{
  "index": "2024-B-2",
  "type": "GEO",
  "tag": [
    "GEO",
    "COMB"
  ],
  "difficulty": "",
  "question": "Two convex quadrilaterals are called \\emph{partners} if they have three vertices in common and they can be labeled $ABCD$ and $ABCE$ so that $E$ is the reflection of $D$ across the perpendicular bisector of the diagonal $\\overline{AC}$. Is there an infinite sequence of convex quadrilaterals such that each quadrilateral is a partner of its successor and no two elements of the sequence are congruent?\n[A diagram has been omitted.]",
  "solution": "No, there is no such sequence. In other words, any sequence of convex quadrilaterals with the property that any two consecutive terms are partners must be finite.\n\n\\noindent\n\\textbf{First solution.}\n\n\\begin{lemma*}\nGiven five positive real numbers $a,b,c,d,K$, there are only finitely many convex quadrilaterals with side lengths $a,b,c,d$ in that order and area $K$. \n\\end{lemma*}\n\\begin{proof}\nLet $PQRS$ be a convex quadrilateral with \n\\[\n\\overline{PQ} = a, \\overline{QR} = b, \\overline{RS} = c, \\overline{SP} = d.\n\\]\nThen the congruence class of $PQRS$ is uniquely determined by the length of the diagonal $f := \\overline{PR}$.\nMoreover, as $f$ increases, the angles $\\angle RPQ$ and $\\angle RPS$ are both strictly decreasing, so $\\angle SPQ$ is decreasing; by the same logic, $\\angle QRS$ is decreasing. \n\nWe next recall \\emph{Bretschneider's formula}: for $s = (a+b+c+d)/2$,\n\\[\nK^2 = (s-a)(s-b)(s-c)(s-d) - abcd \\cos^2 \\frac{\\angle SPQ + \\angle QRS}{2}.\n\\]\nConsequently, fixing $K$ also fixes $\\cos^2 \\frac{\\angle SPQ + \\angle QRS}{2}$,\nand thus limits $\\angle SPQ + \\angle QRS$ to one of two values. By the previous paragraph, this leaves at most two possible congruence classes for the triangle.\n\\end{proof}\n\nReturning to our original sequence, note that any two consecutive quadrilaterals in the sequence have the same area and the same unordered list of side lengths. The latter can occur as an ordered list in at most six different ways (up to cyclic shift); for each of these, we can have only finitely many distinct congruence classes of quadrilaterals in our sequence with that area and ordered list of side lengths. We deduce that our sequence must be finite.\n\n\\noindent\n\\textbf{Remark.}\nVarious proofs of the lemma are possible; for example, here is one using Cartesian coordinates. We \nfirst specify \n\\[\nP = (0,0), Q = (a, 0).\n\\]\nFor two additional points $R = (x,y),S = (z,w)$, the conditions $\\overline{QR} = b$, $\\overline{SP} = d$ restrict $R$ and $S$ to the circles\n\\[\n(x-a)^2 + y^2 = b^2, \\quad\nz^2+w^2 = d^2\n\\]\nrespectively. Since we want a convex quadrilateral, we may assume without loss of generality that $y,w > 0$.\nThe area of the quadrilateral is $\\frac{1}{2} a(y+w)$, which we also want to fix; we may thus regard $w$ as a function of $y$ (possibly restricting $y$ to a range for which $w>0$). After splitting the semicircles on which $R$ and $S$ lie into two arcs each, we may also regard $x$ and $w$ as functions of $y$. It now suffices to observe that $\\overline{RS}^2 = (z-x)^2 + (w-y)^2$\nis a nonconstant algebraic function of $y$, so it takes any given value only finitely many times.\n\n\\noindent\n\\textbf{Second solution.}\nLet $ABCD$ be the first quadrilateral in the sequence.\nSince the quadrilateral is convex, the diagonals $\\overline{AC}$ and $\\overline{BD}$ intersect. In particular they are not parallel, so their perpendicular bisectors are not parallel either; let $O$ be the intersection\nof the bisectors.\n\nWe claim that the point $O$ remains fixed throughout the sequence, as do the distances $OA, OB, OC, OD$. To see this, we check this for two partners as described in the problem statement: the diagonal $\\overline{BD}$ gets reflected across the perpendicular bisector of $\\overline{AC}$, so its perpendicular bisector also gets reflected; the point $O$ is the unique point on the perpendicular bisector of $\\overline{BD}$ fixed by the reflection. In particular, the segments $\\overline{OD}$ and $\\overline{OE}$ are mirror images across the perpendicular bisector of $\\overline{AC}$, so their lengths coincide.\n\nAs noted in the first solution, the unordered list of side lengths of the quadrilateral also remains invariant throughout the sequence. Consequently, the unordered list of side lengths of each of the triangles $\\triangle OAB, \\triangle OBC, \\triangle OCD, \\triangle ODA$ is limited to a finite set;\neach such list uniquely determines the unoriented congruence class of the corresponding triangle,\nand limits the oriented congruence class to two possibilities. Given the oriented congruence classes of the four triangles we can reconstruct the quadrilateral $ABCD$ up to oriented congruence (even up to rotation around $O$); this proves that the sequence must be finite.",
