{"id":1119,"date":"2016-08-09T22:18:42","date_gmt":"2016-08-09T22:18:42","guid":{"rendered":"http:\/\/matematika.besaba.com\/?p=1119"},"modified":"2016-08-12T20:08:26","modified_gmt":"2016-08-12T20:08:26","slug":"sutazne-ulohy-kategorie-a-b-a-c-3","status":"publish","type":"post","link":"https:\/\/matematika.besaba.com\/?p=1119","title":{"rendered":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3rie A, B a C"},"content":{"rendered":"<p class=\"hh2\"><a name=\"top\"><\/a>65. ro\u010dn\u00edk matematickej olympi\u00e1dy 2015-2016<br \/>S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3rie A, B a C<\/p>\n<p><span class=\"povinne\">aktualizovan\u00e9 12.8.2016 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=\"\/65\/65_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<hr \/>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=1494\" target=\"_blank\"><input name=\"buttc1z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=1746\" target=\"_blank\"><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 \/>N\u00e1jdite v\u0161etky mo\u017en\u00e9 hodnoty s\u00fa\u010dinu prvo\u010d\u00edsel <em>p, q, r<\/em>, pre ktor\u00e9 plat\u00ed<\/p>\n<div class=\"pc\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"65\/65ca1.png\" alt=\"\" width=\"141\" height=\"17\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Vojtech B\u00e1lint, Jarom\u00edr \u0160im\u0161a)<\/div>\n<p><strong>C-I-2<\/strong><br \/>Ur\u010dte, ko\u013ek\u00fdmi sp\u00f4sobmi mo\u017eno k jednotliv\u00fdm vrcholom kocky <em>ABCDEFGH<\/em> prip\u00edsa\u0165 \u010d\u00edsla 1, 3, 3, 3, 4, 4, 4, 4 tak, aby s\u00fa\u010din \u010d\u00edsel prip\u00edsan\u00fdch \u013eubovo\u013en\u00fdm trom vrcholom ka\u017edej zo stien kocky bol p\u00e1rny.<\/p>\n<div class=\"pri\">(Jaroslav \u0160vr\u010dek)<\/div>\n<p><strong>C-I-3<\/strong><br \/>Uva\u017eujme v\u00fdraz<\/p>\n<div class=\"pc\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"65\/65ca2.png\" alt=\"\" width=\"179\" height=\"16\" border=\"0\" \/><\/div>\n<p>\u00a0\u00a0\u00a0\u00a0\u00a0a) N\u00e1jdite v\u0161etky re\u00e1lne \u010d\u00edsla <em>x<\/em> a <em>y<\/em>, pre ktor\u00e9 dan\u00fd v\u00fdraz nadob\u00fada svoju najmen\u0161iu hodnotu.<\/p>\n<p>\u00a0\u00a0\u00a0\u00a0\u00a0b) Ur\u010dte v\u0161etky dvojice cel\u00fdch nez\u00e1porn\u00fdch \u010d\u00edsel <em>x<\/em> a <em>y<\/em>, pre ktor\u00e9 je hodnota dan\u00e9ho v\u00fdrazu rovn\u00e1 \u010d\u00edslu 16.<\/p>\n<div class=\"pri\">(Ale\u0161 Kobza)<\/div>\n<p><strong>C-I-4<\/strong><br \/>Vn\u00fatri str\u00e1n <em>AB<\/em>, <em>AC<\/em> dan\u00e9ho trojuholn\u00edka <em>ABC<\/em> s\u00fa zvolen\u00e9 postupne body <em>E<\/em>, <em>F<\/em>, pri\u010dom <em>EF<\/em>\u2551<em>BC<\/em>. \u00dase\u010dka <em>EF<\/em> je potom rozdelen\u00e1 bodom <em>D<\/em> tak, \u017ee plat\u00ed<\/p>\n<div class=\"pc\"><em>p<\/em> =|ED|:|DF|=|BE|:|EA|.<\/div>\n<p>\u00a0\u00a0\u00a0\u00a0\u00a0a) Uk\u00e1\u017ete, \u017ee pomer obsahov trojuholn\u00edkov <em>ABC<\/em> a <em>ABD<\/em> je pre <em>p<\/em> = 2:3 rovnak\u00fd ako pre <em>p = 3:2.