{"id":969,"date":"2014-09-16T19:39:43","date_gmt":"2014-09-16T19:39:43","guid":{"rendered":"http:\/\/matematika.okamzite.eu\/?p=969"},"modified":"2015-04-07T19:39:08","modified_gmt":"2015-04-07T19:39:08","slug":"sutazne-ulohy-kategorie-a-b-a-c-2","status":"publish","type":"post","link":"https:\/\/matematika.besaba.com\/?p=969","title":{"rendered":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3rie A, B a C"},"content":{"rendered":"<p class=\"hh2\"><a name=\"top\"><\/a>64. ro\u010dn\u00edk matematickej olympi\u00e1dy 2014-2015<br \/>S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3rie A, B a C<\/p>\n<p><span class=\"povinne\">aktualizovan\u00e9 7.4.2015 21:35<\/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=\"\/64\/64_ABC.pdf\" target=\"_blank\"><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<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=1338\" target=\"_blank\"><input name=\"buttc1\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><a name=\"c1\"><\/a><\/p>\n<p><strong>C-I-1<\/strong><br \/> Ur\u010dte v\u0161etky dvojice (x, y) re\u00e1lnych \u010d\u00edsel, ktor\u00e9 vyhovuj\u00fa s\u00fastave rovn\u00edc<\/p>\n<div class=\"pc\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"64\/64c1a.png\" alt=\"\" width=\"119\" height=\"42\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Jaroslav \u0160vr\u010dek)<\/div>\n<p><strong>C-I-2<\/strong><br \/> Peter m\u00e1 zvl\u00e1\u0161tne hodinky s tromi ru\u010di\u010dkami &#8211; prv\u00e1 z nich obehne kruhov\u00fd cifern\u00edk za min\u00fatu, druh\u00e1 za 3 min\u00faty a tretia za 15 min\u00fat. Na za\u010diatku s\u00fa v\u0161etky ru\u010di\u010dky v rovnakej polohe. Ur\u010dte, za ako dlho bud\u00fa ru\u010di\u010dky rozde\u013eova\u0165 cifern\u00edk na tri zhodn\u00e9 \u010dasti. N\u00e1jdite v\u0161etky rie\u0161enia.<\/p>\n<div class=\"pri\">(Tom\u00e1\u0161 Jur\u00edk)<\/div>\n<p><strong>C-I-3<\/strong><br \/> Simona a Lenka hraj\u00fa hru. Pre dan\u00e9 cel\u00e9 \u010d\u00edslo k tak\u00e9, \u017ee 0 \u2264 <em>k<\/em> \u2264 64, vyberie Simona <em>k<\/em> pol\u00ed\u010dok \u0161achovnice 8 x 8 a ka\u017ed\u00e9 z nich ozna\u010d\u00ed kr\u00ed\u017eikom. Lenka potom \u0161achovnicu nejak\u00fdm sp\u00f4sobom vypln\u00ed tridsiatimi dvoma dominov\u00fdmi kockami. Ak je po\u010det kociek pokr\u00fdvaj\u00facich dva kr\u00ed\u017eiky nep\u00e1rny, vyhr\u00e1va Lenka, inak vyhr\u00e1va Simona. V z\u00e1vislosti od <em>k<\/em> ur\u010dte, ktor\u00e9 z diev\u010dat m\u00e1 vyhr\u00e1vaj\u00facu strat\u00e9giu.<\/p>\n<div class=\"pri\">(Michal Rol\u00ednek)<\/div>\n<p><strong>C-I-4<\/strong><br \/> Ozna\u010dme <em>E<\/em> stred z\u00e1kladne <em>AB<\/em> lichobe\u017en\u00edka <em>ABCD<\/em>, v ktorom plat\u00ed |<em>AB<\/em>|:|<em>CD<\/em>| = 3:1. Uhloprie\u010dka <em>AC<\/em> pret\u00edna \u00fase\u010dky <em>ED<\/em>, <em>BD<\/em> postupne v bodoch <em>F<\/em>, <em>G<\/em>. Ur\u010dte postupn\u00fd pomer<\/p>\n<div class=\"r\">|<em>AF<\/em>| : |<em>FG<\/em>| : |<em>GC<\/em>|.