  "vars": [
    "A",
    "B",
    "C",
    "D",
    "E",
    "P",
    "Q",
    "R",
    "S",
    "O",
    "f",
    "s",
    "x",
    "y",
    "z",
    "w"
  ],
  "params": [
    "a",
    "b",
    "c",
    "d",
    "K"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "A": "pointa",
        "B": "pointb",
        "C": "pointc",
        "D": "pointd",
        "E": "pointe",
        "P": "pointp",
        "Q": "pointq",
        "R": "pointr",
        "S": "points",
        "O": "centero",
        "f": "diaglen",
        "s": "semiper",
        "x": "coordx",
        "y": "coordy",
        "z": "coordz",
        "w": "coordw",
        "a": "sidelena",
        "b": "sidelenb",
        "c": "sidelenc",
        "d": "sidelend",
        "K": "areaval"
      },
      "question": "Two convex quadrilaterals are called \\emph{partners} if they have three vertices in common and they can be labeled $pointapointbpointcpointd$ and $pointapointbpointcpointe$ so that $pointe$ is the reflection of $pointd$ across the perpendicular bisector of the diagonal $\\overline{pointapointc}$. Is there an infinite sequence of convex quadrilaterals such that each quadrilateral is a partner of its successor and no two elements of the sequence are congruent?\n[A diagram has been omitted.]",
      "solution": "No, there is no such sequence. In other words, any sequence of convex quadrilaterals with the property that any two consecutive terms are partners must be finite.\n\n\\noindent\n\\textbf{First solution.}\n\n\\begin{lemma*}\nGiven five positive real numbers $sidelena, sidelenb, sidelenc, sidelend, areaval$, there are only finitely many convex quadrilaterals with side lengths $sidelena, sidelenb, sidelenc, sidelend$ in that order and area $areaval$. \n\\end{lemma*}\n\\begin{proof}\nLet $pointp pointq pointr points$ be a convex quadrilateral with \n\\[\n\\overline{pointppointq} = sidelena, \\overline{pointqpointr} = sidelenb, \\overline{pointrpoints} = sidelenc, \\overline{pointspointp} = sidelend.\n\\]\nThen the congruence class of $pointp pointq pointr points$ is uniquely determined by the length of the diagonal $diaglen := \\overline{pointppointr}$.\nMoreover, as $diaglen$ increases, the angles $\\angle pointr pointp pointq$ and $\\angle pointr pointp points$ are both strictly decreasing, so $\\angle points pointp pointq$ is decreasing; by the same logic, $\\angle pointq pointr points$ is decreasing. \n\nWe next recall \\emph{Bretschneider's formula}: for $semiper = (sidelena + sidelenb + sidelenc + sidelend)/2$,\\[\nareaval^{2} = (semiper - sidelena)(semiper - sidelenb)(semiper - sidelenc)(semiper - sidelend) - sidelena \\, sidelenb \\, sidelenc \\, sidelend \\cos^{2} \\frac{\\angle points pointp pointq + \\angle pointq pointr points}{2}.\n\\]\nConsequently, fixing $areaval$ also fixes $\\cos^{2} \\dfrac{\\angle points pointp pointq + \\angle pointq pointr points}{2}$,\nand thus limits $\\angle points pointp pointq + \\angle pointq pointr points$ to one of two values. By the previous paragraph, this leaves at most two possible congruence classes for the triangle.\n\\end{proof}\n\nReturning to our original sequence, note that any two consecutive quadrilaterals in the sequence have the same area and the same unordered list of side lengths. The latter can occur as an ordered list in at most six different ways (up to cyclic shift); for each of these, we can have only finitely many distinct congruence classes of quadrilaterals in our sequence with that area and ordered list of side lengths. We deduce that our sequence must be finite.\n\n\\noindent\n\\textbf{Remark.}\nVarious proofs of the lemma are possible; for example, here is one using Cartesian coordinates. We \nfirst specify \n\\[\npointp = (0,0), \\quad pointq = (sidelena, 0).\n\\]\nFor two additional points $pointr = (coordx, coordy), \\; points = (coordz, coordw)$, the conditions $\\overline{pointqpointr} = sidelenb$, $\\overline{pointspointp} = sidelend$ restrict $pointr$ and $points$ to the circles\n\\[\n(coordx - sidelena)^{2} + coordy^{2} = sidelenb^{2}, \\quad\ncoordz^{2} + coordw^{2} = sidelend^{2}\n\\]\nrespectively. Since we want a convex quadrilateral, we may assume without loss of generality that $coordy, coordw > 0$.