<\/em><\/p>\n<p>\u00a0\u00a0\u00a0\u00a0\u00a0b) Zd\u00f4vodnite, pre\u010do pomer obsahov trojuholn\u00edkov <em>ABC<\/em> a <em>ABD<\/em> m\u00e1 hodnotu aspo\u0148 4.<\/p>\n<div class=\"pri\">(Vojt\u011bch \u017d\u00e1dn\u00edk)<\/div>\n<p><strong>C-I-5<\/strong><br \/>M\u00e1me karti\u010dky s \u010d\u00edslami 5, 6, 7, &#8230;, 55 (na ka\u017edej karti\u010dke je jedno \u010d\u00edslo). Ko\u013eko najviac karti\u010diek m\u00f4\u017eeme vybra\u0165 tak, aby s\u00fa\u010det \u010d\u00edsel na \u017eiadnych dvoch vybran\u00fdch karti\u010dk\u00e1ch nebol palindr\u00f3m? (<em>Palindr\u00f3m je \u010d\u00edslo, ktor\u00e9 je rovnak\u00e9 pri \u010d\u00edtan\u00ed z\u013eava doprava i sprava do\u013eava.<\/em>)<\/p>\n<div class=\"pri\">(Tom\u00e1\u0161 Jur\u00edk)<\/div>\n<p><strong>C-I-6<\/strong><br \/>Dan\u00e1 je kru\u017enica <em>k<\/em><sub>1<\/sub>(<em>A<\/em>; 4 cm), jej bod <em>B<\/em> a kru\u017enica <em>k<\/em><sub>2<\/sub>(<em>B<\/em>; 2 cm). Bod <em>C<\/em> je stredom \u00fase\u010dky <em>AB<\/em> a bod <em>K<\/em> je stredom \u00fase\u010dky <em>AC<\/em>. Vypo\u010d\u00edtajte obsah pravouhl\u00e9ho trojuholn\u00edka <em>KLM<\/em>, ktor\u00e9ho vrchol <em>L<\/em> je jeden z priese\u010dn\u00edkov kru\u017en\u00edc <em>k<\/em><sub>1<\/sub>, <em>k<\/em><sub>2<\/sub> a ktor\u00e9ho prepona <em>KM<\/em> le\u017e\u00ed na priamke <em>AB<\/em>.<\/p>\n<div class=\"pri\">(\u0160\u00e1rka Gergelitsov\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 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=1752\" target=\"_blank\"><input name=\"buttc2z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=1754\" target=\"_blank\"><input name=\"buttc2r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>C-S-1<\/strong><br \/>N\u00e1jdite v\u0161etky \u0161tvorcifern\u00e9 \u010d\u00edsla <span style=\"text-decoration: overline;\"><em>abcd<\/em><\/span>, pre ktor\u00e9 plat\u00ed <span style=\"text-decoration: overline;\"><em>abcd<\/em><\/span> = 20 \u00b7 <span style=\"text-decoration: overline;\"><em>ab<\/em><\/span> + 16 \u00b7 <span style=\"text-decoration: overline;\"><em>cd<\/em><\/span>.<\/p>\n<div class=\"pri\">(Tom\u00e1\u0161 Jur\u00edk)<\/div>\n<p><strong>C-S-2<\/strong><br \/>Pri stole sed\u00ed nieko\u013eko \u013eud\u00ed (aspo\u0148 dvaja) a hraj\u00fa tak\u00fato hru: V ka\u017edom kole tajn\u00fdm hlasovan\u00edm ka\u017ed\u00fd hr\u00e1\u010d udel\u00ed hlas jedn\u00e9mu hr\u00e1\u010dovi (m\u00f4\u017ee aj s\u00e1m sebe). Potom sa kolo vyhodnot\u00ed: ka\u017ed\u00fd hr\u00e1\u010d, ktor\u00fd dostal pr\u00e1ve jeden hlas, z hry vypad\u00e1va.<\/p>\n<p>\u00a0\u00a0\u00a0\u00a0\u00a0a) Ko\u013eko \u013eud\u00ed mohlo sedie\u0165 pri stole na za\u010diatku, ak v prvom kole vypadol z hry pr\u00e1ve jeden hr\u00e1\u010d?<\/p>\n<p>\u00a0\u00a0\u00a0\u00a0\u00a0b) Mohla ma\u0165 hra jedin\u00e9ho v\u00ed\u0165aza, teda \u010dloveka, ktor\u00fd po ur\u010ditom po\u010dte k\u00f4l zostal v hre s\u00e1m?