{
  "index": "1947-A-3",
  "type": "COMB",
  "tag": [
    "COMB",
    "GEO"
  ],
  "difficulty": "",
  "question": "3. Given this figure (see p.23) and any two points \\( Q_{1}, Q_{2} \\) in the plane not lying on any of the segments \\( s_{1}, s_{2}, \\ldots, s_{6} \\), show that there does not exist a polygonal line \\( P \\) joining \\( Q_{1} \\) and \\( Q_{2} \\) such that:\n(i) \\( P \\) crosses each \\( s_{i}, i=1,2, \\ldots, 6 \\), exactly once;\n(ii) \\( \\boldsymbol{P} \\) does not intersect itself;\n(iii) \\( P \\) does not pass through any vertex \\( V_{1}, V_{2}, V_{3}, V_{4} \\).",
  "solution": "Solution. We begin with a lemma.\nLemma. If a polygonal line in the plane of a triangle passes through no vertex of the triangle, crosses each side exactly once, and has neither endpoint on the boundary of the triangle, then one of its endpoints is interior to the triangle.\n\nSuppose there exists a polygonal line \\( P \\) as described in the problem. \\( P \\) crosses each side of each of the triangles \\( V V_{1} V_{2}, V V_{2} V_{3} \\), and \\( V V_{3} V_{1} \\) exactly once, so it must have one end in the interior of each of these triangles. But the interiors of these triangles are disjoint and \\( P \\) has only two ends, so this is impossible.\n\nProof of the Lemma. Let \\( T \\) be the triangle, and suppose the polygonal line \\( P \\) with endpoints \\( Q_{1} \\) and \\( Q_{2} \\) satisfies the hypothesis.\n\nIf \\( Q_{1} \\) is in the interior of \\( T \\), there is nothing to prove, so we assume \\( Q_{1} \\) is in the exterior of \\( T \\). As we move along \\( P \\) from \\( Q_{1} \\) toward \\( Q_{2} \\) we encounter only exterior points until we cross the first side of \\( T \\); then we encounter only interior points until we cross the second side of \\( T \\). After that we encounter exterior points until we cross the third side of \\( T \\) and from there on all points, including the endpoint \\( Q_{2} \\), are in the interior. Thus \\( \\boldsymbol{Q}_{2} \\) is an interior point.\n\nDiscussion. The solution is easy enough and this is, no doubt, what the examining committee had in mind. However, the entire argument can be criticized because, for example, nowhere is it made clear what we mean by \" \\( P \\) crosses a segment exactly once.\" The definition of this phrase must be carefully worded so as not to count the spurious crossing shown in Figure 1, but to allow for the crossing of Figure 2.\n\nThe restriction to non-self-intersecting polygonal lines is not actually necessary but was probably intended to obviate another technicality sug. gested by Figure 3.\n\nFig. 1.\nFig. 2.\n\nFig. 3.\n\nSuch technicalities can all be avoided by restricting consideration to polygonal lines that are in general position with respect to the figure, that is, no vertex of \\( P \\) should lie on any of the segments \\( s_{i} \\). With this restriction, the number of times \\( P \\) crosses a segment \\( s \\) is just the number of segments of \\( P \\) that meet \\( s \\).\n\nThe problem is a simplification of the famous bridges-of-Königsberg problem that is discussed in many expository books on mathematics. See, for example, S. K. Stein, Mathematics, the Man-Made Universe, Freeman, San Francisco, 1963.",
  "vars": [
    "P",
    "Q_1",
    "Q_2",
    "T",
    "s",
    "i"
  ],
  "params": [
    "s_1",
    "s_2",
    "s_3",
    "s_4",
    "s_5",
    "s_6",
    "V",
    "V_1",
    "V_2",
    "V_3",
    "V_4"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "P": "polygonpath",
        "Q_1": "startpoint",
        "Q_2": "endpoint",
        "T": "triangle",
        "s": "segment",
        "i": "indexvar",
        "s_1": "segmentone",
        "s_2": "segmenttwo",
        "s_3": "segmentthree",
