{"id":1268,"date":"2020-01-31T12:15:49","date_gmt":"2020-01-31T12:15:49","guid":{"rendered":"http:\/\/matematika.besaba.com\/?p=1268"},"modified":"2023-11-02T18:52:44","modified_gmt":"2023-11-02T18:52:44","slug":"sutazne-ulohy-kategorie-a-b-a-c-5","status":"publish","type":"post","link":"https:\/\/matematika.besaba.com\/?p=1268","title":{"rendered":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3rie A, B a C"},"content":{"rendered":"<p class=\"hh2\"><a name=\"top\"><\/a>67. ro\u010dn\u00edk matematickej olympi\u00e1dy 2017-2018<br \/>\nS\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed A, B a C<\/p>\n<p><span class=\"povinne\">aktualizovan\u00e9 31.1.2019 13:30<\/span><\/p>\n<p><center><script type=\"text\/javascript\" async=\"true\" charset=\"utf-8\" src=\"http:\/\/sk.search.etargetnet.com\/generic\/advert.php?g=ref:48528,area:600x70,tabl:4,divid:,design_name:green,border_color:ffffff,border_style:none,background_opacity:100,background_color:ffffff,title_color:426000,text_color:000000,url_color:426000,h_title_color:426000,h_text_color:000000,h_url_color:426000,freespace:0,logo_type:3,logo:1,title_underline:0,url_underline:0,h_title_underline:1,h_url_underline:1,nourl:,fsi:11,font:verdana,require_login_pass:,login_pass:,show_in_fb:,show_in_fb_text:,allow_youtube_ads:1\" ><\/script><\/center><\/p>\n<p><!--more--><\/p>\n<p><center><script type=\"text\/javascript\" async=\"true\" charset=\"utf-8\" src=\"http:\/\/sk.search.etargetnet.com\/generic\/advert.php?g=ref:48528,area:600x70,tabl:4,divid:,design_name:green,border_color:ffffff,border_style:none,background_opacity:100,background_color:ffffff,title_color:426000,text_color:000000,url_color:426000,h_title_color:426000,h_text_color:000000,h_url_color:426000,freespace:0,logo_type:3,logo:1,title_underline:0,url_underline:0,h_title_underline:1,h_url_underline:1,nourl:,fsi:11,font:verdana,require_login_pass:,login_pass:,show_in_fb:,show_in_fb_text:,allow_youtube_ads:1\" ><\/script><\/center><\/p>\n<table style=\"width: 98%; text-align: center;\" border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr class=\"a\" bgcolor=\"#c0c0c0\">\n<td>C<\/td>\n<td>B<\/td>\n<td>A<\/td>\n<\/tr>\n<tr align=\"center\">\n<td><span class=\"a\"><a href=\"#c1\">dom\u00e1ce kolo<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#b1\">dom\u00e1ce kolo<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#a1\">dom\u00e1ce kolo<\/a><\/span><\/td>\n<\/tr>\n<tr align=\"center\">\n<td><span class=\"a\"><a href=\"#c2\">\u0161kolsk\u00e9 kolo<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#b2\">\u0161kolsk\u00e9 kolo<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#a2\">\u0161kolsk\u00e9 kolo<\/a><\/span><\/td>\n<\/tr>\n<tr align=\"center\">\n<td><span class=\"a\"><a href=\"#c3\">krajsk\u00e9 kolo<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#b3\">krajsk\u00e9 kolo<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#a3\">krajsk\u00e9 kolo<\/a><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" bgcolor=\"#c0c0c0\">\u00a0<\/td>\n<td align=\"center\"><span class=\"a\">celo\u0161t\u00e1tne kolo<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"background-color: #ffff99;\" colspan=\"3\" align=\"center\"><span class=\"rs\">Zadania \u00faloh dom\u00e1ceho kola vo form\u00e1te