Bariz - Problem

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Tags: Iran, IGO, 2018 igo, geometry

bgn

20.09.2018 10:27

There are some segments on the plane such that no two of them intersect each other (even at the ending points). We say segment $AB$ breaks segment $CD$ if the extension of $AB$ cuts $CD$ at some point between $C$ and $D$. [asy][asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(4cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -5.267474904743955, xmax = 11.572179069738377, ymin = -10.642621257034536, ymax = 4.543526642434019; /* image dimensions */ /* draw figures */ draw((-4,-2)--(1.08,-2.03), linewidth(2)); draw(shift((-2.1866176795507295,-2.0107089507113147))*scale(0.21166666666666667)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */ draw((-0.16981767035094117,3.225314210196242)--(-2.1866176795507295,-2.0107089507113147), linewidth(2) + linetype("4 4")); draw((-0.16981767035094117,3.225314210196242)--(-0.8194002739586808,1.538865607509914), linewidth(2)); label("$A$",(-1.2684397405642523,3.860690076971137),SE*labelscalefactor,fontsize(16)); label("$B$",(-1.9211395070170559,2.002590777612728),SE*labelscalefactor,fontsize(16)); label("$C$",(-4.971261820527631,-1.6571211388676117),SE*labelscalefactor,fontsize(16)); label("$D$",(1.08925640451367566,-1.6571211388676117),SE*labelscalefactor,fontsize(16)); /* dots and labels */ dot((-4,-2),dotstyle); dot((1.08,-2.03),dotstyle); dot((-0.16981767035094117,3.225314210196242),dotstyle); dot((-0.8194002739586808,1.538865607509914),dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy][/asy] $a)$ Is it possible that each segment when extended from both ends, breaks exactly one other segment from each way? [asy][asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(4cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -6.8, xmax = 8.68, ymin = -10.32, ymax = 3.64; /* image dimensions */ /* draw figures */ draw((-2.56,1.24)--(-0.36,1.4), linewidth(2)); draw((-3.32,-2.68)--(-1.24,-3.08), linewidth(2)); draw(shift((-2.551651190956802,-2.8277593863544612))*scale(0.17638888888888887)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */ draw(shift((-0.8889576602618603,1.3615303519809556))*scale(0.17638888888888887)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */ draw((-2.551651190956802,-2.8277593863544612)--(-0.8889576602618603,1.3615303519809556), linewidth(2) + linetype("4 4")); draw((-1.4097008194020806,0.049476186483185636)--(-1.8514772275312024,-1.0636149148218605), linewidth(2)); /* dots and labels */ dot((-2.56,1.24),dotstyle); dot((-0.36,1.4),dotstyle); dot((-3.32,-2.68),dotstyle); dot((-1.24,-3.08),dotstyle); dot((-1.4097008194020806,0.049476186483185636),dotstyle); dot((-1.8514772275312024,-1.0636149148218605),dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy][/asy] $b)$ A segment is called surrounded if from both sides of it, there is exactly one segment that breaks it. (e.g. segment $AB$ in the figure.) Is it possible to have all segments to be surrounded? [asy][asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(7cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -10.70976151557872, xmax = 18.64292748469251, ymin = -16.354300717041443, ymax = 9.136192362141452; /* image dimensions */ /* draw figures */ draw((1.0313140845297686,0.748205038977829)--(-1.3,-4), linewidth(2.8)); draw((-5.780195085389632,-2.13088646583346)--(-2.549994860479401,-2.13088646583346), linewidth(2.8)); draw((4.121070821400425,-3.816208322308361)--(1.78,-1.88), linewidth(2.8)); draw(shift((-0.38228674372374466,-2.13088646583346))*scale(0.21166666666666667)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */ draw((-2.549994860479401,-2.13088646583346)--(-0.38228674372374466,-2.13088646583346), linewidth(2.8) + linetype("4 4")); draw(shift((0.32979226045261084,-0.6805897691262632))*scale(0.21166666666666667)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */ draw((4.121070821400425,-3.816208322308361)--(0.32979226045261084,-0.6805897691262632), linewidth(2.8) + linetype("4 4")); draw((-3.6313140845297687,-8.74820503897783)--(3.600422205681574,5.980726991931396), linewidth(2.8) + linetype("2 2")); label("$A$",(-0.397698406272906,1.754593418658662),SE*labelscalefactor,fontsize(16)); label("$B$",(-2.6377720405041316,-3.266261278756151),SE*labelscalefactor,fontsize(16)); /* dots and labels */ dot((1.0313140845297686,0.748205038977829),linewidth(6pt) + dotstyle); dot((-1.3,-4),linewidth(6pt) + dotstyle); dot((-5.780195085389632,-2.13088646583346),linewidth(6pt) + dotstyle); dot((-2.549994860479401,-2.13088646583346),linewidth(6pt) + dotstyle); dot((4.121070821400425,-3.816208322308361),linewidth(6pt) + dotstyle); dot((1.78,-1.88),linewidth(6pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy][/asy] Proposed by Morteza Saghafian