        "s_4": "segmentfour",
        "s_5": "segmentfive",
        "s_6": "segmentsix",
        "V": "centervertex",
        "V_1": "vertexone",
        "V_2": "vertextwo",
        "V_3": "vertexthree",
        "V_4": "vertexfour"
      },
      "question": "3. Given this figure (see p.23) and any two points \\( startpoint, endpoint \\) in the plane not lying on any of the segments \\( segmentone, segmenttwo, \\ldots, segmentsix \\), show that there does not exist a polygonal line \\( polygonpath \\) joining \\( startpoint \\) and \\( endpoint \\) such that:\n(i) \\( polygonpath \\) crosses each \\( segment_{indexvar}, indexvar=1,2, \\ldots, 6 \\), exactly once;\n(ii) \\( \\boldsymbol{polygonpath} \\) does not intersect itself;\n(iii) \\( polygonpath \\) does not pass through any vertex \\( vertexone, vertextwo, vertexthree, vertexfour \\).",
      "solution": "Solution. We begin with a lemma.\nLemma. If a polygonal line in the plane of a triangle passes through no vertex of the triangle, crosses each side exactly once, and has neither endpoint on the boundary of the triangle, then one of its endpoints is interior to the triangle.\n\nSuppose there exists a polygonal line \\( polygonpath \\) as described in the problem. \\( polygonpath \\) crosses each side of each of the triangles \\( centervertex vertexone vertextwo, centervertex vertextwo vertexthree, and centervertex vertexthree vertexone \\) exactly once, so it must have one end in the interior of each of these triangles. But the interiors of these triangles are disjoint and \\( polygonpath \\) has only two ends, so this is impossible.\n\nProof of the Lemma. Let \\( triangle \\) be the triangle, and suppose the polygonal line \\( polygonpath \\) with endpoints \\( startpoint \\) and \\( endpoint \\) satisfies the hypothesis.\n\nIf \\( startpoint \\) is in the interior of \\( triangle \\), there is nothing to prove, so we assume \\( startpoint \\) is in the exterior of \\( triangle \\). As we move along \\( polygonpath \\) from \\( startpoint \\) toward \\( endpoint \\) we encounter only exterior points until we cross the first side of \\( triangle \\); then we encounter only interior points until we cross the second side of \\( triangle \\). After that we encounter exterior points until we cross the third side of \\( triangle \\) and from there on all points, including the endpoint \\( endpoint \\), are in the interior. Thus \\( \\boldsymbol{endpoint} \\) is an interior point.\n\nDiscussion. The solution is easy enough and this is, no doubt, what the examining committee had in mind. However, the entire argument can be criticized because, for example, nowhere is it made clear what we mean by \" \\( polygonpath \\) crosses a segment exactly once.\" The definition of this phrase must be carefully worded so as not to count the spurious crossing shown in Figure 1, but to allow for the crossing of Figure 2.\n\nThe restriction to non-self-intersecting polygonal lines is not actually necessary but was probably intended to obviate another technicality suggested by Figure 3.\n\nFig. 1.\nFig. 2.\n\nFig. 3.\n\nSuch technicalities can all be avoided by restricting consideration to polygonal lines that are in general position with respect to the figure, that is, no vertex of \\( polygonpath \\) should lie on any of the segments \\( segment_{indexvar} \\). With this restriction, the number of times \\( polygonpath \\) crosses a segment \\( segment \\) is just the number of segments of \\( polygonpath \\) that meet \\( segment \\).\n\nThe problem is a simplification of the famous bridges-of-Königsberg problem that is discussed in many expository books on mathematics. See, for example, S. K. Stein, Mathematics, the Man-Made Universe, Freeman, San Francisco, 1963."