<\/div>\n<div class=\"pri\">(Jaroslav Zhouf)<\/div>\n<p><strong>C-I-5<\/strong><br \/> Rozdiel dvoch prirodzen\u00fdch \u010d\u00edsel je 2 010 a ich najv\u00e4\u010d\u0161\u00ed spolo\u010dn\u00fd delite\u013e je 2 014-kr\u00e1t men\u0161\u00ed ako ich najmen\u0161\u00ed spolo\u010dn\u00fd n\u00e1sobok. Ur\u010dte v\u0161etky tak\u00e9 dvojice \u010d\u00edsel.<\/p>\n<div class=\"pri\">(Jarom\u00edr \u0160im\u0161a)<\/div>\n<p><strong>C-I-6<\/strong><br \/> N\u00e1jdite najmen\u0161ie prirodzen\u00e9 \u010d\u00edslo <em>n<\/em> tak\u00e9, \u017ee v z\u00e1pise iracion\u00e1lneho \u010d\u00edsla <img loading=\"lazy\" decoding=\"async\" src=\"64\/64c1b.png\" alt=\"\" width=\"19\" height=\"12\" border=\"0\" \/> nasleduj\u00fa bezprostredne za desatinnou \u010diarkou dve deviatky.<\/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=\"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=1340\" target=\"_blank\"><input name=\"buttc2z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=1342\" target=\"_blank\"><input name=\"buttc2r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>C-S-1<\/strong><br \/> V obore re\u00e1lnych \u010d\u00edsel vyrie\u0161te s\u00fastavu rovn\u00edc<\/p>\n<div class=\"r\">|1 \u2212 x| = y + 1,<br \/>|1 + y| = z \u2212 2,<br \/>\u00a0\u00a0|2 \u2212 z| = x \u2212 x<sup>2<\/sup>.<\/div>\n<div class=\"pri\">(Jaroslav \u0160vr\u010dek)<\/div>\n<p><strong>C-S-2<\/strong><br \/> Ozna\u010dme <em>K<\/em> a <em>L<\/em> postupne body str\u00e1n <em>BC<\/em> a <em>AC<\/em> trojuholn\u00edka <em>ABC<\/em>, pre ktor\u00e9 plat\u00ed |<em>BK<\/em>|\u00a0=\u00a0<sup>1<\/sup>\/<sub>3<\/sub>|<em>BC<\/em>|, |<em>AL<\/em>|\u00a0=\u00a0<sup>1<\/sup>\/<sub>3<\/sub>|<em>AC<\/em>|. Nech <em>M<\/em> je priese\u010dn\u00edk \u00fase\u010diek <em>AK<\/em> a <em>BL<\/em>. Vypo\u010d\u00edtajte pomer obsahov trojuholn\u00edkov <em>ABM<\/em> a <em>ABC<\/em>.<\/p>\n<div class=\"pri\">(Pavel Novotn\u00fd)<\/div>\n<p><strong>C-S-3<\/strong><br \/> N\u00e1jdite najmen\u0161ie prirodzen\u00e9 \u010d\u00edslo <em>n<\/em> s cifern\u00fdm s\u00fa\u010dtom 8, ktor\u00e9 sa rovn\u00e1 s\u00fa\u010dinu troch r\u00f4znych prvo\u010d\u00edsel, pri\u010dom rozdiel dvoch najmen\u0161\u00edch z nich je 8.<\/p>\n<div class=\"pri\">(Tom\u00e1\u0161 Jur\u00edk)<\/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<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=1370\" target=\"_blank\"><input name=\"buttc3z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=1372\" target=\"_blank\"><input name=\"buttc3r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>C-II-1<\/strong><\/p>\n<p><strong>C-II-2<\/strong><\/p>\n<p><strong>C-II-3<\/strong><\/p>\n<p><strong>C-II-4<\/strong><\/p>\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=1339\" target=\"_blank\"><input name=\"buttb1\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>B-I-1<\/strong><br \/> V obore re\u00e1lnych \u010d\u00edsel vyrie\u0161te s\u00fastavu rovn\u00edc<\/p>\n<div class=\"r\">|<em>x &#8211; 5<\/em>| + |<em>y &#8211; 9<\/em>| = 6,<br \/> |<em>x<sup>2<\/sup> &#8211; 9<\/em>| + |<em>y<sup>2<\/sup> &#8211; 5<\/em>| = 52.