\nThe area of the quadrilateral is $\\frac{1}{2}\\, sidelena (coordy + coordw)$, which we also want to fix; we may thus regard $coordw$ as a function of $coordy$ (possibly restricting $coordy$ to a range for which $coordw>0$). After splitting the semicircles on which $pointr$ and $points$ lie into two arcs each, we may also regard $coordx$ and $coordw$ as functions of $coordy$. It now suffices to observe that $\\overline{pointrpoints}^{2} = (coordz - coordx)^{2} + (coordw - coordy)^{2}$\nis a nonconstant algebraic function of $coordy$, so it takes any given value only finitely many times.\n\n\\noindent\n\\textbf{Second solution.}\nLet $pointapointbpointcpointd$ be the first quadrilateral in the sequence.\nSince the quadrilateral is convex, the diagonals $\\overline{pointapointc}$ and $\\overline{pointbpointd}$ intersect. In particular they are not parallel, so their perpendicular bisectors are not parallel either; let $centero$ be the intersection\nof the bisectors.\n\nWe claim that the point $centero$ remains fixed throughout the sequence, as do the distances $centero pointa, centero pointb, centero pointc, centero pointd$. To see this, we check this for two partners as described in the problem statement: the diagonal $\\overline{pointbpointd}$ gets reflected across the perpendicular bisector of $\\overline{pointapointc}$, so its perpendicular bisector also gets reflected; the point $centero$ is the unique point on the perpendicular bisector of $\\overline{pointbpointd}$ fixed by the reflection. In particular, the segments $\\overline{centeropointd}$ and $\\overline{centeropointe}$ are mirror images across the perpendicular bisector of $\\overline{pointapointc}$, so their lengths coincide.\n\nAs noted in the first solution, the unordered list of side lengths of the quadrilateral also remains invariant throughout the sequence. Consequently, the unordered list of side lengths of each of the triangles $\\triangle centero pointa pointb, \\triangle centero pointb pointc, \\triangle centero pointc pointd, \\triangle centero pointd pointa$ is limited to a finite set;\neach such list uniquely determines the unoriented congruence class of the corresponding triangle,\nand limits the oriented congruence class to two possibilities. Given the oriented congruence classes of the four triangles we can reconstruct the quadrilateral $pointapointbpointcpointd$ up to oriented congruence (even up to rotation around $centero$); this proves that the sequence must be finite."
    },
    "descriptive_long_confusing": {
      "map": {
        "A": "afterglow",
        "B": "brushwood",
        "C": "clocktower",
        "D": "driftwood",
        "E": "evergreens",
        "P": "patchwork",
        "Q": "quicksand",
        "R": "rainstorm",
        "S": "sandstone",
        "O": "overgrowth",
        "f": "feathered",
        "s": "sandpaper",
        "x": "xylophone",
        "y": "yardstick",
        "z": "zookeeper",
        "w": "windchime",
        "a": "applecart",
        "b": "buttercup",
        "c": "cornstalk",
        "d": "daydreams",
        "K": "kingfisher"
      },
      "question": "Two convex quadrilaterals are called \\emph{partners} if they have three vertices in common and they can be labeled $afterglowbrushwoodclocktowerdriftwood$ and $afterglowbrushwoodclocktowerevergreens$ so that $evergreens$ is the reflection of $driftwood$ across the perpendicular bisector of the diagonal $\\overline{afterglowclocktower}$. Is there an infinite sequence of convex quadrilaterals such that each quadrilateral is a partner of its successor and no two elements of the sequence are congruent?\n[A diagram has been omitted.]",
      "solution": "No, there is no such sequence. In other words, any sequence of convex quadrilaterals with the property that any two consecutive terms are partners must be finite.\n\n\\noindent\n\\textbf{First solution.}\n\n\\begin{lemma*}\nGiven five positive real numbers $applecart,buttercup,cornstalk,daydreams,kingfisher$, there are only finitely many convex quadrilaterals with side lengths $applecart,buttercup,cornstalk,daydreams$ in that order and area $kingfisher$. \n\\end{lemma*}\n\\begin{proof}\nLet $patchworkquicksandrainstormsandstone$ be a convex quadrilateral with \n\\[\n\\overline{patchworkquicksand}=applecart,\\;\\overline{quicksandrainstorm}=buttercup,\\;\\overline{rainstormsandstone}=cornstalk,\\;\\overline{sandstonepatchwork}=daydreams.\n\\]\nThen the congruence class of $patchworkquicksandrainstormsandstone$ is uniquely determined by the length of the diagonal $feathered := \\overline{patchworkrainstorm}$.  