    },
    "descriptive_long_confusing": {
      "map": {
        "P": "blueprint",
        "Q_1": "cinnamon",
        "Q_2": "isoprene",
        "T": "lighthouse",
        "s": "marigolds",
        "i": "nectarine",
        "s_1": "parchment",
        "s_2": "quartzite",
        "s_3": "rainstorm",
        "s_4": "sandpaper",
        "s_5": "tangerine",
        "s_6": "unicycle",
        "V": "victorian",
        "V_1": "waterfall",
        "V_2": "xylophone",
        "V_3": "yardstick",
        "V_4": "zeppelin"
      },
      "question": "3. Given this figure (see p.23) and any two points \\( cinnamon, isoprene \\) in the plane not lying on any of the segments \\( parchment, quartzite, \\ldots, unicycle \\), show that there does not exist a polygonal line \\( blueprint \\) joining \\( cinnamon \\) and \\( isoprene \\) such that:\n(i) \\( blueprint \\) crosses each \\( marigolds_{nectarine},\\; nectarine=1,2, \\ldots, 6 \\), exactly once;\n(ii) \\( \\boldsymbol{blueprint} \\) does not intersect itself;\n(iii) \\( blueprint \\) does not pass through any vertex \\( waterfall, xylophone, yardstick, zeppelin \\).",
      "solution": "Solution. We begin with a lemma.\nLemma. If a polygonal line in the plane of a triangle passes through no vertex of the triangle, crosses each side exactly once, and has neither endpoint on the boundary of the triangle, then one of its endpoints is interior to the triangle.\n\nSuppose there exists a polygonal line \\( blueprint \\) as described in the problem. \\( blueprint \\) crosses each side of each of the triangles \\( victorian waterfall xylophone, victorian xylophone yardstick \\), and \\( victorian yardstick waterfall \\) exactly once, so it must have one end in the interior of each of these triangles. But the interiors of these triangles are disjoint and \\( blueprint \\) has only two ends, so this is impossible.\n\nProof of the Lemma. Let \\( lighthouse \\) be the triangle, and suppose the polygonal line \\( blueprint \\) with endpoints \\( cinnamon \\) and \\( isoprene \\) satisfies the hypothesis.\n\nIf \\( cinnamon \\) is in the interior of \\( lighthouse \\), there is nothing to prove, so we assume \\( cinnamon \\) is in the exterior of \\( lighthouse \\). As we move along \\( blueprint \\) from \\( cinnamon \\) toward \\( isoprene \\) we encounter only exterior points until we cross the first side of \\( lighthouse \\); then we encounter only interior points until we cross the second side of \\( lighthouse \\). After that we encounter exterior points until we cross the third side of \\( lighthouse \\) and from there on all points, including the endpoint \\( isoprene \\), are in the interior. Thus \\( \\boldsymbol{isoprene} \\) is an interior point.\n\nDiscussion. The solution is easy enough and this is, no doubt, what the examining committee had in mind. However, the entire argument can be criticized because, for example, nowhere is it made clear what we mean by \" \\( blueprint \\) crosses a segment exactly once.\" The definition of this phrase must be carefully worded so as not to count the spurious crossing shown in Figure 1, but to allow for the crossing of Figure 2.\n\nThe restriction to non-self-intersecting polygonal lines is not actually necessary but was probably intended to obviate another technicality suggested by Figure 3.\n\nFig. 1.\nFig. 2.\n\nFig. 3.\n\nSuch technicalities can all be avoided by restricting consideration to polygonal lines that are in general position with respect to the figure, that is, no vertex of \\( blueprint \\) should lie on any of the segments \\( marigolds_{nectarine} \\). With this restriction, the number of times \\( blueprint \\) crosses a segment \\( marigolds \\) is just the number of segments of \\( blueprint \\) that meet \\( marigolds \\).\n\nThe problem is a simplification of the famous bridges-of-Königsberg problem that is discussed in many expository books on mathematics. See, for example, S. K. Stein, Mathematics, the Man-Made Universe, Freeman, San Francisco, 1963."