<\/p>\n<div class=\"pri\">(R\u00f3bert T\u00f3th)<\/div>\n<p><strong>C-S-3<\/strong><br \/>V kru\u017enici so stredom <em>S<\/em> zostroj\u00edme priemer <em>AB<\/em> a \u013eubovo\u013en\u00fa na\u0148 kolm\u00fa tetivu <em>CD<\/em>. Zd\u00f4vodnite, pre\u010do je obvod trojuholn\u00edka <em>ACD<\/em> men\u0161\u00ed ako dvojn\u00e1sobok obvodu trojuholn\u00edka <em>SBC<\/em>.<\/p>\n<div class=\"pri\">(\u0160\u00e1rka Gergelitsov\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 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=1902\" target=\"_blank\"><input name=\"buttc3z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=1904\" target=\"_blank\"><input name=\"buttc3r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>C-II-1<\/strong><br \/>N\u00e1jdite najmen\u0161iu mo\u017en\u00fa hodnotu v\u00fdrazu<\/p>\n<div class=\"pc\">3<em>x<\/em><sup>2<\/sup> &#8211; 12<em>xy<\/em> + <em>y<\/em><sup>4<\/sup>,<\/div>\n<p>v ktorom <em>x<\/em> a <em>y<\/em> s\u00fa \u013eubovo\u013en\u00e9 cel\u00e9 nez\u00e1porn\u00e9 \u010d\u00edsla.<\/p>\n<div class=\"pri\">(Jarom\u00edr \u0160im\u0161a)<\/div>\n<p><strong>C-II-2<\/strong><br \/>Ur\u010dte, ko\u013ek\u00fdmi sp\u00f4sobmi mo\u017eno v\u0161etky hrany kocky <em>ABCDEFGH<\/em> ofarbi\u0165 \u0161tyrmi dan\u00fdmi farbami (cel\u00fa hranu bez krajn\u00fdch bodov v\u017edy jednou farbou), aby pritom ka\u017ed\u00e1 stena kocky mala hrany v\u0161etk\u00fdch \u0161tyroch farieb.<\/p>\n<div class=\"pri\">(Jaroslav \u0160vr\u010dek)<\/div>\n<p><strong>C-II-3<\/strong><br \/>V pravouhlom lichobe\u017en\u00edku <em>ABCD<\/em> s prav\u00fdm uhlom pri vrchole <em>A<\/em> z\u00e1kladne <em>AB<\/em> je bod <em>K<\/em> priese\u010dn\u00edkom v\u00fd\u0161ky <em>CP<\/em> lichobe\u017en\u00edka s jeho uhloprie\u010dkou <em>BD<\/em>. Obsah \u0161tvoruholn\u00edka <em>APCD<\/em> je polovicou obsahu lichobe\u017en\u00edka <em>ABCD<\/em>. Ur\u010dte, ak\u00fa \u010das\u0165 obsahu trojuholn\u00edka <em>ABC<\/em> zaber\u00e1 trojuholn\u00edk <em>BCK<\/em>.<\/p>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\u00e1)<\/div>\n<p><strong>C-II-4<\/strong><br \/>Adam s Barborou hraj\u00fa so zlomkom<\/p>\n<div class=\"pc\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"65\/65cc2.png\" alt=\"\" width=\"55\" height=\"36\" border=\"0\" \/><\/div>\n<p>tak\u00fato hru na \u0161tyri \u0165ahy: Hr\u00e1\u010di striedavo nahr\u00e1dzaj\u00fa \u013eubovo\u013en\u00e9 z doposia\u013e neur\u010den\u00fdch p\u00edsmen <em>a, b, c, d<\/em> nejakou cifrou od 1 do 9. Barbora vyhr\u00e1, ke\u010f v\u00fdsledn\u00fd zlomok bude rovn\u00fd bu\u010f cel\u00e9mu \u010d\u00edslu, alebo \u010d\u00edslu s kone\u010dn\u00fdm po\u010dtom desatinn\u00fdch miest; inak vyhr\u00e1 Adam (napr\u00edklad ke\u010f vznikne zlomok <img loading=\"lazy\" decoding=\"async\" src=\"65\/65cc3.png\" alt=\"\" width=\"12\" height=\"15\" align=\"absmiddle\" border=\"0\" \/>). Ak za\u010d\u00edna Adam, ako m\u00e1 hra\u0165 Barbora, aby zaru\u010dene vyhrala? Ak za\u010d\u00edna Barbora, je mo\u017en\u00e9 poradi\u0165 Adamovi tak, aby v\u017edy vyhral?