steadydrift_{2}=\\cdots \\). But \\( constantseqnext^{2}-constantseq constantseqfuture=1=constantseq^{2}-constantseqprev constantseqnext \\), so\n\\[\n\\begin{aligned}\nconstantseqnext\\left(constantseqnext+constantseqprev\\right) & =constantseq\\left(constantseqfuture+constantseq\\right) \\\\\n\\frac{constantseqnext+constantseqprev}{constantseq} & =\\frac{constantseqfuture+constantseq}{constantseqnext} \\\\\nsteadydrift_{timeless} & =steadydrift_{timeless+1}\n\\end{aligned}\n\\]\nsince \\( constantseq, constantseqnext \\neq 0 \\). The result follows by induction.\n\nSolution 2. Given \\( constantseqzero, constantseqone, constantseqtwo \\), let \\( variable=\\left(constantseqzero+constantseqtwo\\right) / constantseqone \\). From \\( constantseqone^{2}-constantseqzero constantseqtwo=1 \\), we get \\( constantseqone^{2}-variable constantseqone constantseqzero+constantseqzero^{2}=1 \\).\n\nNow\n\\[\n\\begin{aligned}\nconstantseqtwo^{2}-constantseqone\\left(variable constantseqtwo-constantseqone\\right) & =\\left(variable constantseqone-constantseqzero\\right)^{2}-constantseqone\\left(variable\\left(variable constantseqone-constantseqzero\\right)-constantseqone\\right) \\\\\n& =constantseqone^{2}-variable constantseqone constantseqzero+constantseqzero^{2}=1,\n\\end{aligned}\n\\]\nso \\( constantseqthree=variable constantseqtwo-constantseqone \\).\nBy essentially the same algebra (and a simple induction), \\( constantseqnext=variable constantseq-constantseqprev \\) for all \\( timeless \\geq 1 \\).\n\nSolution 3. Let \\( variable=\\left(constantseqzero+constantseqtwo\\right) / constantseqone \\). Consider a sequence defined by \\( stationarystreamnext-variable\\;stationarystream+stationarystreamprev=0 \\) where \\( stationarystream_{motionless}=constantseq_{motionless} \\) for \\( motionless=0,1,2 \\); we wish to show that \\( stationarystream_{motionless}=constantseq_{motionless} \\) for all nonnegative integers \\( motionless \\).\n\nIf \\( variable \\neq \\pm 2 \\), then by the theory of linear recursive sequences (discussed after the solution to 1988A5), the general solution is\n\\[\nstationarystream=flexible\\, branchless^{timeless}+shifting\\, stemless^{timeless}\n\\]\nwhere \\( flexible \\) and \\( shifting \\) are constants, and \\( branchless \\) and \\( stemless \\) are the roots of the characteristic equation \\( staticval^{2}-variable\\, staticval+1=0 \\). Then \\( branchless\\, stemless=1 \\), and \\( (branchless-stemless)^{2}=variable^{2}-4 \\) by the quadratic formula or by the identity \\( (branchless-stemless)^{2}=(branchless+stemless)^{2}-4\\, branchless\\, stemless \\). Now\n\\[\n\\begin{aligned}\nstationarystream^{2}-stationarystreamnext\\, stationarystreamprev & =\\left(flexible\\, branchless^{timeless}+shifting\\, stemless^{timeless}\\right)^{2}-\\left(flexible\\, branchless^{timeless-1}+shifting\\, stemless^{timeless-1}\\right)\\left(flexible\\, branchless^{timeless+1}+shifting\\, stemless^{timeless+1}\\right) \\\\\n& =-flexible\\, shifting(branchless\\, stemless)^{timeless-1}(branchless-stemless)^{2} \\\\\n& =-flexible\\, shifting\\left(variable^{2}-4\\right)\n\\end{aligned}\n\\]\nwhich is independent of \\( timeless \\). Thus \\( stationarystream^{2}-stationarystreamnext\\, stationarystreamprev=stationarystreamone^{2}-stationarystreamtwo\\, stationarystreamzero=1 \\) for all \\( timeless \\), so \\( stationarystream=constantseq \\) by induction on \\( timeless \\).\n\nIf \\( variable= \\pm 2 \\), then the general solution is\n\\[\nstationarystream=(flexible+shifting\\, timeless)( \\pm 1)^{timeless}\n\\]\n\nThen\n\\[\nstationarystream^{2}-stationarystreamnext\\, stationarystreamprev=shifting^{2}\n\\]\nwhich is independent of \\( timeless \\). Hence as before, \\( stationarystream^{2}-stationarystreamnext\\, stationarystreamprev=stationarystreamone^{2}-stationarystreamtwo\\, stationarystreamzero=1 \\) for all \\( timeless \\), so \\( stationarystream=constantseq \\) for all \\( timeless \\) as well.\n\nSolution 4. For all \\( timeless \\),\n\\[\n\\begin{aligned}\n\\operatorname{det}\\left(\\begin{array}{cc}\nconstantseqprev+constantseqnext & constantseq+constantseqfuture \\\\\nconstantseq & constantseqnext\n\\end{array}\\right) & =\\operatorname{det}\\left(\\begin{array}{cc}\nconstantseqprev & constantseq \\\\\nconstantseq & constantseqnext\n\\end{array}\\right)+\\operatorname{det}\\left(\\begin{array}{cc}\nconstantseqnext & constantseqfuture \\\\\nconstantseq & constantseqnext\n\\end{array}\\right) \\\\\n& =-1+1=0\n\\end{aligned}\n\\]\n\nThus \\( \\left(constantseqprev+constantseqnext, constantseq+constantseqfuture\\right)=steadydrift_{timeless}\\left(constantseq, constantseqnext\\right) \\) for some scalar \\( steadydrift_{timeless} \\). Hence\n\\[\n\\frac{constantseqnext+constantseqprev}{constantseq}=\\frac{constantseqfuture+constantseq}{constantseqnext}\n\\]\nfor all \\( timeless \\geq 1 \\), since \\( constantseq, constantseqnext \\neq 0 \\), so we are done by induction."