    },
    "descriptive_long_misleading": {
      "map": {
        "P": "straightline",
        "Q_1": "interiorpointone",
        "Q_2": "interiorpointtwo",
        "T": "circularshape",
        "s": "anglezone",
        "i": "cofactor",
        "s_1": "angleone",
        "s_2": "angletwo",
        "s_3": "anglethree",
        "s_4": "anglefour",
        "s_5": "anglefive",
        "s_6": "anglesix",
        "V": "edgenode",
        "V_1": "edgenodeone",
        "V_2": "edgenodetwo",
        "V_3": "edgenodethree",
        "V_4": "edgenodefour"
      },
      "question": "3. Given this figure (see p.23) and any two points \\( interiorpointone, interiorpointtwo \\) in the plane not lying on any of the segments \\( angleone, angletwo, \\ldots, anglesix \\), show that there does not exist a polygonal line \\( straightline \\) joining \\( interiorpointone \\) and \\( interiorpointtwo \\) such that:\n(i) \\( straightline \\) crosses each \\( anglezone_{cofactor}, cofactor=1,2, \\ldots, 6 \\), exactly once;\n(ii) \\( \\boldsymbol{straightline} \\) does not intersect itself;\n(iii) \\( straightline \\) does not pass through any vertex \\( edgenodeone, edgenodetwo, edgenodethree, edgenodefour \\).",
      "solution": "Solution. We begin with a lemma.\nLemma. If a polygonal line in the plane of a triangle passes through no vertex of the triangle, crosses each side exactly once, and has neither endpoint on the boundary of the triangle, then one of its endpoints is interior to the triangle.\n\nSuppose there exists a polygonal line \\( straightline \\) as described in the problem. \\( straightline \\) crosses each side of each of the triangles \\( edgenode\\, edgenodeone\\, edgenodetwo, edgenode\\, edgenodetwo\\, edgenodethree \\), and \\( edgenode\\, edgenodethree\\, edgenodeone \\) exactly once, so it must have one end in the interior of each of these triangles. But the interiors of these triangles are disjoint and \\( straightline \\) has only two ends, so this is impossible.\n\nProof of the Lemma. Let \\( circularshape \\) be the triangle, and suppose the polygonal line \\( straightline \\) with endpoints \\( interiorpointone \\) and \\( interiorpointtwo \\) satisfies the hypothesis.\n\nIf \\( interiorpointone \\) is in the interior of \\( circularshape \\), there is nothing to prove, so we assume \\( interiorpointone \\) is in the exterior of \\( circularshape \\). As we move along \\( straightline \\) from \\( interiorpointone \\) toward \\( interiorpointtwo \\) we encounter only exterior points until we cross the first side of \\( circularshape \\); then we encounter only interior points until we cross the second side of \\( circularshape \\). After that we encounter exterior points until we cross the third side of \\( circularshape \\) and from there on all points, including the endpoint \\( interiorpointtwo \\), are in the interior. Thus \\( \\boldsymbol{interiorpointtwo} \\) is an interior point.\n\nDiscussion. The solution is easy enough and this is, no doubt, what the examining committee had in mind. However, the entire argument can be criticized because, for example, nowhere is it made clear what we mean by \" \\( straightline \\) crosses a segment exactly once.\" The definition of this phrase must be carefully worded so as not to count the spurious crossing shown in Figure 1, but to allow for the crossing of Figure 2.\n\nThe restriction to non-self-intersecting polygonal lines is not actually necessary but was probably intended to obviate another technicality suggested by Figure 3.\n\nFig. 1.\nFig. 2.\n\nFig. 3.\n\nSuch technicalities can all be avoided by restricting consideration to polygonal lines that are in general position with respect to the figure, that is, no vertex of \\( straightline \\) should lie on any of the segments \\( anglezone_{cofactor} \\). With this restriction, the number of times \\( straightline \\) crosses a segment \\( anglezone \\) is just the number of segments of \\( straightline \\) that meet \\( anglezone \\).\n\nThe problem is a simplification of the famous bridges-of-Königsberg problem that is discussed in many expository books on mathematics. See, for example, S. K. Stein, Mathematics, the Man-Made Universe, Freeman, San Francisco, 1963."