<\/div>\n<div class=\"pri\">(Pavel Cal\u00e1bek)<\/div>\n<p><strong>B-I-2<\/strong><br \/> Drak m\u00e1 <em>n<\/em> hl\u00e1v, po jednej na ka\u017edom z <em>n<\/em> krkov usporiadan\u00fdch do kruhu. Rytier dok\u00e1\u017ee jedn\u00fdm \u00faderom sekn\u00fa\u0165 <em>k<\/em> susedn\u00fdch krkov a hlavy na nich s\u0165a\u0165. Ak drakovi po \u00fadere zostane aspo\u0148 jedna hlava, m\u00f4\u017ee si necha\u0165 niektor\u00fa z ch\u00fdbaj\u00facich hl\u00e1v dor\u00e1s\u0165. Dok\u00e1\u017ete, \u017ee ak pre dan\u00e9 \u010d\u00edsla <em>n<\/em> a <em>k<\/em> m\u00f4\u017ee rytier draka zbavi\u0165 v\u0161etk\u00fdch hl\u00e1v bez oh\u013eadu na to, ako mu dorastaj\u00fa, dok\u00e1\u017ee to urobi\u0165 najviac tromi \u00fadermi.<\/p>\n<div class=\"pri\">(J\u00e1n Maz\u00e1k)<\/div>\n<p><strong>B-I-3<\/strong><br \/> V trojuholn\u00edku <em>ABC<\/em> ozna\u010dme <em>U<\/em> stred strany <em>AB<\/em> a <em>V<\/em> stred strany <em>AC<\/em>. V polrovine opa\u010dnej k polrovine <em>BCA<\/em> uva\u017eujme \u013eubovo\u013en\u00fd rovnobe\u017en\u00edk <em>BCDE<\/em>. Ozna\u010dme <em>X<\/em> priese\u010dn\u00edk priamok <em>UD<\/em> a <em>VE<\/em>. Dok\u00e1\u017ete, \u017ee priamka <em>AX<\/em> del\u00ed rovnobe\u017en\u00edk <em>BCDE<\/em> na dve \u010dasti s rovnak\u00fdm obsahom.<\/p>\n<div class=\"pri\">(Michal Rol\u00ednek)<\/div>\n<p><strong>B-I-4<\/strong><br \/> Nech <em>m<\/em> je prirodzen\u00e9 \u010d\u00edslo, ktor\u00e9 m\u00e1 7 kladn\u00fdch delite\u013eov, a <em>n<\/em> je prirodzen\u00e9 \u010d\u00edslo, ktor\u00e9 m\u00e1 9 kladn\u00fdch delite\u013eov. Ko\u013eko delite\u013eov m\u00f4\u017ee ma\u0165 s\u00fa\u010din m \u00b7 n?<\/p>\n<div class=\"pri\">(Eva Pat\u00e1kov\u00e1)<\/div>\n<p><strong>B-I-5<\/strong><br \/> Nech <em>S<\/em> je stred prepony <em>AB<\/em> pravouhl\u00e9ho trojuholn\u00edka <em>ABC<\/em>, ktor\u00fd nie je rovnoramenn\u00fd. Ozna\u010dme <em>D<\/em> p\u00e4tu v\u00fd\u0161ky z vrcholu <em>C<\/em> a <em>R<\/em> priese\u010dn\u00edk osi vn\u00fatorn\u00e9ho uhla pri vrchole <em>C<\/em> s preponou <em>AB<\/em>. Ur\u010dte ve\u013ekosti vn\u00fatorn\u00fdch uhlov tohto trojuholn\u00edka, ak plat\u00ed |<em>SR<\/em>| = 2|<em>DR<\/em>|.<\/p>\n<div class=\"pri\">(Jaroslav \u0160vr\u010dek)<\/div>\n<p><strong>B-I-6<\/strong><br \/> Dok\u00e1\u017ete, \u017ee pre \u013eubovo\u013en\u00e9 kladn\u00e9 re\u00e1lne \u010d\u00edsla <em>a, b, c<\/em> plat\u00ed<\/p>\n<div class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"64\/64b1a.png\" alt=\"\" width=\"416\" height=\"35\" border=\"0\" \/><\/div>\n<p>Ur\u010dte, kedy nast\u00e1va rovnos\u0165.