    },
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        "t": "nvasdukj"
      },
      "question": "such that $qzxwvtnp^{2} - hjgrksla bdfmjchy = 1$ for $clbrxwno=1,2,3,\\dots$. Prove there\nexists a real number $ujmtokle$ such that $bdfmjchy = ujmtokle qzxwvtnp - hjgrksla$ for all $clbrxwno\n\\geq 1$.",
      "solution": "Solution 1. Since the terms are nonzero, we can define \\( ytrnlmso=\\left(bdfmjchy+hjgrksla\\right) / qzxwvtnp \\) for \\( clbrxwno \\geq 1 \\). It suffices to show \\( a_{1}=a_{2}=\\cdots \\). But \\( bdfmjchy^{2}-qzxwvtnp plxrogew=1=qzxwvtnp^{2}-hjgrksla bdfmjchy \\), so\n\\[\n\\begin{aligned}\nbdfmjchy\\left(bdfmjchy+hjgrksla\\right) & =qzxwvtnp\\left(plxrogew+qzxwvtnp\\right) \\\\\n\\frac{bdfmjchy+hjgrksla}{qzxwvtnp} & =\\frac{plxrogew+qzxwvtnp}{bdfmjchy} \\\\\nytrnlmso & =a_{n+1}\n\\end{aligned}\n\\]\nsince \\( qzxwvtnp, bdfmjchy \\neq 0 \\). The result follows by induction.\n\nSolution 2. Given \\( kdasfjhg, ovmxpqaz, sdyclmab \\), let \\( ujmtokle=\\left(kdasfjhg+sdyclmab\\right) / ovmxpqaz \\). From \\( ovmxpqaz^{2}-kdasfjhg sdyclmab=1 \\), we get \\( ovmxpqaz^{2}-ujmtokle ovmxpqaz kdasfjhg+kdasfjhg^{2}=1 \\).\n\nNow\n\\[\n\\begin{aligned}\nsdyclmab^{2}-ovmxpqaz\\left(ujmtokle sdyclmab-ovmxpqaz\\right) & =\\left(ujmtokle ovmxpqaz-kdasfjhg\\right)^{2}-ovmxpqaz\\left(ujmtokle\\left(ujmtokle ovmxpqaz-kdasfjhg\\right)-ovmxpqaz\\right) \\\\\n& =ovmxpqaz^{2}-ujmtokle ovmxpqaz kdasfjhg+kdasfjhg^{2}=1,\n\\end{aligned}\n\\]\nso \\( trqnezip=ujmtokle sdyclmab-ovmxpqaz \\).\nBy essentially the same algebra (and a simple induction), \\( bdfmjchy=ujmtokle qzxwvtnp-hjgrksla \\) for all \\( clbrxwno \\geq 1 \\).\n\nSolution 3. Let \\( ujmtokle=\\left(kdasfjhg+sdyclmab\\right) / ovmxpqaz \\). Consider a sequence defined by \\( zxlpqywe-ujmtokle vghkpsam+ mncovrut=0 \\) where \\( y_{dfxqgpei}=x_{dfxqgpei} \\) for \\( dfxqgpei=0,1,2 \\); we wish to show that \\( y_{dfxqgpei}=x_{dfxqgpei} \\) for all nonnegative integers \\( dfxqgpei \\).