Moreover, as $feathered$ increases, the angles $\\angle rainstormpatchworkquicksand$ and $\\angle rainstormpatchworksandstone$ are both strictly decreasing, so $\\angle sandstonepatchworkquicksand$ is decreasing; by the same logic, $\\angle quicksandrainstormsandstone$ is decreasing. \n\nWe next recall \\emph{Bretschneider's formula}: for $sandpaper=(applecart+buttercup+cornstalk+daydreams)/2$,\n\\[\nkingfisher^2=(sandpaper-applecart)(sandpaper-buttercup)(sandpaper-cornstalk)(sandpaper-daydreams)-applecartbuttercupcornstalkdaydreams\\cos^2\\frac{\\angle sandstonepatchworkquicksand+\\angle quicksandrainstormsandstone}{2}.\n\\]\nConsequently, fixing $kingfisher$ also fixes $\\cos^2\\frac{\\angle sandstonepatchworkquicksand+\\angle quicksandrainstormsandstone}{2}$, and thus limits $\\angle sandstonepatchworkquicksand+\\angle quicksandrainstormsandstone$ to one of two values. By the previous paragraph, this leaves at most two possible congruence classes for the triangle.\n\\end{proof}\n\nReturning to our original sequence, note that any two consecutive quadrilaterals in the sequence have the same area and the same unordered list of side lengths. The latter can occur as an ordered list in at most six different ways (up to cyclic shift); for each of these, we can have only finitely many distinct congruence classes of quadrilaterals in our sequence with that area and ordered list of side lengths. We deduce that our sequence must be finite.\n\n\\noindent\n\\textbf{Remark.}\nVarious proofs of the lemma are possible; for example, here is one using Cartesian coordinates. We first specify \n\\[\npatchwork=(0,0),\\;quicksand=(applecart,0).\n\\]\nFor two additional points $rainstorm=(xylophone,yardstick),\\;sandstone=(zookeeper,windchime)$, the conditions $\\overline{quicksandrainstorm}=buttercup$, $\\overline{sandstonepatchwork}=daydreams$ restrict $rainstorm$ and $sandstone$ to the circles\n\\[\n(xylophone-applecart)^2+yardstick^2=buttercup^2,\\quad zookeeper^2+windchime^2=daydreams^2\n\\]\nrespectively. Since we want a convex quadrilateral, we may assume without loss of generality that $yardstick,windchime>0$. The area of the quadrilateral is $\\frac12\\,applecart(yardstick+windchime)$, which we also want to fix; we may thus regard $windchime$ as a function of $yardstick$ (possibly restricting $yardstick$ to a range for which $windchime>0$). After splitting the semicircles on which $rainstorm$ and $sandstone$ lie into two arcs each, we may also regard $xylophone$ and $zookeeper$ as functions of $yardstick$. It now suffices to observe that $\\overline{rainstormsandstone}^2=(zookeeper-xylophone)^2+(windchime-yardstick)^2$ is a nonconstant algebraic function of $yardstick$, so it takes any given value only finitely many times.\n\n\\noindent\n\\textbf{Second solution.}\nLet $afterglowbrushwoodclocktowerdriftwood$ be the first quadrilateral in the sequence. Since the quadrilateral is convex, the diagonals $\\overline{afterglowclocktower}$ and $\\overline{brushwooddriftwood}$ intersect. In particular they are not parallel, so their perpendicular bisectors are not parallel either; let $overgrowth$ be the intersection of the bisectors.\n\nWe claim that the point $overgrowth$ remains fixed throughout the sequence, as do the distances $\\overline{overgrowthafterglow},\\;\\overline{overgrowthbrushwood},\\;\\overline{overgrowthclocktower},\\;\\overline{overgrowthdriftwood}$. To see this, we check this for two partners as described in the problem statement: the diagonal $\\overline{brushwooddriftwood}$ gets reflected across the perpendicular bisector of $\\overline{afterglowclocktower}$, so its perpendicular bisector also gets reflected; the point $overgrowth$ is the unique point on the perpendicular bisector of $\\overline{brushwooddriftwood}$ fixed by the reflection. In particular, the segments $\\overline{overgrowthdriftwood}$ and $\\overline{overgrowthevergreens}$ are mirror images across the perpendicular bisector of $\\overline{afterglowclocktower}$, so their lengths coincide.\n\nAs noted in the first solution, the unordered list of side lengths of the quadrilateral also remains invariant throughout the sequence. Consequently, the unordered list of side lengths of each of the triangles $\\triangle overgrowthafterglowbrushwood$, $\\triangle overgrowthbrushwoodclocktower$, $\\triangle overgrowthclocktowerdriftwood$, $\\triangle overgrowthdriftwoodafterglow$ is limited to a finite set; each such list uniquely determines the unoriented congruence class of the corresponding triangle, and limits the oriented congruence class to two possibilities. Given the oriented congruence classes of the four triangles we can reconstruct the quadrilateral $afterglowbrushwoodclocktowerdriftwood$ up to oriented congruence (even up to rotation around $overgrowth$); this proves that the sequence must be finite."