<\/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=\"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=1493\"><input name=\"buttb1z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=1745\"><input name=\"buttb1r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>B-I-1<\/strong><br \/>Pre prirodzen\u00e9 \u010d\u00edsla <em>k, l, m<\/em> plat\u00ed<\/p>\n<div class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"65\/65ba1.png\" alt=\"\" width=\"154\" height=\"35\" border=\"0\" \/><\/div>\n<p>Ur\u010dte v\u0161etky mo\u017en\u00e9 hodnoty s\u00fa\u010dinu <em>klm<\/em>.<\/p>\n<div class=\"pri\">(Ale\u0161 Kobza)<\/div>\n<p><strong>B-I-2<\/strong><br \/>Do \u0161tvorcovej tabu\u013eky 11\u00a0\u00d7 11 sme vp\u00edsali prirodzen\u00e9 \u010d\u00edsla 1, 2, \u2026, 121 postupne po riadkoch z\u013eava doprava a zhora nadol. \u0160tvorcovou do\u0161ti\u010dkou 4\u00a0\u00d7 4 sme v\u0161etk\u00fdmi mo\u017en\u00fdmi sp\u00f4sobmi zakryli pr\u00e1ve 16 pol\u00ed\u010dok. Ko\u013ekokr\u00e1t bol s\u00fa\u010det zakryt\u00fdch 16 \u010d\u00edsel druhou mocninou cel\u00e9ho \u010d\u00edsla?<\/p>\n<div class=\"pri\">(Vojtech B\u00e1lint, Tom\u00e1\u0161 Jur\u00edk)<\/div>\n<p><strong>B-I-3<\/strong><br \/>V pravouhlom trojuholn\u00edku <em>ABC<\/em> s preponou <em>AB<\/em> a odvesnami d\u013a\u017eok |<em>AC<\/em>|= 4 cm a |<em>BC<\/em>|= 3 cm le\u017eia navz\u00e1jom sa dot\u00fdkaj\u00face kru\u017enice k<sub>1<\/sub>(S<sub>1<\/sub>; r<sub>1<\/sub>) a k<sub>2<\/sub>(S<sub>2<\/sub>; r<sub>2<\/sub>) tak, \u017ee k<sub>1<\/sub> sa dot\u00fdka str\u00e1n <em>AB<\/em> a <em>AC<\/em>, zatia\u013e \u010do k<sub>2<\/sub> sa dot\u00fdka str\u00e1n <em>AB<\/em> a <em>BC<\/em>. Ur\u010dte najmen\u0161iu a najv\u00e4\u010d\u0161iu mo\u017en\u00fa hodnotu polomeru r<sub>2<\/sub>.<\/p>\n<div class=\"pri\">(Pavel Novotn\u00fd)<\/div>\n<p><strong>B-I-4<\/strong><br \/>Po\u010det v\u0161etk\u00fdch p\u00e1rnych delite\u013eov niektor\u00e9ho prirodzen\u00e9ho \u010d\u00edsla je o 3 v\u00e4\u010d\u0161\u00ed ako po\u010det v\u0161etk\u00fdch jeho nep\u00e1rnych delite\u013eov. Ak\u00fd je podiel s\u00fa\u010dtu v\u0161etk\u00fdch jeho p\u00e1rnych delite\u013eov a s\u00fa\u010dtu v\u0161etk\u00fdch jeho nep\u00e1rnych delite\u013eov? N\u00e1jdite v\u0161etky mo\u017en\u00e9 odpovede.<\/p>\n<div class=\"pri\">(Erika Novotn\u00e1)<\/div>\n<p><strong>B-I-5<\/strong><br \/>Vrcholy konvexn\u00e9ho \u0161es\u0165uholn\u00edka <em>ABCDEF<\/em> le\u017eia na kru\u017enici, pri\u010dom |<em>AB<\/em>|=|<em>CD<\/em>|.\u00dase\u010dky <em>AE<\/em> a <em>CF<\/em> sa pret\u00ednaj\u00fa v bode <em>G<\/em> a \u00fase\u010dky <em>BE<\/em> a <em>DF<\/em> sa pret\u00ednaj\u00fa v bode <em>H<\/em>. Dok\u00e1\u017ete, \u017ee \u00fase\u010dky <em>GH<\/em>, <em>AD<\/em> a <em>BC<\/em> s\u00fa navz\u00e1jom rovnobe\u017en\u00e9.