\n\nIf \\( ujmtokle \\neq \\pm 2 \\), then by the theory of linear recursive sequences (discussed after the solution to 1988A5), the general solution is\n\\[\nvghkpsam=prvxdhgw xdazlqwe^{clbrxwno}+oltmcakd frykcmsp^{clbrxwno}\n\\]\nwhere \\( prvxdhgw \\) and \\( oltmcakd \\) are constants, and \\( xdazlqwe \\) and \\( frykcmsp \\) are the roots of the characteristic equation \\( nvasdukj^{2}-ujmtokle nvasdukj+1=0 \\). Then \\( xdazlqwe frykcmsp=1 \\), and \\( (xdazlqwe-frykcmsp)^{2}=ujmtokle^{2}-4 \\) by the quadratic formula or by the identity \\( (xdazlqwe-frykcmsp)^{2}=(xdazlqwe+frykcmsp)^{2}-4 xdazlqwe frykcmsp \\). Now\n\\[\n\\begin{aligned}\nvghkpsam^{2}-zxlpqywe mncovrut & =\\left(prvxdhgw xdazlqwe^{clbrxwno}+oltmcakd frykcmsp^{clbrxwno}\\right)^{2}-\\left(prvxdhgw xdazlqwe^{clbrxwno-1}+oltmcakd frykcmsp^{clbrxwno-1}\\right)\\left(prvxdhgw xdazlqwe^{clbrxwno+1}+oltmcakd frykcmsp^{clbrxwno+1}\\right) \\\\\n& =-prvxdhgw oltmcakd(xdazlqwe frykcmsp)^{clbrxwno-1}(xdazlqwe-frykcmsp)^{2} \\\\\n& =-prvxdhgw oltmcakd\\left(ujmtokle^{2}-4\\right)\n\\end{aligned}\n\\]\nwhich is independent of \\( clbrxwno \\). Thus \\( vghkpsam^{2}-zxlpqywe mncovrut=lskvprte^{2}-gbeixkuo wqndfjaz=1 \\) for all \\( clbrxwno \\), so \\( vghkpsam=qzxwvtnp \\) by induction on \\( clbrxwno \\).\n\nIf \\( ujmtokle= \\pm 2 \\), then the general solution is\n\\[\nvghkpsam=(prvxdhgw+oltmcakd clbrxwno)( \\pm 1)^{clbrxwno}\n\\]\n\nThen\n\\[\nvghkpsam^{2}-zxlpqywe mncovrut=oltmcakd^{2}\n\\]\nwhich is independent of \\( clbrxwno \\). Hence as before, \\( vghkpsam^{2}-zxlpqywe mncovrut=lskvprte^{2}-gbeixkuo wqndfjaz=1 \\) for all \\( clbrxwno \\), so \\( vghkpsam=qzxwvtnp \\) for all \\( clbrxwno \\) as well.\n\nSolution 4. For all \\( clbrxwno \\),\n\\[\n\\begin{aligned}\n\\operatorname{det}\\left(\\begin{array}{cc}\nhjgrksla+bdfmjchy & qzxwvtnp+plxrogew \\\\\nqzxwvtnp & bdfmjchy\n\\end{array}\\right) & =\\operatorname{det}\\left(\\begin{array}{cc}\nhjgrksla & qzxwvtnp \\\\\nqzxwvtnp & bdfmjchy\n\\end{array}\\right)+\\operatorname{det}\\left(\\begin{array}{cc}\nbdfmjchy & plxrogew \\\\\nqzxwvtnp & bdfmjchy\n\\end{array}\\right) \\\\\n& =-1+1=0\n\\end{aligned}\n\\]\n\nThus \\( \\left(hjgrksla+bdfmjchy, qzxwvtnp+plxrogew\\right)=ytrnlmso\\left(qzxwvtnp, bdfmjchy\\right) \\) for some scalar \\( ytrnlmso \\). Hence\n\\[\n\\frac{bdfmjchy+hjgrksla}{qzxwvtnp}=\\frac{plxrogew+qzxwvtnp}{bdfmjchy}\n\\]\nfor all \\( clbrxwno \\geq 1 \\), since \\( qzxwvtnp, bdfmjchy \\neq 0 \\), so we are done by induction."