    },
    "garbled_string": {
      "map": {
        "P": "hjgrksla",
        "Q_1": "ziboqtuv",
        "Q_2": "xynumbrq",
        "T": "kpewacdl",
        "s": "vruecmtg",
        "i": "wpyjfzso",
        "s_1": "florkejb",
        "s_2": "ruscenqa",
        "s_3": "vamoptds",
        "s_4": "qlizhnrw",
        "s_5": "nyetcagb",
        "s_6": "dplskhfi",
        "V": "qxrimine",
        "V_1": "ltmredsx",
        "V_2": "bhcqygzo",
        "V_3": "awvizkun",
        "V_4": "ixmoldea"
      },
      "question": "3. Given this figure (see p.23) and any two points \\( ziboqtuv, xynumbrq \\) in the plane not lying on any of the segments \\( florkejb, ruscenqa, \\ldots, dplskhfi \\), show that there does not exist a polygonal line \\( hjgrksla \\) joining \\( ziboqtuv \\) and \\( xynumbrq \\) such that:\n(i) \\( hjgrksla \\) crosses each \\( vruecmtg_{wpyjfzso}, wpyjfzso=1,2, \\ldots, 6 \\), exactly once;\n(ii) \\( \\boldsymbol{hjgrksla} \\) does not intersect itself;\n(iii) \\( hjgrksla \\) does not pass through any vertex \\( ltmredsx, bhcqygzo, awvizkun, ixmoldea \\).",
      "solution": "Solution. We begin with a lemma.\nLemma. If a polygonal line in the plane of a triangle passes through no vertex of the triangle, crosses each side exactly once, and has neither endpoint on the boundary of the triangle, then one of its endpoints is interior to the triangle.\n\nSuppose there exists a polygonal line \\( hjgrksla \\) as described in the problem. \\( hjgrksla \\) crosses each side of each of the triangles \\( qxrimine ltmredsx bhcqygzo, qxrimine bhcqygzo awvizkun \\), and \\( qxrimine awvizkun ltmredsx \\) exactly once, so it must have one end in the interior of each of these triangles. But the interiors of these triangles are disjoint and \\( hjgrksla \\) has only two ends, so this is impossible.\n\nProof of the Lemma. Let \\( kpewacdl \\) be the triangle, and suppose the polygonal line \\( hjgrksla \\) with endpoints \\( ziboqtuv \\) and \\( xynumbrq \\) satisfies the hypothesis.\n\nIf \\( ziboqtuv \\) is in the interior of \\( kpewacdl \\), there is nothing to prove, so we assume \\( ziboqtuv \\) is in the exterior of \\( kpewacdl \\). As we move along \\( hjgrksla \\) from \\( ziboqtuv \\) toward \\( xynumbrq \\) we encounter only exterior points until we cross the first side of \\( kpewacdl \\); then we encounter only interior points until we cross the second side of \\( kpewacdl \\). After that we encounter exterior points until we cross the third side of \\( kpewacdl \\) and from there on all points, including the endpoint \\( xynumbrq \\), are in the interior. Thus \\( \\boldsymbol{xynumbrq} \\) is an interior point.\n\nDiscussion. The solution is easy enough and this is, no doubt, what the examining committee had in mind. However, the entire argument can be criticized because, for example, nowhere is it made clear what we mean by \" \\( hjgrksla \\) crosses a segment exactly once.\" The definition of this phrase must be carefully worded so as not to count the spurious crossing shown in Figure 1, but to allow for the crossing of Figure 2.\n\nThe restriction to non-self-intersecting polygonal lines is not actually necessary but was probably intended to obviate another technicality sug. gested by Figure 3.\n\nFig. 1.\nFig. 2.\n\nFig. 3.\n\nSuch technicalities can all be avoided by restricting consideration to polygonal lines that are in general position with respect to the figure, that is, no vertex of \\( hjgrksla \\) should lie on any of the segments \\( vruecmtg_{wpyjfzso} \\). With this restriction, the number of times \\( hjgrksla \\) crosses a segment \\( vruecmtg \\) is just the number of segments of \\( hjgrksla \\) that meet \\( vruecmtg \\).\n\nThe problem is a simplification of the famous bridges-of-Königsberg problem that is discussed in many expository books on mathematics. See, for example, S. K. Stein, Mathematics, the Man-Made Universe, Freeman, San Francisco, 1963."