    },
    "descriptive_long_misleading": {
      "map": {
        "A": "infiniteplane",
        "B": "boundlessvoid",
        "C": "endlessspace",
        "D": "massivebulk",
        "E": "continuousline",
        "P": "entireuniverse",
        "Q": "sprawlingarea",
        "R": "limitlessfield",
        "S": "extensivezone",
        "O": "grandexpanse",
        "f": "zerolength",
        "s": "interiorcore",
        "x": "nowhereaxis",
        "y": "nullordinate",
        "z": "absentheight",
        "w": "nullbreadth",
        "a": "colossalspan",
        "b": "tremendousreach",
        "c": "boundlessstretch",
        "d": "limitlessextent",
        "K": "emptiness"
      },
      "question": "Two convex quadrilaterals are called \\emph{partners} if they have three vertices in common and they can be labeled $infiniteplaneboundlessvoidendlessspacemassivebulk$ and $infiniteplaneboundlessvoidendlessspacecontinuousline$ so that $continuousline$ is the reflection of $massivebulk$ across the perpendicular bisector of the diagonal $\\overline{infiniteplane\\ endlessspace}$. Is there an infinite sequence of convex quadrilaterals such that each quadrilateral is a partner of its successor and no two elements of the sequence are congruent?\n[A diagram has been omitted.]",
      "solution": "No, there is no such sequence. In other words, any sequence of convex quadrilaterals with the property that any two consecutive terms are partners must be finite.\n\n\\noindent\n\\textbf{First solution.}\n\n\\begin{lemma*}\nGiven five positive real numbers $colossalspan,tremendousreach,boundlessstretch,limitlessextent,emptiness$, there are only finitely many convex quadrilaterals with side lengths $colossalspan,tremendousreach,boundlessstretch,limitlessextent$ in that order and area $emptiness$. \n\\end{lemma*}\n\\begin{proof}\nLet $entireuniversesprawlingarealimitlessfieldextensivezone$ be a convex quadrilateral with \n\\[\n\\overline{entireuniverse\\ sprawlingarea} = colossalspan, \\overline{sprawlingarea\\ limitlessfield} = tremendousreach, \\overline{limitlessfield\\ extensivezone} = boundlessstretch, \\overline{extensivezone\\ entireuniverse} = limitlessextent.\n\\]\nThen the congruence class of $entireuniversesprawlingarealimitlessfieldextensivezone$ is uniquely determined by the length of the diagonal $zerolength := \\overline{entireuniverse\\ limitlessfield}$.\nMoreover, as $zerolength$ increases, the angles $\\angle limitlessfieldentireuniversesprawlingarea$ and $\\angle limitlessfieldentireuniverseextensivezone$ are both strictly decreasing, so $\\angle extensivezoneentireuniversesprawlingarea$ is decreasing; by the same logic, $\\angle sprawlingarealimitlessfieldextensivezone$ is decreasing. \n\nWe next recall \\emph{Bretschneider's formula}: for $interiorcore = (colossalspan+tremendousreach+boundlessstretch+limitlessextent)/2$,\n\\[\nemptiness^2 = (interiorcore-colossalspan)(interiorcore-tremendousreach)(interiorcore-boundlessstretch)(interiorcore-limitlessextent) - colossalspan\\tremendousreach\\boundlessstretch\\limitlessextent \\cos^2 \\frac{\\angle extensivezoneentireuniversesprawlingarea + \\angle sprawlingarealimitlessfieldextensivezone}{2}.\n\\]\nConsequently, fixing $emptiness$ also fixes $\\cos^2 \\frac{\\angle extensivezoneentireuniversesprawlingarea + \\angle sprawlingarealimitlessfieldextensivezone}{2}$,\nand thus limits $\\angle extensivezoneentireuniversesprawlingarea + \\angle sprawlingarealimitlessfieldextensivezone$ to one of two values. By the previous paragraph, this leaves at most two possible congruence classes for the triangle.\n\\end{proof}\n\nReturning to our original sequence, note that any two consecutive quadrilaterals in the sequence have the same area and the same unordered list of side lengths. The latter can occur as an ordered list in at most six different ways (up to cyclic shift); for each of these, we can have only finitely many distinct congruence classes of quadrilaterals in our sequence with that area and ordered list of side lengths. We deduce that our sequence must be finite.\n\n\\noindent\n\\textbf{Remark.}\nVarious proofs of the lemma are possible; for example, here is one using Cartesian coordinates. We \nfirst specify \n\\[\nentireuniverse = (0,0), \\quad sprawlingarea = (colossalspan, 0).\n\\]\nFor two additional points $limitlessfield = (nowhereaxis,nullordinate),\\; extensivezone = (absentheight,nullbreadth)$, the conditions $\\overline{sprawlingarea\\ limitlessfield} = tremendousreach$, $\\overline{extensivezone\\ entireuniverse} = limitlessextent$ restrict $limitlessfield$ and $extensivezone$ to the circles\n\\[\n(nowhereaxis-colossalspan)^2 + nullordinate^2 = tremendousreach^2, \\quad\nabsentheight^2 + nullbreadth^2 = limitlessextent^2\n\\]\nrespectively. Since we want a convex quadrilateral, we may assume without loss of generality that $nullordinate,nullbreadth > 0$.