    },
    "kernel_variant": {
      "question": "Let $(x_n)_{n\\in\\mathbb Z}$ be a doubly–infinite sequence of non–zero complex numbers and form the bi-infinite Hankel matrix  \n\\[H=\\bigl(x_{\\,i+j}\\bigr)_{\\,i,j\\in\\mathbb Z} .\\]\n\nFor every $n\\in\\mathbb Z$ set  \n\\[\n\\Delta _n \\;=\\;\n\\det\\!\\begin{pmatrix}\nx_{\\,n-1}&x_{\\,n}&x_{\\,n+1}\\\\\nx_{\\,n  }&x_{\\,n+1}&x_{\\,n+2}\\\\\nx_{\\,n+1}&x_{\\,n+2}&x_{\\,n+3}\n\\end{pmatrix},\n\\qquad   \nG_n \\;=\\;\n\\det\\!\\begin{pmatrix}\nx_{\\,n}&x_{\\,n+1}\\\\\nx_{\\,n+1}&x_{\\,n+2}\n\\end{pmatrix}\n      =x_{\\,n}x_{\\,n+2}-x_{\\,n+1}^{2}.\n\\]\n\nAssume  \n\n(A) every $3\\times3$ Hankel minor of $H$ vanishes (i.e. $\\Delta _n=0$ for all $n$);  \n\n(B) no centred $2\\times2$ minor vanishes: $G_n\\neq0$ for every $n$.\n\nBecause of (B) the quotient  \n\\[\n\\rho_n:=\\frac{G_{\\,n+1}}{G_{\\,n}}\\qquad(n\\in\\mathbb Z)\n\\]\nis well–defined.\n\n1.  Show that the numbers $\\rho_n$ are independent of $n$; denote their common non-zero value by $\\rho$.\n\n2.  Prove that $\\operatorname{rank}H\\le2$ and hence that there exist UNIQUE complex constants $a,q$ such that  \n\\[\n\\boxed{\\;x_{\\,n+2}=a\\,x_{\\,n+1}+q\\,x_{\\,n}\\qquad(n\\in\\mathbb Z). \\;}\n\\]\n\n3.  Prove the relation $q=-\\rho$ and deduce that $(G_n)$ is the (possibly null) geometric progression $G_n=G_0\\,(-q)^{\\,n}$.\n\n4.  Specialise: if, in addition, the minors $G_n$ are all equal to the same non–zero constant $D$, then necessarily $q=-1$ and the recurrence becomes  \n\\[\nx_{\\,n+2}=a\\,x_{\\,n+1}-x_{\\,n}\\quad(n\\in\\mathbb Z).\n\\]\n\n5.  Converse.  Show that every bi-infinite sequence of non-zero complex numbers satisfying  \n\\[\nx_{\\,n+2}=a\\,x_{\\,n+1}+q\\,x_{\\,n}\\qquad(q\\neq0)\n\\]\nhas $\\Delta _n\\equiv0$ and $G_{n+1}=-q\\,G_n$ for all $n$ (hence either $G_n\\equiv0$ or $G_n$ is a non-zero geometric progression).\n\n————————————————————————————————————————",
      "solution": "We treat the five items in order.  Throughout, $n$ is an arbitrary integer.\n\n1.  Constantness of the ratios $\\rho_n$\n   ------------------------------------\n\n   Expanding $\\Delta _n$ along the first row gives\n   \\[\n   \\Delta _n\n      =x_{\\,n-1}\\bigl(x_{\\,n+1}x_{\\,n+3}-x_{\\,n+2}^{2}\\bigr)\n       -x_{\\,n  }\\bigl(x_{\\,n  }x_{\\,n+3}-x_{\\,n+1}x_{\\,n+2}\\bigr)\n       +x_{\\,n+1}\\bigl(x_{\\,n  }x_{\\,n+2}-x_{\\,n+1}^{2}\\bigr).\n   \\]\n   Recognising the $2\\times2$ determinants $G_{n-1},G_n,G_{n+1}$ we obtain\n   \\[\n      x_{\\,n+1}\\,\\Delta _n =G_{\\,n-1}G_{\\,n+1}-G_{\\,n}^{2}. \\tag{1}\n   \\]\n   Equation (1) is the $k=3$ Jacobi (Sylvester) identity for contiguous minors of a Hankel matrix.\n\n   Hypothesis (A) gives $\\Delta _n=0$, and because $x_{\\,n+1}\\neq0$ we have\n   \\[\n     \\boxed{\\,G_{\\,n-1}G_{\\,n+1}=G_{\\,n}^{2}\\,.} \\tag{2}\n   \\]\n   Divide (2) by the non–zero product $G_{\\,n-1}G_{\\,n}$:\n   \\[\n      \\frac{G_{\\,n+1}}{G_{\\,n}}\n      =\\frac{G_{\\,n}}{G_{\\,n-1}}\n      \\quad\\Longrightarrow\\quad\n      \\rho_{n}= \\rho_{\\,n-1}.\n   \\]\n   By induction $\\rho_n$ is independent of $n$; write\n   \\[\n     \\boxed{\\rho_n\\equiv\\rho\\neq0.}\n   \\]\n   Consequently $G_n=G_0\\,\\rho^{\\,n}$ for all integers $n$.\n\n2.  