PDF: <a href=\"\/67\/mo_67abc_domace_.pdf\" target=\"_blank\" rel=\"noopener noreferrer\"><img loading=\"lazy\" decoding=\"async\" src=\"\/img\/down.png\" alt=\"\" width=\"48\" height=\"48\" align=\"absmiddle\" border=\"0\" \/><\/a> (ulo\u017ei\u0165 ako &#8230;)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<hr \/>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=2040\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"buttc1z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2241\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"buttc1r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><a name=\"c1\"><\/a><\/p>\n<p><strong>C-I-1<\/strong><br \/>\nDok\u00e1\u017ete, \u017ee pre \u013eubovo\u013en\u00e9 re\u00e1lne \u010d\u00edslo <em>a<\/em> plat\u00ed nerovnos\u0165<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"66\/66c1a.png\" alt=\"\" \/><\/div>\n<div class=\"pri\">(Jaroslav \u0160vr\u010dek)<\/div>\n<p><strong>C-I-2<\/strong><br \/>\nN\u00e1jdite najv\u00e4\u010d\u0161ie prirodzen\u00e9 \u010d\u00edslo <em>d<\/em>, ktor\u00e9 m\u00e1 t\u00fa vlastnos\u0165, \u017ee pre \u013eubovo\u013en\u00e9 prirodzen\u00e9 \u010d\u00edslo <em>n<\/em> je hodnota v\u00fdrazu<\/p>\n<div class=\"pc\">V(<em>n<\/em>) = <em>n<\/em><sup>4<\/sup> + 11<em>n<\/em><sup>2<\/sup> &#8211; 12<\/div>\n<p>delite\u013en\u00e1 \u010d\u00edslom <em>d<\/em>.<\/p>\n<div class=\"pri\">(Ale\u0161 Kobza)<\/div>\n<p><strong>C-I-3<\/strong><br \/>\nP\u00e4ta v\u00fd\u0161ky z vrcholu <em>C<\/em> v trojuholn\u00edku <em>ABC<\/em> del\u00ed stranu <em>AB<\/em> v pomere 1 : 2. Dok\u00e1\u017ete, \u017ee pri zvy\u010dajnom ozna\u010den\u00ed d\u013a\u017eok str\u00e1n trojuholn\u00edka ABC plat\u00ed nerovnos\u0165<\/p>\n<div class=\"pc\">3|<em>a &#8211; b<\/em>|&lt; <em>c<\/em>.<\/div>\n<div class=\"pri\">(Jaroslav \u0160vr\u010dek)<\/div>\n<p><strong>C-I-4<\/strong><br \/>\nN\u00e1jdite v\u0161etky troj\u010dleny <em>P<\/em>(<em>x<\/em>) = <em>ax<\/em><sup>2<\/sup> + <em>bx<\/em> + <em>c<\/em> s celo\u010d\u00edseln\u00fdmi koeficientmi <em>a<\/em>, <em>b<\/em>, <em>c<\/em>, pre ktor\u00e9 plat\u00ed <em>P<\/em>(1) &lt; <em>P<\/em>(2) &lt; <em>P<\/em>(3) a z\u00e1rove\u0148<\/p>\n<div class=\"pc\">(<em>P<\/em>(1))<sup>2<\/sup> + (<em>P<\/em>(2))<sup>2<\/sup> + (<em>P<\/em>(3))<sup>2<\/sup> = 22.<\/div>\n<div class=\"pri\">(Tom\u00e1\u0161 Jur\u00edk)<\/div>\n<p><strong>C-I-5<\/strong><br \/>\nV danom trojuholn\u00edku <em>ABC<\/em> zvo\u013eme vn\u00fatri strany <em>AC<\/em> body <em>K<\/em>, <em>M<\/em> a vn\u00fatri strany <em>BC<\/em> body <em>L<\/em>, <em>N<\/em> tak, \u017ee<\/p>\n<div class=\"pc\">|<em>AK<\/em>| = |<em>KM<\/em>| = |<em>MC<\/em>|, |<em>BL<\/em>| = |<em>LN<\/em>| = |<em>NC<\/em>|.<\/div>\n<p>\u010ealej ozna\u010dme <em>E<\/em> priese\u010dn\u00edk uhloprie\u010dok lichobe\u017en\u00edka <em>ABLK<\/em>, <em>F<\/em> priese\u010dn\u00edk uhloprie\u010dok lichobe\u017en\u00edka <em>KLNM<\/em> a <em>G<\/em> priese\u010dn\u00edk uhloprie\u010dok lichobe\u017en\u00edka <em>ABNM<\/em>. Dok\u00e1\u017ete, \u017ee body <em>E<\/em>, <em>F<\/em> a <em>G<\/em> le\u017eia na \u0165a\u017enici trojuholn\u00edka <em>ABC<\/em> z vrcholu <em>C<\/em> a ur\u010dte pomer |<em>GF<\/em>|:|<em>EF<\/em>|.