{"id":1205,"date":"2017-02-18T16:39:31","date_gmt":"2017-02-18T16:39:31","guid":{"rendered":"http:\/\/matematika.besaba.com\/?p=1205"},"modified":"2023-11-02T18:51:52","modified_gmt":"2023-11-02T18:51:52","slug":"sutazne-ulohy-kategorie-a-b-a-c-4","status":"publish","type":"post","link":"https:\/\/matematika.besaba.com\/?p=1205","title":{"rendered":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3rie A, B a C"},"content":{"rendered":"<ol style=\"list-style-type: lower-alpha;\">\n<li class=\"hh2\"><a name=\"top\"><\/a>66. ro\u010dn\u00edk matematickej olympi\u00e1dy 2016-2017<br \/>S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3rie A, B a C<\/li>\n<\/ol>\n<p><span class=\"povinne\">aktualizovan\u00e9 18.2.2017 15:35<\/span><\/p>\n<p style=\"text-align: center;\"><script type=\"text\/javascript\"><!--\r\ngoogle_ad_client = \"pub-0508472359151885\";\r\n\/* 468x60, bola vytvoren\u00e1 12.11.2010 *\/\r\ngoogle_ad_slot = \"3127737040\";\r\ngoogle_ad_width = 468;\r\ngoogle_ad_height = 60;\r\n\/\/-->\r\n<\/script>\r\n<script type=\"text\/javascript\"\r\nsrc=\"http:\/\/pagead2.googlesyndication.com\/pagead\/show_ads.js\">\r\n<\/script><\/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=\"\/66\/66_ABC.pdf\" target=\"_blank\" rel=\"noopener\"><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\"><input name=\"buttc1z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2241\" target=\"_blank\" rel=\"noopener\"><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 \/>Dok\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 \/>N\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 \/>P\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>|< <em>c<\/em>.<\/div>\n<div class=\"pri\">(Jaroslav \u0160vr\u010dek)<\/div>\n<p><strong>C-I-4<\/strong><br \/>N\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) < <em>P<\/em>(2) < <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 \/>V 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>|,\u00a0\u00a0\u00a0\u00a0|<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\"><input name=\"buttc2z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2245\" target=\"_blank\" rel=\"noopener\"><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\"><input name=\"buttb2z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2246\" target=\"_blank\" rel=\"noopener\"><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=2038\"><input name=\"butta1z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2195\"><input name=\"butta1r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>A-I-1<\/strong><br \/>N\u00e1jdite v\u0161etky prvo\u010d\u00edsla p, pre ktor\u00e9 existuje prirodzen\u00e9 \u010d\u00edslo n tak\u00e9, \u017ee p<sup>n<\/sup> + 1 je tre\u0165ou mocninou niektor\u00e9ho prirodzen\u00e9ho \u010d\u00edsla.<\/p>\n<div class=\"pri\">(J\u00e1n Maz\u00e1k, R\u00f3bert T\u00f3th)<\/div>\n<div>\n<p><strong>A-I-2<\/strong><br \/>M\u00e1me n<sup>2<\/sup> pr\u00e1zdnych \u0161kat\u00fa\u013e; ka\u017ed\u00e1 z nich m\u00e1 \u0161tvorcov\u00e9 dno. V\u00fd\u0161ka aj \u0161\u00edrka ka\u017edej \u0161katule je prirodzen\u00e9 \u010d\u00edslo z mno\u017einy {1,2,&#8230; ,n}. Ka\u017ed\u00e9 dve \u0161katule sa l\u00ed\u0161ia aspo\u0148 v jednom z t\u00fdchto dvoch rozmerov. Jednu \u0161katu\u013eu je dovolen\u00e9 vlo\u017ei\u0165 do druhej, ak m\u00e1 oba rozmery men\u0161ie a aspo\u0148 jeden z rozmerov m\u00e1 aspo\u0148 o 2 men\u0161\u00ed. Takto m\u00f4\u017eeme vytvori\u0165 postupnos\u0165 \u0161kat\u00fa\u013e vlo\u017een\u00fdch navz\u00e1jom do seba (t. j. prv\u00e1 \u0161katu\u013ea je vn\u00fatri druhej, druh\u00e1 \u0161katu\u013ea je vn\u00fatri tretej at\u010f.). Ka\u017ed\u00fa tak\u00fato sadu ulo\u017e\u00edme na in\u00fa poli\u010dku. Ur\u010dte najmen\u0161\u00ed mo\u017en\u00fd po\u010det poli\u010diek potrebn\u00fd na uskladnenie v\u0161etk\u00fdch n<sup>2<\/sup> \u0161kat\u00fa\u013e.