Vanishing of all $3\\times3$ Hankel minors  $\\Longrightarrow$  $\\operatorname{rank}H\\le2$\n   ----------------------------------------------------------------------------------------\n\n   By definition of the rank of a matrix, if every $3\\times3$ minor is $0$ then no $3\\times3$ submatrix has full rank, hence  \n   \\[\n       \\operatorname{rank}H\\le 2. \\tag{3}\n   \\]\n   On the other hand, hypothesis (B) states that some (indeed every) centred $2\\times2$ minor is non-zero, so $\\operatorname{rank}H\\ge2$.  Therefore  \n   \\[\n       \\operatorname{rank}H = 2.\n   \\]\n\n   Pick any three consecutive columns of $H$, say $C_{0},C_{1},C_{2}$.  Because the column-space is $2$-dimensional there exist scalars $\\alpha,\\beta,\\gamma$, not all $0$, such that  \n   \\[\n        \\alpha C_{0}+\\beta C_{1}+\\gamma C_{2}=0.\n   \\]\n   Taking the $n$-th entry of this column equation yields a *global* linear recurrence\n   \\[\n        \\alpha x_{\\,n}+\\beta x_{\\,n+1}+\\gamma x_{\\,n+2}=0\n        \\qquad(n\\in\\mathbb Z). \\tag{4}\n   \\]\n\n   If $\\gamma=0$, then (4) reduces to $x_{\\,n+1}=-(\\alpha/\\beta)\\,x_{\\,n}$, so each $G_n=x_{\\,n}x_{\\,n+2}-x_{\\,n+1}^{2}$ vanishes, contradicting (B).  Hence $\\gamma\\neq0$, and we may normalise $\\gamma=-1$ to obtain UNIQUE constants\n   \\[\n        a:=-\\frac{\\beta}{\\gamma},\\qquad q:=-\\frac{\\alpha}{\\gamma}\n   \\]\n   such that\n   \\[\n       \\boxed{x_{\\,n+2}=a\\,x_{\\,n+1}+q\\,x_{\\,n}\\qquad(n\\in\\mathbb Z).}\n   \\]\n\n3.  The coefficient $q$ equals $-\\rho$\n   -----------------------------------\n\n   Write  \n   \\[\n        \\Phi_n:=x_{\\,n+1}^{2}-a\\,x_{\\,n+1}x_{\\,n}-q\\,x_{\\,n}^{2}.\n   \\]\n\n   First compute $G_n$ with the recurrence:\n   \\[\n   \\begin{aligned}\n        G_n&=x_{\\,n}x_{\\,n+2}-x_{\\,n+1}^2\n             =x_{\\,n}(a x_{\\,n+1}+q x_{\\,n})-x_{\\,n+1}^2  \\\\\n            &=-\\bigl(x_{\\,n+1}^{2}-a x_{\\,n+1}x_{\\,n}-q x_{\\,n}^{2}\\bigr)\n             =-\\Phi_n.                                       \\tag{5}\n   \\end{aligned}\n   \\]\n\n   Next compute $G_{n+1}$:\n   \\[\n   \\begin{aligned}\n        x_{\\,n+3}&=a x_{\\,n+2}+q x_{\\,n+1}\n                  =a(a x_{\\,n+1}+q x_{\\,n})+q x_{\\,n+1}  \\\\\n                 &=(a^{2}+q)\\,x_{\\,n+1}+a q\\,x_{\\,n},\\\\[2pt]\n        G_{n+1}&=x_{\\,n+1}x_{\\,n+3}-x_{\\,n+2}^{2}\\\\\n               &=x_{\\,n+1}\\bigl((a^{2}+q)x_{\\,n+1}+a q x_n\\bigr)\n                 -(a x_{\\,n+1}+q x_{\\,n})^{2}\\\\\n               &=q\\bigl(x_{\\,n+1}^{2}-a x_{\\,n+1}x_{\\,n}-q x_{\\,n}^{2}\\bigr)\n                 =q\\,\\Phi_n.                                 \\tag{6}\n   \\end{aligned}\n   \\]\n\n   Combining (5) and (6) we have\n   \\[\n        \\frac{G_{\\,n+1}}{G_{\\,n}}=\\frac{q\\,\\Phi_n}{-\\Phi_n}=-q,\n   \\]\n   so the constant ratio found in Part 1 satisfies\n   \\[\n        \\boxed{q=-\\rho\\neq0.}\n   \\]\n   Hence $G_n=G_0(-q)^{\\,n}$; if $G_0=0$ then $G_n\\equiv0$, while for $G_0\\neq0$ the sequence $(G_n)$ is a non-zero geometric progression.\n\n4.  The special case $G_n\\equiv D\\neq0$\n   ------------------------------------\n\n   If $G_n$ is constant, then $\\rho=1$, whence $q=-1$ by Part 3.  The recurrence therefore assumes the classical “Chebyshev–type” form  \n   \\[\n        x_{\\,n+2}=a\\,x_{\\,n+1}-x_{\\,n}\\qquad(n\\in\\mathbb Z).\n   \\]\n\n5.  Converse\n   ---------\n\n   Assume $x_{\\,n+2}=a x_{\\,n+1}+q x_{\\,n}$ with $q\\neq0$.  In every $3\\times3$ Hankel minor the third column is a linear combination ($a,q$) of the first two, so $\\Delta _n\\equiv0$ for all $n$.  The calculation of Part 3, read backwards, shows $G_{\\,n+1}=-q\\,G_{\\,n}$ for every $n$.  Thus either $G_n\\equiv0$, or else $G_n=G_0(-q)^{\\,n}\\neq0$.\n\n\\hfill$\\square$\n\n————————————————————————————————————————",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.729241",