<\/p>\n<div class=\"pri\">(Jaroslav \u0160vr\u010dek)<\/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=1339\" target=\"_blank\"><input name=\"buttb2z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=1341\" target=\"_blank\"><input name=\"buttb2r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>B-S-1<\/strong><br \/>Predpokladajme, \u017ee prirodzen\u00e9 \u010d\u00edslo <em>a<\/em> m\u00e1 15 kladn\u00fdch delite\u013eov. Ko\u013eko ich m\u00f4\u017ee ma\u0165 prirodzen\u00e9 \u010d\u00edslo <em>b<\/em>, ak najmen\u0161\u00ed spolo\u010dn\u00fd n\u00e1sobok \u010d\u00edsel <em>a<\/em> a <em>b<\/em> m\u00e1 20 kladn\u00fdch delite\u013eov?<\/p>\n<div class=\"pri\">(Jarom\u00edr \u0160im\u0161a)<\/div>\n<p><strong>B-S-2<\/strong><br \/>Ozna\u010dme <em>P<\/em> priese\u010dn\u00edk uhloprie\u010dok konvexn\u00e9ho \u0161tvoruholn\u00edka <em>ABCD<\/em>. Vypo\u010d\u00edtajte jeho obsah, ak obsahy trojuholn\u00edkov <em>ABC<\/em>, <em>BCD<\/em> a <em>DAP<\/em> s\u00fa postupne 8 cm<sup>2<\/sup>, 9 cm<sup>2<\/sup>, 10 cm<sup>2<\/sup>.<\/p>\n<div class=\"pri\">(Pavel Novotn\u00fd)<\/div>\n<p><strong>B-S-3<\/strong><br \/>Dok\u00e1\u017ete, \u017ee pre \u013eubovo\u013en\u00e9 kladn\u00e9 re\u00e1lne \u010d\u00edsla <em>a, b, c<\/em> plat\u00ed<\/p>\n<div class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"64\/64bb1.png\" alt=\"\" width=\"330\" height=\"35\" border=\"0\" \/><\/div>\n<p>Ur\u010dte, kedy nast\u00e1va rovnos\u0165.<\/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=\"b3\"><\/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=1369\" target=\"_blank\"><input name=\"buttb3z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=1371\" target=\"_blank\"><input name=\"buttb3r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>B-II-1<\/strong><br \/>S\u00fa\u010din v\u0161etk\u00fdch kladn\u00fdch delite\u013eov prirodzen\u00e9ho \u010d\u00edsla <em>n<\/em> je 20<sup>15<\/sup>. Ur\u010dte <em>n<\/em>.<\/p>\n<div class=\"pri\">(Pavel Cal\u00e1bek)<\/div>\n<p><strong>B-II-2<\/strong><br \/>Ur\u010dte najmen\u0161iu hodnotu v\u00fdrazu<\/p>\n<div class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"64\/64b3b.png\" alt=\"\" width=\"128\" height=\"34\" \/><\/div>\n<p>pri\u010dom <em>x<\/em> je \u013eubovo\u013en\u00e9 re\u00e1lne \u010d\u00edslo. Pre ktor\u00e9 <em>x<\/em> v\u00fdraz V t\u00fato hodnotu nadob\u00fada?<\/p>\n<div class=\"pri\">(Jaroslav \u0160vr\u010dek)<\/div>\n<p><strong>B-II-3<\/strong><br \/>Dok\u00e1\u017ete, \u017ee priese\u010dn\u00edk v\u00fd\u0161ok a \u0165a\u017eisko dan\u00e9ho ostrouhl\u00e9ho trojuholn\u00edka <em>ABC<\/em> maj\u00fa rovnak\u00fa vzdialenos\u0165 od strany <em>AB<\/em> pr\u00e1ve vtedy, ke\u010f pre vn\u00fatorn\u00e9 uhly <em>\u03b1<\/em>,\u00a0\u03b2 pri vrcholoch <em>A<\/em>, <em>B<\/em> plat\u00ed rovnos\u0165<strong> tg <em>\u03b1<\/em> \u00b7 tg \u03b2 = 3<\/strong>.