\nThe area of the quadrilateral is $\\frac{1}{2} colossalspan(nullordinate+nullbreadth)$, which we also want to fix; we may thus regard $nullbreadth$ as a function of $nullordinate$ (possibly restricting $nullordinate$ to a range for which $nullbreadth>0$). After splitting the semicircles on which $limitlessfield$ and $extensivezone$ lie into two arcs each, we may also regard $nowhereaxis$ and $nullbreadth$ as functions of $nullordinate$. It now suffices to observe that $\\overline{limitlessfield\\ extensivezone}^2 = (absentheight-nowhereaxis)^2 + (nullbreadth-nullordinate)^2$\nis a nonconstant algebraic function of $nullordinate$, so it takes any given value only finitely many times.\n\n\\noindent\n\\textbf{Second solution.}\nLet $infiniteplaneboundlessvoidendlessspacemassivebulk$ be the first quadrilateral in the sequence.\nSince the quadrilateral is convex, the diagonals $\\overline{infiniteplane\\ endlessspace}$ and $\\overline{boundlessvoid\\ massivebulk}$ intersect. In particular they are not parallel, so their perpendicular bisectors are not parallel either; let $grandexpanse$ be the intersection\nof the bisectors.\n\nWe claim that the point $grandexpanse$ remains fixed throughout the sequence, as do the distances $grandexpanse infiniteplane, grandexpanse boundlessvoid, grandexpanse endlessspace, grandexpanse massivebulk$. To see this, we check this for two partners as described in the problem statement: the diagonal $\\overline{boundlessvoid\\ massivebulk}$ gets reflected across the perpendicular bisector of $\\overline{infiniteplane\\ endlessspace}$, so its perpendicular bisector also gets reflected; the point $grandexpanse$ is the unique point on the perpendicular bisector of $\\overline{boundlessvoid\\ massivebulk}$ fixed by the reflection. In particular, the segments $\\overline{grandexpanse\\ massivebulk}$ and $\\overline{grandexpanse\\ continuousline}$ are mirror images across the perpendicular bisector of $\\overline{infiniteplane\\ endlessspace}$, so their lengths coincide.\n\nAs noted in the first solution, the unordered list of side lengths of the quadrilateral also remains invariant throughout the sequence. Consequently, the unordered list of side lengths of each of the triangles $\\triangle grandexpanse\\ infiniteplane\\ boundlessvoid, \\triangle grandexpanse\\ boundlessvoid\\ endlessspace, \\triangle grandexpanse\\ endlessspace\\ massivebulk, \\triangle grandexpanse\\ massivebulk\\ infiniteplane$ is limited to a finite set;\neach such list uniquely determines the unoriented congruence class of the corresponding triangle,\nand limits the oriented congruence class to two possibilities. Given the oriented congruence classes of the four triangles we can reconstruct the quadrilateral $infiniteplaneboundlessvoidendlessspacemassivebulk$ up to oriented congruence (even up to rotation around $grandexpanse$); this proves that the sequence must be finite."
    },
    "garbled_string": {
      "map": {
        "A": "qzxwvtnp",
        "B": "hjgrksla",
        "C": "plmdvcek",
        "D": "srowfgha",
        "E": "ycbntrwe",
        "P": "asvnjkei",
        "Q": "dflpzoim",
        "R": "wqkemnrb",
        "S": "tifghcva",
        "O": "bxrjzsqu",
        "f": "cgeadklb",
        "s": "mznhyqow",
        "x": "ulgefrto",
        "y": "nvaqpsie",
        "z": "tmrakbwo",
        "w": "csluipav",
        "a": "kjhwemct",
        "b": "gqzxalmd",
        "c": "lkhpferu",
        "d": "qbstynev",
        "K": "rvxosdip"
      },
      "question": "Two convex quadrilaterals are called \\emph{partners} if they have three vertices in common and they can be labeled $qzxwvtnphjgrkslaplmdvceksrowfgha$ and $qzxwvtnphjgrkslaplmdvcekycbntrwe$ so that $ycbntrwe$ is the reflection of $srowfgha$ across the perpendicular bisector of the diagonal $\\overline{qzxwvtnpplmdvcek}$. Is there an infinite sequence of convex quadrilaterals such that each quadrilateral is a partner of its successor and no two elements of the sequence are congruent?\n[A diagram has been omitted.]",
      "solution": "No, there is no such sequence. In other words, any sequence of convex quadrilaterals with the property that any two consecutive terms are partners must be finite.\n\n\\noindent\n\\textbf{First solution.}\n\n\\begin{lemma*}\nGiven five positive real numbers $kjhwemct,gqzxalmd,lkhpferu,qbstynev,rvxosdip$, there are only finitely many convex quadrilaterals with side lengths $kjhwemct,gqzxalmd,lkhpferu,qbstynev$ in that order and area $rvxosdip$. \n\\end{lemma*}\n\\begin{proof}\nLet $asvnjkeidflpzoimwqkemnrbtifghcva$ be a convex quadrilateral with \n\\[\n\\overline{asvnjkeidflpzoim} = kjhwemct, \\overline{dflpzoimwqkemnrb} = gqzxalmd, \\overline{wqkemnrbtifghcva} = lkhpferu, \\overline{tifghcvaasvnjkei} = qbstynev.