        "was_fixed": false,
        "difficulty_analysis": "1.  Higher-order invariant – the original problem keeps a $2\\times2$ determinant constant; the new one fixes a $3\\times3$ determinant, producing cubic expressions such as (1).  Handling them requires significantly heavier algebra and familiarity with Hankel determinants.\n\n2.  Deeper theory – the proof uses the sliding‐determinant (Sylvester) identity to relate $\\Delta_n$ and $\\Delta_{n-1}$, demands a careful expansion, and then translates the result into an invariance property for the rational sequence $(F_n)$.\n\n3.  More steps – the solution needs (i) determinant expansion, (ii) telescoping via $\\Delta_n-\\Delta_{n-1}$, (iii) introduction of a cleverly chosen auxiliary quantity $F_n$, (iv) a proof of its constancy, and finally (v) derivation of the linear second-order recurrence.  Each stage is appreciably more intricate than the single short manipulation used in the original variant.\n\n4.  No pattern matching – the cubic identity (1) offers no immediate hint of a linear recurrence; the solver must invent the right normalization and recognise hidden cancellations, unlike the straightforward ratio comparison in the original problem.\n\nConsequently, both the technical workload and the conceptual depth are substantially increased, making this kernel variant markedly harder than the given ones."
      }
    },
    "original_kernel_variant": {
      "question": "Let $(x_n)_{n\\in\\mathbb Z}$ be a doubly–infinite sequence of non–zero complex numbers and form the bi-infinite Hankel matrix  \n\\[H=\\bigl(x_{\\,i+j}\\bigr)_{\\,i,j\\in\\mathbb Z} .\\]\n\nFor every $n\\in\\mathbb Z$ set  \n\\[\n\\Delta _n \\;=\\;\n\\det\\!\\begin{pmatrix}\nx_{\\,n-1}&x_{\\,n}&x_{\\,n+1}\\\\\nx_{\\,n  }&x_{\\,n+1}&x_{\\,n+2}\\\\\nx_{\\,n+1}&x_{\\,n+2}&x_{\\,n+3}\n\\end{pmatrix},\n\\qquad   \nG_n \\;=\\;\n\\det\\!\\begin{pmatrix}\nx_{\\,n}&x_{\\,n+1}\\\\\nx_{\\,n+1}&x_{\\,n+2}\n\\end{pmatrix}\n      =x_{\\,n}x_{\\,n+2}-x_{\\,n+1}^{2}.\n\\]\n\nAssume  \n\n(A) every $3\\times3$ Hankel minor of $H$ vanishes (i.e. $\\Delta _n=0$ for all $n$);  \n\n(B) no centred $2\\times2$ minor vanishes: $G_n\\neq0$ for every $n$.\n\nBecause of (B) the quotient  \n\\[\n\\rho_n:=\\frac{G_{\\,n+1}}{G_{\\,n}}\\qquad(n\\in\\mathbb Z)\n\\]\nis well–defined.\n\n1.  Show that the numbers $\\rho_n$ are independent of $n$; denote their common non-zero value by $\\rho$.\n\n2.  Prove that $\\operatorname{rank}H\\le2$ and hence that there exist UNIQUE complex constants $a,q$ such that  \n\\[\n\\boxed{\\;x_{\\,n+2}=a\\,x_{\\,n+1}+q\\,x_{\\,n}\\qquad(n\\in\\mathbb Z). \\;}\n\\]\n\n3.  Prove the relation $q=-\\rho$ and deduce that $(G_n)$ is the (possibly null) geometric progression $G_n=G_0\\,(-q)^{\\,n}$.\n\n4.  Specialise: if, in addition, the minors $G_n$ are all equal to the same non–zero constant $D$, then necessarily $q=-1$ and the recurrence becomes  \n\\[\nx_{\\,n+2}=a\\,x_{\\,n+1}-x_{\\,n}\\quad(n\\in\\mathbb Z).\n\\]\n\n5.  Converse.  Show that every bi-infinite sequence of non-zero complex numbers satisfying  \n\\[\nx_{\\,n+2}=a\\,x_{\\,n+1}+q\\,x_{\\,n}\\qquad(q\\neq0)\n\\]\nhas $\\Delta _n\\equiv0$ and $G_{n+1}=-q\\,G_n$ for all $n$ (hence either $G_n\\equiv0$ or $G_n$ is a non-zero geometric progression).\n\n————————————————————————————————————————",