<\/p>\n<div class=\"pri\">(Jarom\u00edr \u0160im\u0161a)<\/div>\n<p><strong>B-II-4<\/strong><br \/>Na tabuli je zoznam \u010d\u00edsel 1, 2, 3, 4, 5, 6 a &#8222;rovnica&#8220;<\/p>\n<div class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"64\/64b3a.png\" alt=\"\" width=\"154\" height=\"38\" \/><\/div>\n<p>Marek s Tom\u00e1\u0161om hraj\u00fa nasleduj\u00facu hru. Najsk\u00f4r Marek vyberie \u013eubovo\u013en\u00e9 \u010d\u00edslo zo zoznamu, nap\u00ed\u0161e ho do jedn\u00e9ho z pr\u00e1zdnych pol\u00ed\u010dok v &#8222;rovnici&#8220; a \u010d\u00edslo zo zoznamu zotrie. Potom Tom\u00e1\u0161 vyberie niektor\u00e9 zo zvy\u0161n\u00fdch \u010d\u00edsel, nap\u00ed\u0161e ho do in\u00e9ho pr\u00e1zdneho pol\u00ed\u010dka a v zozname ho zotrie. Nato Marek urob\u00ed to ist\u00e9 a nakoniec Tom\u00e1\u0161 dopln\u00ed tri zvy\u0161n\u00e9 \u010d\u00edsla na tri zvy\u0161n\u00e9 vo\u013en\u00e9 pol\u00ed\u010dka v &#8222;rovnici&#8220;. Marek vyhr\u00e1, ak vzniknut\u00e1 kvadratick\u00e1 rovnica s racion\u00e1lnymi koeficientmi bude ma\u0165 dva r\u00f4zne re\u00e1lne korene, inak vyhr\u00e1 Tom\u00e1\u0161. Rozhodnite, ktor\u00fd z hr\u00e1\u010dov m\u00f4\u017ee vyhra\u0165 nez\u00e1visle na postupe druh\u00e9ho hr\u00e1\u010da.<\/p>\n<div class=\"pri\">(Pavel Cal\u00e1bek)<\/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=1246\"><input name=\"butta1\" type=\"button\" value=\"Zadanie\" \/><\/a><input name=\"butta1\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>A-I-1<\/strong><br \/> Dan\u00e9 je prirodzen\u00e9 \u010d\u00edslo <em>n<\/em>. \u0160tvorec so stranou d\u013a\u017eky <em>n<\/em> je rozdelen\u00fd na <em>n<\/em><sup>2<\/sup> jednotkov\u00fdch \u0161tvor\u010dekov. Za vzdialenos\u0165 dvoch \u0161tvor\u010dekov pova\u017eujeme vzdialenos\u0165 ich stredov. Ur\u010dte po\u010det dvoj\u00edc \u0161tvor\u010dekov, ktor\u00fdch vzdialenos\u0165 je 5.<\/p>\n<div class=\"pri\">(Jaroslav Zhouf)<\/div>\n<p><strong>A-I-2<\/strong><br \/> Dan\u00fd je trojuholn\u00edk <em>ABC<\/em>, v ktorom je <em>BC<\/em> najkrat\u0161ia strana. Jej stred ozna\u010dme <em>M<\/em>. Na stran\u00e1ch <em>AB<\/em> a <em>AC<\/em> ur\u010d\u00edme postupne body <em>X<\/em> a <em>Y<\/em> tak, aby platilo |<em>BX<\/em>| = |<em>BC<\/em>| = |<em>CY<\/em>|. Priese\u010dn\u00edk priamok <em>CX<\/em> a <em>BY<\/em> ozna\u010dme <em>Z<\/em>. Dok\u00e1\u017ete, \u017ee priamka <em>ZM<\/em> prech\u00e1dza stredom kru\u017enice prip\u00edsanej ku strane <em>BC<\/em> dan\u00e9ho trojuholn\u00edka.<\/p>\n<div class=\"pri\">(Michal Rol\u00ednek)<\/div>\n<p><strong>A-I-3<\/strong><br \/> N\u00e1jdite v\u0161etky cel\u00e9 \u010d\u00edsla <em>k<\/em> \u2265 2, pre ktor\u00e9 existuje <em>k<\/em>-prvkov\u00e1 mno\u017eina <em>M<\/em> cel\u00fdch kladn\u00fdch \u010d\u00edsel tak\u00e1, \u017ee s\u00fa\u010din v\u0161etk\u00fdch \u010d\u00edsel z <em>M<\/em> je delite\u013en\u00fd s\u00fa\u010dtom \u013eubovo\u013en\u00fdch dvoch (r\u00f4znych) \u010d\u00edsel z <em>M<\/em>.