\n\\]\nThen the congruence class of $asvnjkeidflpzoimwqkemnrbtifghcva$ is uniquely determined by the length of the diagonal $cgeadklb := \\overline{asvnjkeiwqkemnrb}$. Moreover, as $cgeadklb$ increases, the angles $\\angle wqkemnrbasvnjkeidflpzoim$ and $\\angle wqkemnrbasvnjkeitifghcva$ are both strictly decreasing, so $\\angle tifghcvaasvnjkeidflpzoim$ is decreasing; by the same logic, $\\angle dflpzoimwqkemnrbtifghcva$ is decreasing. \n\nWe next recall \\emph{Bretschneider's formula}: for $mznhyqow = (kjhwemct+gqzxalmd+lkhpferu+qbstynev)/2$,\n\\[\nrvxosdip^2 = (mznhyqow-kjhwemct)(mznhyqow-gqzxalmd)(mznhyqow-lkhpferu)(mznhyqow-qbstynev) - kjhwemct\\,gqzxalmd\\,lkhpferu\\,qbstynev \\cos^2 \\frac{\\angle tifghcvaasvnjkeidflpzoim + \\angle dflpzoimwqkemnrbtifghcva}{2}.\n\\]\nConsequently, fixing $rvxosdip$ also fixes $\\cos^2 \\frac{\\angle tifghcvaasvnjkeidflpzoim + \\angle dflpzoimwqkemnrbtifghcva}{2}$, and thus limits $\\angle tifghcvaasvnjkeidflpzoim + \\angle dflpzoimwqkemnrbtifghcva$ to one of two values. By the previous paragraph, this leaves at most two possible congruence classes for the quadrilateral.\n\\end{proof}\n\nReturning to our original sequence, note that any two consecutive quadrilaterals in the sequence have the same area and the same unordered list of side lengths. The latter can occur as an ordered list in at most six different ways (up to cyclic shift); for each of these, we can have only finitely many distinct congruence classes of quadrilaterals in our sequence with that area and ordered list of side lengths. We deduce that our sequence must be finite.\n\n\\noindent\n\\textbf{Remark.}\nVarious proofs of the lemma are possible; for example, here is one using Cartesian coordinates. We first specify \n\\[\nasvnjkei = (0,0), \\quad dflpzoim = (kjhwemct, 0).\n\\]\nFor two additional points $wqkemnrb = (ulgefrto,nvaqpsie),\\;tifghcva = (tmrakbwo,csluipav)$, the conditions $\\overline{dflpzoimwqkemnrb} = gqzxalmd$, $\\overline{tifghcvaasvnjkei} = qbstynev$ restrict $wqkemnrb$ and $tifghcva$ to the circles\n\\[\n(ulgefrto-kjhwemct)^2 + nvaqpsie^2 = gqzxalmd^2, \\quad tmrakbwo^2 + csluipav^2 = qbstynev^2\n\\]\nrespectively. Since we want a convex quadrilateral, we may assume without loss of generality that $nvaqpsie,csluipav > 0$. The area of the quadrilateral is $\\frac{1}{2} kjhwemct(nvaqpsie+csluipav)$, which we also want to fix; we may thus regard $csluipav$ as a function of $nvaqpsie$ (possibly restricting $nvaqpsie$ to a range for which $csluipav>0$). After splitting the semicircles on which $wqkemnrb$ and $tifghcva$ lie into two arcs each, we may also regard $ulgefrto$ and $csluipav$ as functions of $nvaqpsie$. It now suffices to observe that $\\overline{wqkemnrbtifghcva}^2 = (tmrakbwo-ulgefrto)^2 + (csluipav-nvaqpsie)^2$ is a nonconstant algebraic function of $nvaqpsie$, so it takes any given value only finitely many times.\n\n\\noindent\n\\textbf{Second solution.}\nLet $qzxwvtnphjgrkslaplmdvceksrowfgha$ be the first quadrilateral in the sequence. Since the quadrilateral is convex, the diagonals $\\overline{qzxwvtnpplmdvcek}$ and $\\overline{hjgrkslasrowfgha}$ intersect. In particular they are not parallel, so their perpendicular bisectors are not parallel either; let $bxrjzsqu$ be the intersection of the bisectors.\n\nWe claim that the point $bxrjzsqu$ remains fixed throughout the sequence, as do the distances $bxrjzsqquzxwvtnp, bxrjzsquhjgrksla, bxrjzsquplmdvcek, bxrjzsqusrowfgha$. To see this, we check this for two partners as described in the problem statement: the diagonal $\\overline{hjgrkslasrowfgha}$ gets reflected across the perpendicular bisector of $\\overline{qzxwvtnpplmdvcek}$, so its perpendicular bisector also gets reflected; the point $bxrjzsqu$ is the unique point on the perpendicular bisector of $\\overline{hjgrkslasrowfgha}$ fixed by the reflection. In particular, the segments $\\overline{bxrjzsqusrowfgha}$ and $\\overline{bxrjzsquycbntrwe}$ are mirror images across the perpendicular bisector of $\\overline{qzxwvtnpplmdvcek}$, so their lengths coincide.\n\nAs noted in the first solution, the unordered list of side lengths of the quadrilateral also remains invariant throughout the sequence. Consequently, the unordered list of side lengths of each of the triangles $\\triangle bxrjzsqquzxwvtnphjgrksla, \\triangle bxrjzsquhjgrkslaplmdvcek, \\triangle bxrjzsquplmdvceksrowfgha, \\triangle bxrjzsqusrowfghaqzxwvtnp$ is limited to a finite set; each such list uniquely determines the unoriented congruence class of the corresponding triangle, and limits the oriented congruence class to two possibilities. Given the oriented congruence classes of the four triangles we can reconstruct the quadrilateral $qzxwvtnphjgrkslaplmdvceksrowfgha$ up to oriented congruence (even up to rotation around $bxrjzsqu$); this proves that the sequence must be finite."