<\/p>\n<div class=\"pri\">(\u0160\u00e1rka Gergelitsov\u00e1)<\/div>\n<p><strong>C-I-6<\/strong><br \/>\n<span class=\"mp\">a) Marienka rozmiestni do vrcholov pravideln\u00e9ho osemuholn\u00edka r\u00f4zne po\u010dty od jedn\u00e9ho po osem cukr\u00edkov. Peter si potom m\u00f4\u017ee vybra\u0165, ktor\u00e9 tri k\u00f4pky cukr\u00edkov d\u00e1 Marienke, ostatn\u00e9 si ponech\u00e1. Jedinou podmienkou je, \u017ee tieto tri k\u00f4pky le\u017eia vo vrcholoch rovnoramenn\u00e9ho trojuholn\u00edka. Marienka chce rozmiestni\u0165 cukr\u00edky tak, aby ich dostala \u010do najviac, nech u\u017e Peter trojicu vrcholov vyberie akoko\u013evek. Ko\u013eko ich tak Marienka zaru\u010dene z\u00edska?<br \/>\nb) Rovnak\u00fa \u00falohu vyrie\u0161te aj pre pravideln\u00fd dev\u00e4\u0165uholn\u00edk, do ktor\u00e9ho vrcholov rozmiestni Marienka 1 a\u017e 9 cukr\u00edkov. (Medzi rovnoramenn\u00e9 trojuholn\u00edky zara\u010fujeme aj trojuholn\u00edky rovnostrann\u00e9.)<\/span><\/p>\n<div class=\"pri\">(Jarom\u00edr \u0160im\u0161a)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"c2\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=2243\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"buttc2z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2245\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"buttc2r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>C-S-1<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<p><strong>C-S-2<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<p><strong>C-S-3<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"c3\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>C-II-1<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<p><strong>C-II-2<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<p><strong>C-II-3<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<p><strong>C-II-4<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"b1\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><center><script type=\"text\/javascript\" async=\"true\" charset=\"utf-8\" src=\"http:\/\/sk.search.etargetnet.com\/generic\/advert.php?g=ref:48528,area:600x70,tabl:4,divid:,design_name:green,border_color:ffffff,border_style:none,background_opacity:100,background_color:ffffff,title_color:426000,text_color:000000,url_color:426000,h_title_color:426000,h_text_color:000000,h_url_color:426000,freespace:0,logo_type:3,logo:1,title_underline:0,url_underline:0,h_title_underline:1,h_url_underline:1,nourl:,fsi:11,font:verdana,require_login_pass:,login_pass:,show_in_fb:,show_in_fb_text:,allow_youtube_ads:1\" ><\/script><\/center><\/p>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=2039\"><input name=\"buttb1z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2242\"><input name=\"buttb1r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>B-I-1<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<p><strong>B-I-2<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<p><strong>B-I-3<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<p><strong>B-I-4<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<p><strong>B-I-5<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<p><strong>B-I-6<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"b2\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=2244\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