<\/p>\n<div class=\"pri\">(Peter Novotn\u00fd)<\/div>\n<p><strong>A-I-3<\/strong><br \/>Dan\u00fd je ostrouhl\u00fd trojuholn\u00edk ABC s v\u00fd\u0161kami AK, BL, CM. Dok\u00e1\u017ete, \u017ee trojuholn\u00edk ABC je rovnoramenn\u00fd pr\u00e1ve vtedy, ke\u010f plat\u00ed rovnos\u0165<\/p>\n<div class=\"pc\">|AM| + |BK| + |CL| = |AL| + |BM| + |CK|.<\/div>\n<div class=\"pri\">(Jarom\u00edr \u0160im\u0161a)<\/div>\n<p><strong>A-I-4<\/strong><\/p>\n<div class=\"pri\">\u00a0<\/div>\n<p><strong>A-I-5<\/strong><br \/>Vn\u00fatri z\u00e1kladne <em>AB<\/em> rovnoramenn\u00e9ho trojuholn\u00edka <em>ABC<\/em> le\u017e\u00ed bod <em>D<\/em>. Zvo\u013eme bod <em>E<\/em> tak, aby <em>ADEC<\/em> bol rovnobe\u017en\u00edk. Na polpriamke opa\u010dnej k <em>ED<\/em> le\u017e\u00ed bod <em>F<\/em> tak\u00fd, \u017ee |<em>EB<\/em>| = |<em>EF<\/em>|. Dok\u00e1\u017ete, \u017ee d\u013a\u017eka tetivy, ktor\u00fa vyt\u00edna priamka <em>BE<\/em> na kru\u017enici op\u00edsanej trojuholn\u00edku <em>ABF<\/em>, je dvojn\u00e1sobkom d\u013a\u017eky \u00fase\u010dky <em>AC<\/em>.<\/p>\n<div class=\"pri\">(Jan Kucha\u0159\u00edk, Patrik Bak)<\/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=2196\" target=\"_blank\" rel=\"noopener\"><input name=\"butta2z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2197\" target=\"_blank\" rel=\"noopener\"><input name=\"butta2r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>A-S-1<\/strong><br \/>Zistite, ak\u00e9 najmen\u0161ie kladn\u00e9 cel\u00e9 \u010d\u00edslo mo\u017eno vlo\u017ei\u0165 medzi dvoj\u010d\u00edslia 20 a 16 tak, aby v\u00fdsledn\u00e9 \u010d\u00edslo bolo n\u00e1sobkom \u010d\u00edsla 2016.<\/p>\n<div class=\"pri\">(Radek Horensk\u00fd)<\/div>\n<p><strong>A-S-2<\/strong><br \/>N\u00e1jdite v\u0161etky kladn\u00e9 cel\u00e9 \u010d\u00edsla <em>n<\/em>, pre ktor\u00e9 sa daj\u00fa \u010d\u00edsla 1, 2, &#8230;, <em>n<\/em> rozdeli\u0165 do troch disjunktn\u00fdch nepr\u00e1zdnych mno\u017e\u00edn s navz\u00e1jom r\u00f4znymi po\u010dtami prvkov tak, \u017ee v \u013eubovo\u013enej dvojici mno\u017e\u00edn m\u00e1 t\u00e1 s men\u0161\u00edm po\u010dtom prvkov v\u00e4\u010d\u0161\u00ed s\u00fa\u010det svojich prvkov.<\/p>\n<div class=\"pri\">(Martin Pan\u00e1k)<\/div>\n<p><strong>A-S-3<\/strong><br \/>Body <em>D<\/em> a <em>E<\/em> s\u00fa (v tomto porad\u00ed) p\u00e4tami v\u00fd\u0161ok z vrcholov <em>B<\/em> a <em>C<\/em> ostrouhl\u00e9ho trojuholn\u00edka <em>ABC<\/em>. Predpokladajme, \u017ee plat\u00ed |<em>AE<\/em>|\u2022|<em>AD<\/em>| = |<em>BE<\/em>|\u2022|<em>CD<\/em>|. Ak\u00fa najmen\u0161iu ve\u013ekos\u0165 m\u00f4\u017ee ma\u0165 uhol <em>BAC<\/em>?<\/p>\n<div class=\"pri\">(Patrik Bak)<\/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<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=2198\" target=\"_blank\" rel=\"noopener\"><input name=\"butta3z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2199\" target=\"_blank\" rel=\"noopener\"><input name=\"butta3r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\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<\/div>\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>66. ro\u010dn\u00edk matematickej olympi\u00e1dy 2016-2017S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3rie A, B a C aktualizovan\u00e9 18.2.2017 15:35<\/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":[24],"tags":[],"class_list":["post-1205","post","type-post","status-publish","format-standard","hentry","category-66-rocnik"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.6 - 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