    },
    "kernel_variant": {
      "question": "Let $\\mathcal G$ be a finite straight–line embedding of a connected planar graph in the Euclidean plane such that  \n\n• the boundary of its unbounded (outer) face is a convex polygon;  \n\n• every bounded face is a non–degenerate triangle (so $\\mathcal G$ is a geometric triangulation);  \n\n• the embedding is in general position: no three vertices of $\\mathcal G$ are collinear and no vertex of $\\mathcal G$ lies in the interior of an edge it is not incident with.  \n\nDenote by $\\mathcal E$ the set of straight edges of $\\mathcal G$.  \nFix two distinct points $Q_{1},Q_{2}$ that lie strictly outside the outer face and outside every edge of $\\mathcal G$.\n\nFor a polygonal path $P$ joining $Q_{1}$ to $Q_{2}$ we say that $P$ {\\it crosses} an edge $e\\in\\mathcal E$ {\\it exactly once} if  \n\n(i) no vertex of $P$ lies on $e$, and  \n\n(ii) exactly one segment of $P$ meets $e$ transversally at an interior point of that segment.\n\nProve that there is no simple (i.e. non-self-intersecting) polygonal path $P$ that  \n\n(1) crosses every edge $e\\in\\mathcal E$ exactly once, and  \n\n(2) passes through none of the vertices of $\\mathcal G$.\n\n(Equivalently: a Jordan arc cannot meet each edge of a planar straight-line triangulation exactly once.)",
      "solution": "Throughout we place $P$ in general position with respect to $\\mathcal G$: no vertex of $P$ lies on an edge of $\\mathcal G$, every intersection of $P$ with $\\mathcal G$ is transversal, and no point of the plane is the intersection of three or more relevant segments.  \nMark every point where $P$ meets an edge of $\\mathcal G$; the marked points split $P$ into finitely many open sub–arcs which lie alternately in adjacent faces of the triangulation.\n\nStep 1 – The dual multigraph.  \nForm the (geometric) dual $\\mathcal G^{*}$ as follows.\n\n• Each bounded triangular face $F$ of $\\mathcal G$ gives a vertex $F^{*}$ of $\\mathcal G^{*}$.  \n• The single unbounded face gives the outer vertex $U^{*}$.  \n• For every edge $e$ of $\\mathcal G$ incident with the two (not necessarily distinct) faces $F$ and $G$ we draw a dual edge $e^{*}$ connecting the corresponding dual vertices $F^{*}$ and $G^{*}$, chosen so that $e^{*}$ crosses $e$ exactly once and is otherwise disjoint from $\\mathcal G$.\n\nBecause two distinct edges of $\\mathcal G$ can bound the same ordered pair of faces, $\\mathcal G^{*}$ may contain parallel edges; however, no edge is a loop.  The graph $\\mathcal G^{*}$ is connected, planar, and {\\it cubic} in its interior: every vertex corresponding to a bounded triangular face has degree $3$, while $\\deg(U^{*})=m\\ge 3$, where $m$ is the number of sides of the outer convex polygon.\n\nStep 2 – Translating the motion of $P$ into the dual.  \nTraverse $P$ from $Q_{1}$ to $Q_{2}$ and record the faces of $\\mathcal G$ that the successive sub–arcs of $P$ occupy.  Whenever $P$ crosses a primal edge $e$ that separates two faces $F$ and $G$, the path leaves $F$ and immediately enters $G$.  We therefore step in the dual from $F^{*}$ to $G^{*}$ along the dual edge $e^{*}$.  In this way we obtain a walk  \n\\[\n\\mathcal T \\;=\\; (U^{*}=v_{0},\\,v_{1},\\,\\dots ,\\,v_{k}=U^{*})\n\\]\nin $\\mathcal G^{*}$ with the following properties:\n\n• $\\mathcal T$ starts and ends at the outer vertex $U^{*}$, because $Q_{1}$ and $Q_{2}$ lie in the unbounded face;  \n\n• each dual edge is traversed {\\it exactly once}.  Indeed, $P$ crosses every primal edge precisely once by hypothesis, so the corresponding dual edge appears exactly once in $\\mathcal T$.\n\nConsequently $\\mathcal T$ is an {\\it Eulerian circuit} of the (multi-)graph $\\mathcal G^{*}$.\n\nStep 3 – The parity obstruction.  \nEuler’s classical criterion extends verbatim to finite connected multigraphs without loops: such a graph possesses an Eulerian circuit if and only if every vertex has even degree.  In $\\mathcal G^{*}$ every interior vertex has degree $3$, which is odd, so at least one vertex of $\\mathcal G^{*}$ has odd degree.  Therefore an Eulerian circuit in $\\mathcal G^{*}$ cannot exist.\n\nStep 4 – Contradiction.  \nThe existence of the polygonal path $P$ would force the Eulerian circuit $\\mathcal T$ in $\\mathcal G^{*}$ found in Step 2, contradicting Step 3.  Hence no polygonal path with the stated properties exists.  This completes the proof. $\\square$",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.409768",