Enigma714

22.02.2024 01:28

aWe'll prove it is not possible. Let $A_1B_1,\dots,A_nB_n$ be the segments. Take a circle such that cover all the vertices $A_1,\dots,A_n$ and $B_1,\dots,B_n$. Clearly we can reduce the circle such that a vertex, wlog $A_1$, its on the circumference. [asy][asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(10cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -14.025072561607061, xmax = 14.704173025957557, ymin = -8.044830303919722, ymax = 9.192717048619048; /* image dimensions */ pen ffqqff = rgb(1,0,1); /* draw figures */ draw(circle((-3.05,-0.74), 5.685384771499638), linewidth(1.2) + ffqqff); draw((-4.49,4.76)--(-2.87,4.26), linewidth(2)); draw((-6.89,-0.42)--(-5.01,-2.82), linewidth(2)); draw((-0.35,0.18)--(-1.67,-1.54), linewidth(2)); draw((1.31,-2.34)--(-0.41,-1.38), linewidth(2)); draw((0.37,0)--(0.25,1.66), linewidth(2)); draw((1.11,-3.38)--(1.85,-2.08), linewidth(2)); draw((-5.17,2)--(-5.87,0.46), linewidth(2)); draw((-3.91,-5.04)--(-1.35,-3.88), linewidth(2)); draw((-5.17,-4.14)--(-6.83,-3.3), linewidth(2)); /* dots and labels */ dot((-4.49,4.76),dotstyle); label("$A_1$", (-4.801472662441576,5.236145507173746), NE * labelscalefactor); dot((-2.87,4.26),dotstyle); label("$B_1$", (-2.659379789684565,4.480112728553625), NE * labelscalefactor); dot((-6.89,-0.42),dotstyle); dot((-5.01,-2.82),dotstyle); dot((-0.35,0.18),dotstyle); dot((-1.67,-1.54),dotstyle); dot((1.31,-2.34),dotstyle); dot((-0.41,-1.38),dotstyle); dot((0.37,0),dotstyle); dot((0.25,1.66),dotstyle); dot((1.11,-3.38),dotstyle); dot((1.85,-2.08),dotstyle); dot((-5.17,2),dotstyle); dot((-5.87,0.46),dotstyle); dot((-3.59,-0.88),linewidth(2pt) + dotstyle); dot((-3.07,-0.88),linewidth(2pt) + dotstyle); dot((-2.55,-0.88),linewidth(2pt) + dotstyle); dot((-3.91,-5.04),dotstyle); dot((-1.35,-3.88),dotstyle); dot((-5.17,-4.14),dotstyle); dot((-6.83,-3.3),dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy][/asy] Note that as all vertices are inside (or in the circumference) of the circle, then all the segments are also inside the circle. Suppose $A_1B_1$ breaks some segment, say $A_mB_m$, in the extension from $A_1$ and intersects the segment in point $C$. Then, as $A_1B_1$ is inside and $A_1$ is on the circumference, then $C$ necessarily is outside of the circle. However, $C$ lies on the segment $A_mB_m$ (that is inside) so $C$ should be inside the circle, absurd. Therefore, it is not possible. b)Let's see it is possible. In the next figure we have each segment is clearly surrounded. [asy][asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(15cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -20.426545367053137, xmax = 15.026100359159813, ymin = -11.779964645572868, ymax = 10.535026362031354; /* image dimensions */ /* draw figures */ draw((xmin, 0.00031524385866197176*xmin-3.464477802991963)--(xmax, 0.00031524385866197176*xmax-3.464477802991963), linewidth(1.2) + linetype("4 4") + gray); /* line */ draw((xmin, -1.7307905202742555*xmin-14.960334143085408)--(xmax, -1.7307905202742555*xmax-14.960334143085408), linewidth(1.2) + linetype("4 4") + gray); /* line */ draw((xmin, 1.7333124718954385*xmin + 14.512866609049581)--(xmax, 1.7333124718954385*xmax + 14.512866609049581), linewidth(1.2) + linetype("4 4") + gray); /* line */ draw((xmin, 0.00031524385866197176*xmin + 3.000917359628707)--(xmax, 0.00031524385866197176*xmax + 3.000917359628707), linewidth(1.2) + linetype("4 4") + gray); /* line */ draw((xmin, -1.730790520274257*xmin-2.0366004139980878)--(xmax, -1.730790520274257*xmax-2.0366004139980878), linewidth(1.2) + linetype("4 4") + gray); /* line */ draw((xmin, 1.7333124718954354*xmin + 1.5750119773818703)--(xmax, 1.7333124718954354*xmax + 1.5750119773818703), linewidth(1.2) + linetype("4 4") + gray); /* line */ draw((-6.188145502801059,2.998966584762442)--(-3.364651763794784,2.999856673823634), linewidth(2)); draw((-2.6825499939928483,2.6063316857664987)--(-1.2700322847510856,0.1615594248913467), linewidth(2)); draw((-1.2697840365315651,-0.6259205297520207)--(-2.6807600662960818,-3.0715828796883624), linewidth(2)); draw((-3.3626135878784957,-3.4655378462745947)--(-6.186107326884771,-3.466427935335787), linewidth(2)); draw((-6.868209096686704,-3.0729029472786533)--(-8.280726805928468,-0.6281306864035008), linewidth(2)); draw((-8.280975054147987,0.1593492682398702)--(-6.869999024383477,2.605011618176204), linewidth(2)); draw((xmin, -0.5769300205325729*xmin-2.9883454498997817)--(xmax, -0.5769300205325729*xmax-2.9883454498997817), linewidth(1.2) + linetype("4 