<\/p>\n<div class=\"pri\">(\u0160\u00e1rka Gergelitsov\u00e1)<\/div>\n<p><strong>B-I-6<\/strong><br \/>Kladn\u00e9 re\u00e1lne \u010d\u00edsla <em>a, b, c<\/em> s\u00fa tak\u00e9, \u017ee hodnoty<\/p>\n<div class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"65\/65ba2.png\" alt=\"\" width=\"459\" height=\"38\" border=\"0\" \/><\/div>\n<p>s\u00fa navz\u00e1jom r\u00f4zne. Zap\u00ed\u0161me ich od najmen\u0161ej po najv\u00e4\u010d\u0161iu:<\/p>\n<div class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"65\/65ba3.png\" alt=\"\" width=\"257\" height=\"16\" border=\"0\" \/><\/div>\n<p>Zistite, ko\u013eko r\u00f4znych porad\u00ed (i<sub>1<\/sub>, i<sub>2<\/sub>, \u2026, i<sub>6<\/sub>) indexov 1 a\u017e 6 m\u00f4\u017eeme dosta\u0165, ke\u010f budeme r\u00f4zne voli\u0165 \u010d\u00edsla <em>a, b, c<\/em>.<\/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=\"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=1751\" target=\"_blank\"><input name=\"buttb2z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=1753\" target=\"_blank\"><input name=\"buttb2r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>B-S-1<\/strong><br \/>Ko\u013ek\u00fdmi sp\u00f4sobmi je mo\u017en\u00e9 vyplni\u0165 \u0161tvorcov\u00fa tabu\u013eku 3\u00a0\u00d7 3 \u010d\u00edslami 2, 2, 3, 3, 3, 4, 4, 4, 4 tak, aby s \u00fa\u010det \u010d\u00edsel v ka\u017edom \u0161tvorci 2\u00a0\u00d7 2 tejto tabu\u013eky bol rovn\u00fd 14?<\/p>\n<div class=\"pri\">(Tom\u00e1\u0161 Jur\u00edk)<\/div>\n<p><strong>B-S-2<\/strong><br \/>Dan\u00e1 je \u00fase\u010dka <em>AB<\/em>, jej stred <em>C<\/em> a vn\u00fatri \u00fase\u010dky <em>AB<\/em> bod <em>D<\/em>. Kru\u017enice <em>k<\/em>(<em>C<\/em>; |<em>BC<\/em>|) a\u00a0<em>m<\/em>(<em>B<\/em>; |<em>BD<\/em>|) sa pret\u00ednaj\u00fa v\u00a0bodoch <em>E<\/em> a\u00a0<em>F<\/em> a\u00a0polpriamka <em>FD<\/em> pret\u00edna kru\u017enicu <em>k<\/em> v\u00a0bode <em>K<\/em>, <em>K<\/em>\u00a0\u2260 <em>F<\/em>. Rovnobe\u017eka s\u00a0priamkou <em>AB<\/em> prech\u00e1dzaj\u00faca bodom <em>K<\/em>\u00a0pret\u00edna kru\u017enicu <em>k<\/em> v\u00a0bode <em>L<\/em>, <em>L<\/em>\u00a0\u2260 <em>K<\/em>. Dok\u00e1\u017ete, \u017ee |<em>KL<\/em>|=|<em>BD<\/em>|.<\/p>\n<div class=\"pri\">(\u0160\u00e1rka Gergelitsov\u00e1)<\/div>\n<p><strong>B-S-3<\/strong><br \/>Dan\u00e9 s\u00fa dve r\u00f4zne re\u00e1lne \u010d\u00edsla <em>a, b<\/em> v\u00e4\u010d\u0161ie ako 1. Zap\u00ed\u0161te v\u0161etky mo\u017en\u00e9 poradia hodn\u00f4t v\u00fdrazov<\/p>\n<div class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"65\/65bb1.png\" alt=\"\" width=\"283\" height=\"38\" border=\"0\" \/><\/div>\n<p>od najmen\u0161ej po najv\u00e4\u010d\u0161iu.<\/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=1901\" target=\"_blank\"><input name=\"buttb3z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=1903\" target=\"_blank\"><input name=\"buttb3r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>B-II-1<\/strong><br \/>Ur\u010dte v\u0161etky trojice cel\u00fdch kladn\u00fdch \u010d\u00edsel <em>k<\/em>, <em>l<\/em> a <em>m<\/em>, pre ktor\u00e9 plat\u00ed<\/p>\n<div class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"65\/65bc1.png\" alt=\"\" width=\"173\" height=\"31\" border=\"0\" \/>.