        "was_fixed": false,
        "difficulty_analysis": "1. Scope.  The statement no longer concerns a single quadrilateral with one interior point but an arbitrary planar triangulation; the number of edges is unbounded and their mutual arrangement is far more intricate.  \n\n2. New structures.  The solver must build and exploit the dual graph 𝓖*, connect geometric crossings to combinatorial edge-traversals, and invoke Euler’s necessary-and-sufficient condition for Eulerian circuits.  This mixture of plane topology, geometric graph theory, and classical combinatorics is absent from the original problem.  \n\n3. Global argument.  Local “triangle-endpoint” counting is insufficient; one must translate the entire path into a walk in the dual and use parity of degrees for all vertices simultaneously – a significantly deeper insight.  \n\n4. General position subtleties.  Ensuring that edges, crossings, and faces behave well demands an explicit treatment of transversality and general-position hypotheses, which adds further technical load.  \n\n5. Abstraction.  The claim is formulated for every planar triangulation (with any number of vertices and faces), not for one concrete drawing; purely case-checking or ad-hoc geometric intuition is impossible.  A fully-fledged theorem (Euler’s) must be invoked, elevating the conceptual level.  \n\nFor these reasons the enhanced variant is decisively more sophisticated and substantially harder than both the original olympiad problem and the current kernel variant."
      }
    },
    "original_kernel_variant": {
      "question": "Let $\\mathcal G$ be a finite straight–line embedding of a connected planar graph in the Euclidean plane such that  \n\n• the boundary of its unbounded (outer) face is a convex polygon;  \n\n• every bounded face is a non–degenerate triangle (so $\\mathcal G$ is a geometric triangulation);  \n\n• the embedding is in general position: no three vertices of $\\mathcal G$ are collinear and no vertex of $\\mathcal G$ lies in the interior of an edge it is not incident with.  \n\nDenote by $\\mathcal E$ the set of straight edges of $\\mathcal G$.  \nFix two distinct points $Q_{1},Q_{2}$ that lie strictly outside the outer face and outside every edge of $\\mathcal G$.\n\nFor a polygonal path $P$ joining $Q_{1}$ to $Q_{2}$ we say that $P$ {\\it crosses} an edge $e\\in\\mathcal E$ {\\it exactly once} if  \n\n(i) no vertex of $P$ lies on $e$, and  \n\n(ii) exactly one segment of $P$ meets $e$ transversally at an interior point of that segment.\n\nProve that there is no simple (i.e. non-self-intersecting) polygonal path $P$ that  \n\n(1) crosses every edge $e\\in\\mathcal E$ exactly once, and  \n\n(2) passes through none of the vertices of $\\mathcal G$.\n\n(Equivalently: a Jordan arc cannot meet each edge of a planar straight-line triangulation exactly once.)",
      "solution": "Throughout we place $P$ in general position with respect to $\\mathcal G$: no vertex of $P$ lies on an edge of $\\mathcal G$, every intersection of $P$ with $\\mathcal G$ is transversal, and no point of the plane is the intersection of three or more relevant segments.  \nMark every point where $P$ meets an edge of $\\mathcal G$; the marked points split $P$ into finitely many open sub–arcs which lie alternately in adjacent faces of the triangulation.\n\nStep 1 – The dual multigraph.  \nForm the (geometric) dual $\\mathcal G^{*}$ as follows.\n\n• Each bounded triangular face $F$ of $\\mathcal G$ gives a vertex $F^{*}$ of $\\mathcal G^{*}$.  \n• The single unbounded face gives the outer vertex $U^{*}$.  \n• For every edge $e$ of $\\mathcal G$ incident with the two (not necessarily distinct) faces $F$ and $G$ we draw a dual edge $e^{*}$ connecting the corresponding dual vertices $F^{*}$ and $G^{*}$, chosen so that $e^{*}$ crosses $e$ exactly once and is otherwise disjoint from $\\mathcal G$.\n\nBecause two distinct edges of $\\mathcal G$ can bound the same ordered pair of faces, $\\mathcal G^{*}$ may contain parallel edges; however, no edge is a loop.  The graph $\\mathcal G^{*}$ is connected, planar, and {\\it cubic} in its interior: every vertex corresponding to a bounded triangular face has degree $3$, while $\\deg(U^{*})=m\\ge 3$, where $m$ is the number of sides of the outer convex polygon.\n\nStep 2 – Translating the motion of $P$ into the dual.  \nTraverse $P$ from $Q_{1}$ to $Q_{2}$ and record the faces of $\\mathcal G$ that the successive sub–arcs of $P$ occupy.  Whenever $P$ crosses a primal edge $e$ that separates two faces $F$ and $G$, the path leaves $F$ and immediately enters $G$.  We therefore step in the dual from $F^{*}$ to $G^{*}$ along the dual edge $e^{*}$.  In this way we obtain a walk  \n\\[\n\\mathcal T \\;=\\; (U^{*}=v_{0},\\,v_{1},\\,\\dots ,\\,v_{k}=U^{*})\n\\]\nin $\\mathcal G^{*}$ with the following properties:\n\n• $\\mathcal T$ starts and ends at the outer vertex $U^{*}$, because $Q_{1}$ and $Q_{2}$ lie in the unbounded face;  \n\n• each dual edge is traversed {\\it exactly once}.  Indeed, $P$ crosses every primal edge precisely once by hypothesis, so the corresponding dual edge appears exactly once in $\\mathcal T$.\n\nConsequently $\\mathcal T$ is an {\\it Eulerian circuit} of the (multi-)graph $\\mathcal G^{*}$.\n\nStep 3 – The parity obstruction.  \nEuler’s classical criterion extends verbatim to finite connected multigraphs without loops: such a graph possesses an Eulerian circuit if and only if every vertex has even degree.  In $\\mathcal G^{*}$ every interior vertex has degree $3$, which is odd, so at least one vertex of $\\mathcal G^{*}$ has odd degree.  Therefore an Eulerian circuit in $\\mathcal G^{*}$ cannot exist.\n\nStep 4 – Contradiction.  \nThe existence of the polygonal path $P$ would force the Eulerian circuit $\\mathcal T$ in $\\mathcal G^{*}$ found in Step 2, contradicting Step 3.  Hence no polygonal path with the stated properties exists.  This completes the proof. $\\square$",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.351964",
        "was_fixed": false,
        "difficulty_analysis": "1. Scope.  The statement no longer concerns a single quadrilateral with one interior point but an arbitrary planar triangulation; the number of edges is unbounded and their mutual arrangement is far more intricate.  \n\n2. New structures.  The solver must build and exploit the dual graph 𝓖*, connect geometric crossings to combinatorial edge-traversals, and invoke Euler’s necessary-and-sufficient condition for Eulerian circuits.  This mixture of plane topology, geometric graph theory, and classical combinatorics is absent from the original problem.  \n\n3. Global argument.  Local “triangle-endpoint” counting is insufficient; one must translate the entire path into a walk in the dual and use parity of degrees for all vertices simultaneously – a significantly deeper insight.  \n\n4. General position subtleties.  Ensuring that edges, crossings, and faces behave well demands an explicit treatment of transversality and general-position hypotheses, which adds further technical load.  \n\n5. Abstraction.  The claim is formulated for every planar triangulation (with any number of vertices and faces), not for one concrete drawing; purely case-checking or ad-hoc geometric intuition is impossible.  A fully-fledged theorem (Euler’s) must be invoked, elevating the conceptual level.  \n\nFor these reasons the enhanced variant is decisively more sophisticated and substantially harder than both the original olympiad problem and the current kernel variant."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}