4") + gray); /* line */ draw((xmin, 0.5777706708501866*xmin + 2.525788612719145)--(xmax, 0.5777706708501866*xmax + 2.525788612719145), linewidth(1.2) + linetype("4 4") + gray); /* line */ draw((-4.776873172022336, -3172.147442441616*xmin-15148.441297068335)--(-4.776873172022336, -3172.147442441616*xmax-15148.441297068335), linewidth(1.2) + linetype("4 4") + gray); /* line */ draw(shift((-10.375594533191688,2.997646517172149))*scale(0.17638888888888887)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */ draw((-8.879358238777593,2.1344228811142454)--(-11.871830827605788,3.860870153230057), linewidth(2)); draw((-4.776873172022336,4.504718430286301)--(-4.777962270489793,7.959499348398009), linewidth(2)); draw((-0.672894478584521,2.1370099184159796)--(2.318489011776215,3.8653435644118748), linewidth(2)); draw((2.3210717369262346,-4.327441414742209)--(-0.6714008519019602,-2.600994142626397), linewidth(2)); draw((-4.773885918657218,-4.9712896917984555)--(-4.77279682018976,-8.426070609910163), linewidth(2)); draw((-11.86924810245577,-4.331914825924029)--(-8.877864612095033,-2.603581179928134), linewidth(2)); draw(shift((-4.777417721256066,6.232108889342155))*scale(0.17638888888888887)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */ draw(shift((0.822797266595847,3.001176741413925))*scale(0.17638888888888887)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */ draw(shift((0.8248354425121301,-3.4642177786843042))*scale(0.17638888888888887)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */ draw(shift((-4.773341369423489,-6.6986801508543))*scale(0.17638888888888887)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */ draw(shift((-10.373556357275397,-3.46774800292608))*scale(0.17638888888888887)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */ draw(shift((-4.774360457381632,-3.465982890805191))*scale(0.17638888888888887)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */ draw(shift((-1.9762911393719675,1.3839455553289235))*scale(0.17638888888888887)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */ draw(shift((-4.77639863329792,2.999411629293039))*scale(0.17638888888888887)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */ draw(shift((-7.575487039265734,1.3821804432080378))*scale(0.17638888888888887)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */ draw(shift((-7.574467951307587,-1.8505168168410775))*scale(0.17638888888888887)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */ /* dots and labels */ dot((-6.186107326884771,-3.466427935335787),linewidth(6pt) + dotstyle); dot((-3.3626135878784957,-3.4655378462745947),linewidth(6pt) + dotstyle); dot((-2.6807600662960818,-3.0715828796883624),linewidth(6pt) + dotstyle); dot((-1.2697840365315651,-0.6259205297520207),linewidth(6pt) + dotstyle); dot((-1.2700322847510856,0.1615594248913467),linewidth(6pt) + dotstyle); dot((-2.6825499939928483,2.6063316857664987),linewidth(6pt) + dotstyle); dot((-3.364651763794784,2.999856673823634),linewidth(6pt) + dotstyle); dot((-6.188145502801059,2.998966584762442),linewidth(6pt) + dotstyle); dot((-6.869999024383477,2.605011618176204),linewidth(6pt) + dotstyle); dot((-8.280975054147987,0.1593492682398702),linewidth(6pt) + dotstyle); dot((-8.280726805928468,-0.6281306864035008),linewidth(6pt) + dotstyle); dot((-6.868209096686704,-3.0729029472786533),linewidth(6pt) + dotstyle); dot((-8.879358238777593,2.1344228811142454),dotstyle); dot((-11.871830827605788,3.860870153230057),dotstyle); dot((-4.776873172022336,4.504718430286301),dotstyle); dot((-4.777962270489793,7.959499348398009),dotstyle); dot((2.318489011776215,3.8653435644118748),dotstyle); dot((-0.672894478584521,2.1370099184159796),dotstyle); dot((2.3210717369262346,-4.327441414742209),dotstyle); dot((-0.6714008519019602,-2.600994142626397),dotstyle); dot((-4.773885918657218,-4.9712896917984555),dotstyle); dot((-4.77279682018976,-8.426070609910163),dotstyle); dot((-11.86924810245577,-4.331914825924029),dotstyle); dot((-8.877864612095033,-2.603581179928134),dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy][/asy]