"buttb2z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2246\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"buttb2r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>B-S-1<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<p><strong>B-S-2<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<p><strong>B-S-3<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"b3\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>B-II-1<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<p><strong>B-II-2<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<p><strong>B-II-3<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<p><strong>B-II-4<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"a1\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><center><script type=\"text\/javascript\" async=\"true\" charset=\"utf-8\" src=\"http:\/\/sk.search.etargetnet.com\/generic\/advert.php?g=ref:48528,area:600x70,tabl:4,divid:,design_name:green,border_color:ffffff,border_style:none,background_opacity:100,background_color:ffffff,title_color:426000,text_color:000000,url_color:426000,h_title_color:426000,h_text_color:000000,h_url_color:426000,freespace:0,logo_type:3,logo:1,title_underline:0,url_underline:0,h_title_underline:1,h_url_underline:1,nourl:,fsi:11,font:verdana,require_login_pass:,login_pass:,show_in_fb:,show_in_fb_text:,allow_youtube_ads:1\" ><\/script><\/center><\/p>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=2518\"><input name=\"butta1z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2532\"><input name=\"butta1r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>A-I-1<\/strong><br \/>\nPavol striedavo vpisuje kr\u00ed\u017eiky a kr\u00fa\u017eky do pol\u00ed\u010dok tabu\u013eky (za\u010d\u00edna kr\u00ed\u017eikom). Ke\u010f je tabu\u013eka cel\u00e1 vyplnen\u00e1, v\u00fdsledn\u00e9 sk\u00f3re spo\u010d\u00edta ako rozdiel <em>O \u2014 X<\/em>, pri\u010dom <em>O<\/em> je celkov\u00fd po\u010det riadkov a st\u013apcov obsahuj\u00facich viac kr\u00fa\u017ekov ako kr\u00ed\u017eikov a <em>X<\/em> je celkov\u00fd po\u010det riadkov a st\u013apcov obsahuj\u00facich viac kr\u00ed\u017eikov ako kr\u00fa\u017ekov.<\/p>\n<ol type=\"a\">\n<li>Dok\u00e1\u017ete, \u017ee pre tabu\u013eku <em>2 x n<\/em> bude v\u00fdsledn\u00e9 sk\u00f3re v\u017edy 0.<\/li>\n<li>Ur\u010dte najvy\u0161\u0161ie mo\u017en\u00e9 sk\u00f3re dosiahnute\u013en\u00e9 pre tabu\u013eku <em>(2n + 1) x (2n + 1)<\/em> v z\u00e1vislosti od <em>n<\/em>.<\/li>\n<\/ol>\n<div class=\"pri\">(Josef Tkadlec)<\/div>\n<p><strong>A-I-2<\/strong><br \/>\nDok\u00e1\u017ete, \u017ee ak je s\u00fa\u010det dvoch dan\u00fdch re\u00e1lnych \u010d\u00edsel <em>a, b<\/em> v\u00e4\u010d\u0161\u00ed ako 2, m\u00e1 s\u00fastava nerovn\u00edc<\/p>\n<div class=\"r\">(a &#8211; 1)x + b &lt; x<sup>2<\/sup> &lt; ax + (b &#8211; 1)<\/div>\n<p>nekone\u010dne ve\u013ea rie\u0161en\u00ed x v obore re\u00e1lnych \u010d\u00edsel.<\/p>\n<div class=\"pri\">(Jarom\u00edr \u0160im\u0161a)<\/div>\n<p><strong>A-I-3<\/strong><br \/>\nV rovine s\u00fa dan\u00e9 dve zhodn\u00e9 kru\u017enice s polomerom 1, ktor\u00e9 maj\u00fa vonkaj\u0161\u00ed dotyk. Uva\u017eujme pravouholn\u00edk obsahuj\u00faci obe kru\u017enice, ktor\u00e9ho ka\u017ed\u00e1 strana sa dot\u00fdka aspo\u0148 jednej z nich. Ur\u010dte najv\u00e4\u010d\u0161\u00ed a najmen\u0161\u00ed mo\u017en\u00fd obsah tak\u00e9ho pravouholn\u00edka.