La publicación a continuación ha sido eliminada. Haga clic para cerrar. Esta publicación ha sido eliminada. Haga clic aquí para ver la publicación. bgn 178 publicaciones bgn #1 h 20 de sep. de 2018, 2:27 a. m. • 2 Y Y por Adventure10, Mango247 Existen algunos segmentos en el plano tales que no hay dos de ellos que se intersecten entre sí (incluso en los puntos extremos). Decimos que el segmento $AB$ rompe al segmento $CD$ si la extensión de $AB$ corta a $CD$ en algún punto entre $C$ y $D$. [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] $a)$ ¿Es posible que cada segmento, al ser extendido desde ambos extremos, rompa exactamente a otro segmento por cada lado? [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] $b)$ Un segmento se llama rodeado si, desde ambos lados del mismo, hay exactamente un segmento que lo rompe. (p. ej., el segmento $AB$ en la figura). ¿Es posible que todos los segmentos estén rodeados? [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] Propuesto por Morteza Saghafian Esta publicación ha sido editada 1 vez. Última edición por bgn, 20 de sep. de 2018, 2:38 a. m. Z K Y