<\/p>\n<div class=\"pri\">(Jarom\u00edr \u0160im\u0161a)<\/div>\n<p><strong>A-I-4<\/strong><br \/> Predpokladajme, \u017ee pre re\u00e1lne \u010d\u00edsla <em>x, y, z<\/em> plat\u00ed<\/p>\n<div class=\"r\">15(x + y + z) = 12(xy + yz + zx) = 10(x<sup>2<\/sup> + y<sup>2<\/sup> + z<sup>2<\/sup>)<\/div>\n<p>a \u017ee aspo\u0148 jedno z nich je r\u00f4zne od nuly.<br \/> a)\u00a0\u00a0\u00a0Dok\u00e1\u017ete rovnos\u0165 <em>x + y + z<\/em> = 4.<br \/> b)\u00a0\u00a0N\u00e1jdite najmen\u0161\u00ed interval \u2329a, b\u232a, v ktorom le\u017eia v\u0161etky tri \u010d\u00edsla z \u013eubovo\u013enej trojice (<em>x, y, z<\/em>) vyhovuj\u00facej predpokladom \u00falohy.<\/p>\n<div class=\"pri\">(Jarom\u00edr \u0160im\u0161a)<\/div>\n<p><strong>A-I-5<\/strong><br \/> V danom trojuholn\u00edku <em>ABC<\/em> ozna\u010dme <em>D<\/em> bod dotyku kru\u017enice vp\u00edsanej so stranou <em>BC<\/em>. Kru\u017enica vp\u00edsan\u00e1 do trojuholn\u00edka <em>ABD<\/em> sa dot\u00fdka str\u00e1n <em>AB<\/em> a <em>BD<\/em> v bodoch <em>K<\/em> a <em>L<\/em>. Kru\u017enica vp\u00edsan\u00e1 do trojuholn\u00edka <em>ADC<\/em> sa dot\u00fdka str\u00e1n <em>DC<\/em> a <em>AC<\/em> v bodoch <em>M<\/em> a <em>N<\/em>. Dok\u00e1\u017ete, \u017ee body <em>K, L, M, N<\/em> le\u017eia na jednej kru\u017enici.<\/p>\n<div class=\"pri\">(Josef Tkadlec)<\/div>\n<p><strong>A-I-6<\/strong><br \/> Nech <em>a<\/em>, <em>b<\/em> s\u00fa dan\u00e9 navz\u00e1jom nes\u00fadelite\u013en\u00e9 prirodzen\u00e9 \u010d\u00edsla. Postupnos\u0165 <img loading=\"lazy\" decoding=\"async\" src=\"..\/64\/64a1a.png\" alt=\"\" width=\"47\" height=\"14\" border=\"0\" \/> prirodzen\u00fdch \u010d\u00edsel je zostaven\u00e1 tak, \u017ee pre ka\u017ed\u00e9 <em>n<\/em> > 1 plat\u00ed <em>x<\/em><sub>n<\/sub> = <em>ax<\/em><sub>n-1<\/sub> + <em>b<\/em>. Dok\u00e1\u017ete, \u017ee v \u013eubovo\u013enej takej postupnosti ka\u017ed\u00fd \u010dlen <em>x<\/em><sub>n<\/sub> s indexom <em>n<\/em> > 1 del\u00ed nekone\u010dne ve\u013ea jej \u010fal\u0161\u00edch \u010dlenov. Plat\u00ed toto tvrdenie aj pre <em>n<\/em> = 1?<\/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=\"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><input name=\"butta2\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>A-S-1<\/strong><br \/> Ur\u010dte po\u010det ciest d\u013a\u017eky 14, ktor\u00e9 ved\u00fa po hran\u00e1ch siete na obr\u00e1zku z bodu A do bodu B. D\u013a\u017eka ka\u017edej hrany je rovn\u00e1 1.<br \/> <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"..\/..\/64\/64ab1.png\" alt=\"\" width=\"159\" height=\"138\" border=\"0\" \/><\/p>\n<div class=\"pri\">(Pavel Novotn\u00fd)<\/div>\n<p><strong>A-S-2<\/strong><br \/> Dan\u00fd je rovnobe\u017en\u00edk <em>ABCD<\/em>, pri\u010dom |<em>AB<\/em>|= 2|<em>BC<\/em>|. Ur\u010dte v\u0161etky priamky, ktor\u00e9 delia dan\u00fd rovnobe\u017en\u00edk na dva doty\u010dnicov\u00e9 \u0161tvoruholn\u00edky.