<\/p>\n<div class=\"pri\">(Jaroslav \u0160vr\u010dek)<\/div>\n<p><strong>A-I-4<\/strong><br \/>\nN\u00e1jdite najv\u00e4\u010d\u0161ie prirodzen\u00e9 \u010d\u00edslo <em>n<\/em> tak\u00e9, \u017ee hodnota s\u00fa\u010dtu<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"67\/67a1a.png\" alt=\"\" \/><\/div>\n<p>je prvo\u010d\u00edslo. Z\u00e1pis <img decoding=\"async\" src=\"67\/67a1b.png\" align=\"bottom\" alt=\"\" \/> ozna\u010duje najv\u00e4\u010d\u0161ie cel\u00e9 \u010d\u00edslo, ktor\u00e9 nie je v\u00e4\u010d\u0161ie ako x.<\/p>\n<div class=\"pri\">(Patrik Bak)<\/div>\n<p><strong>A-I-5<\/strong><br \/>\nV konvexnom \u0161tvoruholn\u00edku ABCD plat\u00ed |<img decoding=\"async\" src=\"67\/67_angle.png\" alt=\"\" \/>ABC| = |<img decoding=\"async\" src=\"67\/67_angle.png\" alt=\"\" \/>ACD| a |<img decoding=\"async\" src=\"67\/67_angle.png\" alt=\"\" \/>ACB| = |<img decoding=\"async\" src=\"67\/67_angle.png\" alt=\"\" \/>ADC|.<br \/>\nPredpokladajme, \u017ee stred <em>O<\/em> kru\u017enice op\u00edsanej trojuholn\u00edku <em>BCD<\/em> je r\u00f4zny od bodu <em>A<\/em>. Dok\u00e1\u017ete, \u017ee uhol <em>OAC<\/em> je prav\u00fd.<\/p>\n<div class=\"pri\">(Patrik Bak)<\/div>\n<p><strong>A-I-6<\/strong><br \/>\nN\u00e1jdite najv\u00e4\u010d\u0161\u00ed mo\u017en\u00fd po\u010det prvkov mno\u017einy <strong><em>M<\/em><\/strong> cel\u00fdch \u010d\u00edsel, ktor\u00e1 m\u00e1 nasleduj\u00facu vlastnos\u0165: Z ka\u017edej trojice r\u00f4znych \u010d\u00edsel z <strong><em>M<\/em><\/strong> mo\u017eno vybra\u0165 niektor\u00e9 dve, ktor\u00fdch s\u00fa\u010det je mocninou \u010d\u00edsla 2 s celo\u010d\u00edseln\u00fdm exponentom.<\/p>\n<div class=\"pri\">(J\u00e1n Maz\u00e1k)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"a2\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=2533\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butta2z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2534\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butta2r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>A-S-1<\/strong><br \/>\nUr\u010dte v\u0161etky re\u00e1lne \u010d\u00edsla <em>p, pre ktor\u00e9 m\u00e1 s\u00fastava nerovn\u00edc<\/em><\/p>\n<div class=\"r\"><em>x<\/em><sup>2<\/sup> + (<em>p<\/em> \u2212 1)<em>x<\/em> + p \u2264 0,<br \/>\n<em>x<\/em><sup>2<\/sup> &#8211; (<em>p<\/em> \u2212 1)<em>x<\/em> + p \u2264 0<\/div>\n<p><em>aspo\u0148 jedno rie\u0161enie v obore re\u00e1lnych \u010d\u00edsel.<\/em><\/p>\n<div class=\"pri\">(Jarom\u00edr \u0160im\u0161a)<\/div>\n<p><strong>A-S-2<\/strong><br \/>\nV trojuholn\u00edku <em>ABC<\/em> ozna\u010dme postupne <em>S<sub>b<\/sub><\/em>, <em>S<sub>c<\/sub><\/em> stredy str\u00e1n <em>AC<\/em>, <em>AB<\/em>. Dok\u00e1\u017ete, \u017ee ak |<em>AB<\/em>|&lt;|<em>AC<\/em>|, tak |\u2220<em>BS<sub>c<\/sub>C<\/em>|&lt;|\u2220<em>BS<sub>b<\/sub>C<\/em>|.