    },
    "kernel_variant": {
      "question": "Two convex quadrilaterals are called echoes if they share three vertices and can be labelled\nPQR S and PQR T in such a way that T is the reflection of S in the perpendicular\nbisector of the diagonal PR.  Does there exist an infinite sequence\n\n        Q1 , Q2 , Q3 , …\n\nof pairwise non-congruent convex quadrilaterals in which each quadrilateral is an\necho of its successor?",
      "solution": "Answer.  No such infinite sequence exists.\n\nStep 1.  What the echo operation preserves.\nLet the echoes be PQR S and PQR T with T obtained from S by reflection in the\nperpendicular bisector of PR.\n\n(a)  Area.\nTriangles PRS and PRT have the same base PR.  Because the mirror line is the\nperpendicular bisector of PR, the points S and T are equidistant from the line\nPR; consequently these two triangles have equal heights and therefore equal\nareas.  Adding the common area of triangle PQR to each shows that\narea(PQRS) = area(PQRT).\n\n(b)  Side–length multiset.\nThe reflection is an isometry, so\n    PQ, QR, RS, SP\nare sent to\n    PQ, QR, RT, TP.\nMoreover the reflection swaps P with R, whence RT = SP and TP = RS.  Thus the\nmultiset {PQ, QR, RS, SP} is unchanged.\n\nStep 2.  A finiteness lemma.\nLemma.  Fix positive real numbers p,q,r,s and A.  Up to congruence there are\nonly finitely many convex quadrilaterals whose consecutive side–lengths are\np,q,r,s and whose area is A.\n\nProof.\nLet such a quadrilateral be PQR S with\nPQ = p, QR = q, RS = r, SP = s, and let g = QS.  When p,q,r,s are fixed the\nquadrilateral is determined (up to congruence) by g, because the two triangles\nPQS and RQS are then determined by their three sides (p,s,g) and (q,r,g).\nHence we may regard g as a parameter.\n\nMonotonicity of \\(\\angle QPS+\\angle QRS\\).  In triangle PQS the side opposite\n\\(\\angle QPS\\) is g, so by the law of cosines\n  cos\\,\\angle QPS = (p^2 + s^2 - g^2)/(2ps),\nwhich strictly decreases as g increases; therefore \\(\\angle QPS\\) itself\nstrictly increases with g.  An analogous computation for triangle RQS shows\nthat \\(\\angle QRS\\) also strictly increases with g.  Consequently their sum\nφ(g)=\\(\\angle QPS+\\angle QRS\\) is a strictly increasing function of g.\n\nBretschneider’s formula.  For\n    t = (p+q+r+s)/2\nit states\n      A^2 = (t-p)(t-q)(t-r)(t-s) - pqr s · cos^2(φ(g)/2).\nBecause A is fixed, the right–hand side can attain the given value for only the\nfinitely many g for which cos^2(φ(g)/2) assumes one of at most two possible\nvalues.  Thus only finitely many values of g – and hence only finitely many\ncongruence classes of quadrilaterals – realise the prescribed data (p,q,r,s,A).\n∎\n\nStep 3.  Putting the pieces together.\nAn echo chain keeps the area and the multiset of side–lengths invariant.\nFor a given multiset there are at most 12 cyclic orderings around the perimeter\n(24 permutations up to reversal).  Fixing one such ordering fixes the ordered\nquadruple (p,q,r,s), and by the lemma only finitely many congruence classes of\nquadrilaterals can occur with that ordering and the fixed area.  Hence the\nwhole chain can visit only finitely many congruence classes in total, so an\ninfinite sequence of pairwise non-congruent echoes is impossible.",
      "_meta": {
        "core_steps": [
          "Reflection that defines \"partners\" preserves both the area and the multiset of the four side-lengths.",
          "Lemma: with one ordered 4-tuple of side-lengths and a fixed area, only finitely many convex quadrilaterals exist (proved via varying one diagonal and Bretschneider’s formula).",
          "Because the preserved multiset of side-lengths can be arranged in only finitely many cyclic orders (≤6), the entire partner chain can involve only finitely many congruence classes, so an infinite chain is impossible."
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Exact numerical count of the cyclic orders of four labelled sides; any finite count arising from symmetry would still yield finiteness.",
            "original": "6"
          },
          "slot2": {
            "description": "Concrete symbols/values chosen for the fixed side-lengths and area in the lemma; only their being fixed positive reals is required.",
            "original": "a, b, c, d, K"
          },
          "slot3": {
            "description": "Choice of which diagonal’s length is used as the varying parameter inside the lemma’s proof.",
            "original": "the diagonal PR"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "proof",
  "iteratively_fixed": true
}