<\/p>\n<div class=\"pri\">(Jaroslav \u0160vr\u010dek)<\/div>\n<p><strong>A-S-3<\/strong><br \/> Ur\u010dte v\u0161etky dvojice (<em>p<\/em>, <em>q<\/em>) cel\u00fdch \u010d\u00edsel tak\u00fdch, \u017ee <em>p<\/em> je celo\u010d\u00edseln\u00fdm n\u00e1sobkom \u010d\u00edsla <em>q<\/em> a kvadratick\u00e1 rovnica <em>x<\/em><sup>2<\/sup> + <em>px<\/em> + <em>q<\/em> = 0 m\u00e1 aspo\u0148 jeden celo\u010d\u00edseln\u00fd kore\u0148.<\/p>\n<div class=\"pri\">(Jaroslav \u0160vr\u010dek)<\/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><input name=\"butta2\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>A-II-1<\/strong><br \/> Dan\u00fd je trojuholn\u00edk <em>ABC<\/em> s tup\u00fdm uhlom pri vrchole <em>C<\/em>. Os <em>o<\/em><sub>1<\/sub> \u00fase\u010dky <em>AC<\/em> pret\u00edna stranu <em>AB<\/em> v bode <em>K<\/em>, os <em>o<\/em><sub>2<\/sub> \u00fase\u010dky <em>BC<\/em> pret\u00edna stranu <em>AB<\/em> v bode <em>L<\/em>. Priese\u010dn\u00edk os\u00ed <em>o<\/em><sub>1<\/sub> a <em>o<\/em><sub>2<\/sub> ozna\u010dme <em>O<\/em>. Dok\u00e1\u017ete, \u017ee stred kru\u017enice vp\u00edsanej do trojuholn\u00edka <em>KLC<\/em> le\u017e\u00ed na kru\u017enici op\u00edsanej trojuholn\u00edku <em>OKL<\/em>.<\/p>\n<div class=\"pri\">(Radek Horensk\u00fd)<\/div>\n<p><strong>A-II-2<\/strong><br \/> N\u00e1jdite v\u0161etky dvojice prvo\u010d\u00edsel (<em>p, q<\/em>) tak\u00e9, \u017ee hodnota v\u00fdrazu <em>p<\/em><sup>2<\/sup> + 5<em>pq<\/em> + 4<em>q<\/em><sup>2<\/sup> je druhou mocninou cel\u00e9ho \u010d\u00edsla.<\/p>\n<div class=\"pri\">(Pavel Cal\u00e1bek)<\/div>\n<p><strong>A-II-3<\/strong><br \/> Pre kladn\u00e9 re\u00e1lne \u010d\u00edsla <em>a, b, c<\/em> plat\u00ed<\/p>\n<p class=\"pc\"><em>ab + bc + ca<\/em> = 16, <em>a<\/em> \u2265 3.<\/p>\n<p>N\u00e1jdite najmen\u0161iu mo\u017en\u00fa hodnotu v\u00fdrazu <em>2a + b + c<\/em>.<\/p>\n<div class=\"pri\">(Michal Rol\u00ednek)<\/div>\n<p><strong>A-II-4<\/strong><br \/> Majme <em>n<\/em> bodov v rovine, <em>n<\/em>\u00a0\u2265 3, pri\u010dom \u017eiadne tri z nich nele\u017eia na jednej priamke. Uva\u017eujme vn\u00fatorn\u00e9 uhly v\u0161etk\u00fdch trojuholn\u00edkov s vrcholmi v dan\u00fdch bodoch a ve\u013ekos\u0165 najmen\u0161ieho z nich ozna\u010dme \u03c6. Pre dan\u00e9 <em>n<\/em> n\u00e1jdite najv\u00e4\u010d\u0161ie mo\u017en\u00e9 \u03c6.<\/p>\n<div class=\"pri\">(Stanislava Soj\u00e1kov\u00e1)<\/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<!--CusAds0-->\n<div style=\"font-size: 0px; height: 0px; line-height: 0px; margin: 0; padding: 0; clear: both;\"><\/div>","protected":false},"excerpt":{"rendered":"<p>64. ro\u010dn\u00edk matematickej olympi\u00e1dy 2014-2015S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3rie A, B a C aktualizovan\u00e9 7.4.2015 21:35<\/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":[13],"tags":[],"class_list":["post-969","post","type-post","status-publish","format-standard","hentry","category-64-rocnik"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.6 - 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