<\/p>\n<div class=\"pri\">(Patrik Bak)<\/div>\n<p><strong>A-S-3<\/strong><br \/>\nPavol striedavo vpisuje kr\u00ed\u017eiky a kr\u00fa\u017eky do pol\u00ed\u010dok tabu\u013eky (za\u010d\u00edna kr\u00ed\u017eikom). Ke\u010f je tabu\u013eka cel\u00e1 vyplnen\u00e1, v\u00fdsledn\u00e9 sk\u00f3re spo\u010d\u00edta ako s\u00fa\u010det <em><strong>X + O<\/strong><\/em>, pri\u010dom <em>X<\/em> je po\u010det riadkov obsahuj\u00facich viac kr\u00ed\u017eikov ako kr\u00fa\u017ekov a <em>O<\/em> je po\u010det st\u013apcov obsahuj\u00facich viac kr\u00fa\u017ekov ako kr\u00ed\u017eikov. Ur\u010dte najvy\u0161\u0161ie mo\u017en\u00e9 sk\u00f3re dosiahnute\u013en\u00e9 pre tabu\u013eku (2<em>n<\/em> + 1) \u00d7 (2<em>n<\/em> + 1) v z\u00e1vislosti od prirodzen\u00e9ho \u010d\u00edsla <em>n<\/em>.<\/p>\n<div class=\"pri\">(Josef Tkadlec)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"a3\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>A-II-1<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<p><strong>A-II-2<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<p><strong>A-II-3<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<p><strong>A-II-4<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><center><script type=\"text\/javascript\" async=\"true\" charset=\"utf-8\" src=\"http:\/\/sk.search.etargetnet.com\/generic\/advert.php?g=ref:48528,area:600x70,tabl:4,divid:,design_name:green,border_color:ffffff,border_style:none,background_opacity:100,background_color:ffffff,title_color:426000,text_color:000000,url_color:426000,h_title_color:426000,h_text_color:000000,h_url_color:426000,freespace:0,logo_type:3,logo:1,title_underline:0,url_underline:0,h_title_underline:1,h_url_underline:1,nourl:,fsi:11,font:verdana,require_login_pass:,login_pass:,show_in_fb:,show_in_fb_text:,allow_youtube_ads:1\" ><\/script><\/center><\/p>\n<p><span style=\"border-radius: 2px; text-indent: 20px; width: auto; padding: 0px 4px 0px 0px; text-align: center; font: bold 11px\/20px 'Helvetica Neue',Helvetica,sans-serif; color: #ffffff; background: #bd081c  no-repeat scroll 3px 50% \/ 14px 14px; position: absolute; opacity: 1; z-index: 8675309; display: none; cursor: pointer;\">Ulo\u017ei\u0165<\/span><\/p>\n<p><span style=\"border-radius: 2px; text-indent: 20px; width: auto; padding: 0px 4px 0px 0px; text-align: center; font: bold 11px\/20px 'Helvetica Neue',Helvetica,sans-serif; color: #ffffff; background: #bd081c  no-repeat scroll 3px 50% \/ 14px 14px; position: absolute; opacity: 1; z-index: 8675309; display: none; cursor: pointer;\">Ulo\u017ei\u0165<\/span><\/p>\n<p><span style=\"border-radius: 2px; text-indent: 20px; width: auto; padding: 0px 4px 0px 0px; text-align: center; font: bold 11px\/20px 'Helvetica Neue',Helvetica,sans-serif; color: #ffffff; background: #bd081c  no-repeat scroll 3px 50% \/ 14px 14px; position: absolute; opacity: 1; z-index: 8675309; display: none; cursor: pointer;\">Ulo\u017ei\u0165<\/span><\/p>\n<!--CusAds0-->\n<div style=\"font-size: 0px; height: 0px; line-height: 0px; margin: 0; padding: 0; clear: both;\"><\/div>","protected":false},"excerpt":{"rendered":"<p>67. ro\u010dn\u00edk matematickej olympi\u00e1dy 2017-2018 S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed A, B a C aktualizovan\u00e9 31.1.2019 13:30<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[25],"tags":[],"class_list":["post-1268","post","type-post","status-publish","format-standard","hentry","category-67-rocnik"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.6 - 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