      "solution": "We treat the five items in order.  Throughout, $n$ is an arbitrary integer.\n\n1.  Constantness of the ratios $\\rho_n$\n   ------------------------------------\n\n   Expanding $\\Delta _n$ along the first row gives\n   \\[\n   \\Delta _n\n      =x_{\\,n-1}\\bigl(x_{\\,n+1}x_{\\,n+3}-x_{\\,n+2}^{2}\\bigr)\n       -x_{\\,n  }\\bigl(x_{\\,n  }x_{\\,n+3}-x_{\\,n+1}x_{\\,n+2}\\bigr)\n       +x_{\\,n+1}\\bigl(x_{\\,n  }x_{\\,n+2}-x_{\\,n+1}^{2}\\bigr).\n   \\]\n   Recognising the $2\\times2$ determinants $G_{n-1},G_n,G_{n+1}$ we obtain\n   \\[\n      x_{\\,n+1}\\,\\Delta _n =G_{\\,n-1}G_{\\,n+1}-G_{\\,n}^{2}. \\tag{1}\n   \\]\n   Equation (1) is the $k=3$ Jacobi (Sylvester) identity for contiguous minors of a Hankel matrix.\n\n   Hypothesis (A) gives $\\Delta _n=0$, and because $x_{\\,n+1}\\neq0$ we have\n   \\[\n     \\boxed{\\,G_{\\,n-1}G_{\\,n+1}=G_{\\,n}^{2}\\,.} \\tag{2}\n   \\]\n   Divide (2) by the non–zero product $G_{\\,n-1}G_{\\,n}$:\n   \\[\n      \\frac{G_{\\,n+1}}{G_{\\,n}}\n      =\\frac{G_{\\,n}}{G_{\\,n-1}}\n      \\quad\\Longrightarrow\\quad\n      \\rho_{n}= \\rho_{\\,n-1}.\n   \\]\n   By induction $\\rho_n$ is independent of $n$; write\n   \\[\n     \\boxed{\\rho_n\\equiv\\rho\\neq0.}\n   \\]\n   Consequently $G_n=G_0\\,\\rho^{\\,n}$ for all integers $n$.\n\n2.  Vanishing of all $3\\times3$ Hankel minors  $\\Longrightarrow$  $\\operatorname{rank}H\\le2$\n   ----------------------------------------------------------------------------------------\n\n   By definition of the rank of a matrix, if every $3\\times3$ minor is $0$ then no $3\\times3$ submatrix has full rank, hence  \n   \\[\n       \\operatorname{rank}H\\le 2. \\tag{3}\n   \\]\n   On the other hand, hypothesis (B) states that some (indeed every) centred $2\\times2$ minor is non-zero, so $\\operatorname{rank}H\\ge2$.  Therefore  \n   \\[\n       \\operatorname{rank}H = 2.\n   \\]\n\n   Pick any three consecutive columns of $H$, say $C_{0},C_{1},C_{2}$.  Because the column-space is $2$-dimensional there exist scalars $\\alpha,\\beta,\\gamma$, not all $0$, such that  \n   \\[\n        \\alpha C_{0}+\\beta C_{1}+\\gamma C_{2}=0.\n   \\]\n   Taking the $n$-th entry of this column equation yields a *global* linear recurrence\n   \\[\n        \\alpha x_{\\,n}+\\beta x_{\\,n+1}+\\gamma x_{\\,n+2}=0\n        \\qquad(n\\in\\mathbb Z). \\tag{4}\n   \\]\n\n   If $\\gamma=0$, then (4) reduces to $x_{\\,n+1}=-(\\alpha/\\beta)\\,x_{\\,n}$, so each $G_n=x_{\\,n}x_{\\,n+2}-x_{\\,n+1}^{2}$ vanishes, contradicting (B).  Hence $\\gamma\\neq0$, and we may normalise $\\gamma=-1$ to obtain UNIQUE constants\n   \\[\n        a:=-\\frac{\\beta}{\\gamma},\\qquad q:=-\\frac{\\alpha}{\\gamma}\n   \\]\n   such that\n   \\[\n       \\boxed{x_{\\,n+2}=a\\,x_{\\,n+1}+q\\,x_{\\,n}\\qquad(n\\in\\mathbb Z).}\n   \\]\n\n3.  The coefficient $q$ equals $-\\rho$\n   -----------------------------------\n\n   Write  \n   \\[\n        \\Phi_n:=x_{\\,n+1}^{2}-a\\,x_{\\,n+1}x_{\\,n}-q\\,x_{\\,n}^{2}.\n   \\]\n\n   First compute $G_n$ with the recurrence:\n   \\[\n   \\begin{aligned}\n        G_n&=x_{\\,n}x_{\\,n+2}-x_{\\,n+1}^2\n             =x_{\\,n}(a x_{\\,n+1}+q x_{\\,n})-x_{\\,n+1}^2  \\\\\n            &=-\\bigl(x_{\\,n+1}^{2}-a x_{\\,n+1}x_{\\,n}-q x_{\\,n}^{2}\\bigr)\n             =-\\Phi_n.                                       \\tag{5}\n   \\end{aligned}\n   \\]\n\n   Next compute $G_{n+1}$:\n   \\[\n   \\begin{aligned}\n        x_{\\,n+3}&=a x_{\\,n+2}+q x_{\\,n+1}\n                  =a(a x_{\\,n+1}+q x_{\\,n})+q x_{\\,n+1}  \\\\\n                 &=(a^{2}+q)\\,x_{\\,n+1}+a q\\,x_{\\,n},\\\\[2pt]\n        G_{n+1}&=x_{\\,n+1}x_{\\,n+3}-x_{\\,n+2}^{2}\\\\\n               &=x_{\\,n+1}\\bigl((a^{2}+q)x_{\\,n+1}+a q x_n\\bigr)\n                 -(a x_{\\,n+1}+q x_{\\,n})^{2}\\\\\n               &=q\\bigl(x_{\\,n+1}^{2}-a x_{\\,n+1}x_{\\,n}-q x_{\\,n}^{2}\\bigr)\n                 =q\\,\\Phi_n.                                 \\tag{6}\n   \\end{aligned}\n   \\]\n\n   Combining (5) and (6) we have\n   \\[\n        \\frac{G_{\\,n+1}}{G_{\\,n}}=\\frac{q\\,\\Phi_n}{-\\Phi_n}=-q,\n   \\]\n   so the constant ratio found in Part 1 satisfies\n   \\[\n        \\boxed{q=-\\rho\\neq0.}\n   \\]\n   Hence $G_n=G_0(-q)^{\\,n}$; if $G_0=0$ then $G_n\\equiv0$, while for $G_0\\neq0$ the sequence $(G_n)$ is a non-zero geometric progression.\n\n4.  The special case $G_n\\equiv D\\neq0$\n   ------------------------------------\n\n   If $G_n$ is constant, then $\\rho=1$, whence $q=-1$ by Part 3.  The recurrence therefore assumes the classical “Chebyshev–type” form  \n   \\[\n        x_{\\,n+2}=a\\,x_{\\,n+1}-x_{\\,n}\\qquad(n\\in\\mathbb Z).\n   \\]\n\n5.  Converse\n   ---------\n\n   Assume $x_{\\,n+2}=a x_{\\,n+1}+q x_{\\,n}$ with $q\\neq0$.  In every $3\\times3$ Hankel minor the third column is a linear combination ($a,q$) of the first two, so $\\Delta _n\\equiv0$ for all $n$.  The calculation of Part 3, read backwards, shows $G_{\\,n+1}=-q\\,G_{\\,n}$ for every $n$.  Thus either $G_n\\equiv0$, or else $G_n=G_0(-q)^{\\,n}\\neq0$.\n\n\\hfill$\\square$\n\n————————————————————————————————————————",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.566567",
        "was_fixed": false,
        "difficulty_analysis": "1.  Higher-order invariant – the original problem keeps a $2\\times2$ determinant constant; the new one fixes a $3\\times3$ determinant, producing cubic expressions such as (1).  Handling them requires significantly heavier algebra and familiarity with Hankel determinants.\n\n2.  Deeper theory – the proof uses the sliding‐determinant (Sylvester) identity to relate $\\Delta_n$ and $\\Delta_{n-1}$, demands a careful expansion, and then translates the result into an invariance property for the rational sequence $(F_n)$.\n\n3.  More steps – the solution needs (i) determinant expansion, (ii) telescoping via $\\Delta_n-\\Delta_{n-1}$, (iii) introduction of a cleverly chosen auxiliary quantity $F_n$, (iv) a proof of its constancy, and finally (v) derivation of the linear second-order recurrence.  Each stage is appreciably more intricate than the single short manipulation used in the original variant.\n\n4.  No pattern matching – the cubic identity (1) offers no immediate hint of a linear recurrence; the solver must invent the right normalization and recognise hidden cancellations, unlike the straightforward ratio comparison in the original problem.\n\nConsequently, both the technical workload and the conceptual depth are substantially increased, making this kernel variant markedly harder than the given ones."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}