<\/div>\n<div class=\"pri\">(Jarom\u00edr \u0160im\u0161a)<\/div>\n<p><strong>B-II-2<\/strong><br \/>Dan\u00e1 je \u00fase\u010dka <em>AB<\/em>, jej stred <em>C<\/em> a vn\u00fatri \u00fase\u010dky <em>AB<\/em> bod <em>D<\/em>. Kru\u017enice <em>k<\/em>(<em>C<\/em>; |<em>BC<\/em>|)a <em>m<\/em>(<em>B<\/em>; |<em>BD<\/em>|) sa pret\u00ednaj\u00fa v bodoch <em>E<\/em> a <em>F<\/em>. Zd\u00f4vodnite, pre\u010do je polpriamka <em>FD<\/em> osou uhla <em>AFE<\/em>.<\/p>\n<div class=\"pri\">(\u0160\u00e1rka Gergelitsov\u00e1)<\/div>\n<p><strong>B-II-3<\/strong><br \/>N\u00e1jdite v\u0161etky prirodzen\u00e9 \u010d\u00edsla <em>n<\/em>, ktor\u00e9 maj\u00fa pr\u00e1ve \u0161es\u0165 delite\u013eov, pri\u010dom s\u00fa\u010det druh\u00e9ho najv\u00e4\u010d\u0161ieho a druh\u00e9ho najmen\u0161ieho z nich je 54.<\/p>\n<div class=\"pri\">(Pavel Novotn\u00fd)<\/div>\n<p><strong>B-II-4<\/strong><br \/>Dan\u00e9 je prirodzen\u00e9 \u010d\u00edslo <em>k<\/em>, 4 \u2264 <em>k<\/em> \u2264 900. Adam a Bra\u0148o hraj\u00fa hru: Adam nap\u00ed\u0161e na tabu\u013eu <em>k<\/em> r\u00f4znych trojcifern\u00fdch \u010d\u00edsel, Bra\u0148o si z nich vyberie \u0161tyri r\u00f4zne. Ak rozdiel medzi dvoma najmen\u0161\u00edmi aj rozdiel medzi dvoma najv\u00e4\u010d\u0161\u00edmi vybran\u00fdmi \u010d\u00edslami je nanajv\u00fd\u0161 22, vyhr\u00e1va Bra\u0148o, inak vyhr\u00e1va Adam. V z\u00e1vislosti od hodnoty <em>k<\/em> ur\u010dte, kto m\u00e1 vyhr\u00e1vaj\u00facu strat\u00e9giu.<\/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=\"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=1492\"><input name=\"butta1z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=1728\"><input name=\"butta1r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>A-I-1<\/strong><br \/>V ka\u017edej zo \u0161tyroch miestnost\u00ed je nieko\u013eko predmetov. Nech <em>n<\/em>\u22652 je prirodzen\u00e9 \u010d\u00edslo. Jednu <em>n<\/em>-tinu predmetov z prvej miestnosti prenesieme do druhej miestnosti. N\u00e1sledne jednu <em>n<\/em>-tinu (z nov\u00e9ho po\u010dtu) predmetov prenesieme z druhej miestnosti do tretej. Podobne potom z tretej miestnosti do \u0161tvrtej a zo \u0161tvrtej do prvej. (V\u017edy pritom pren\u00e1\u0161ame cel\u00e9 predmety.) Ak viete, \u017ee na konci bol v ka\u017edej miestnosti rovnak\u00fd po\u010det predmetov, ur\u010dte, ko\u013eko najmenej predmetov mohlo by\u0165 na za\u010diatku v druhej miestnosti. Pre ktor\u00e9 <em>n<\/em> sa tak m\u00f4\u017ee sta\u0165?<\/p>\n<div class=\"pri\">(Vojtech B\u00e1lint, Michal Rol\u00ednek)<\/div>\n<p><strong>A-I-2<\/strong><br \/>N\u00e1jdite najmen\u0161ie re\u00e1lne \u010d\u00edslo <em>m<\/em>, pre ktor\u00e9 mo\u017eno n\u00e1js\u0165 re\u00e1lne \u010d\u00edsla <em>a, b<\/em> tak, aby nerovnos\u0165<\/p>\n<div class=\"r\">|x<sup>2<\/sup> + <em>a<\/em>x + <em>b<\/em>| \u2264 <em>m<\/em><\/div>\n<p>platila pre ka\u017ed\u00e9 x\u2208\u23290, 2\u232a.<\/p>\n<div class=\"pri\">(Michal Rol\u00ednek)<\/div>\n<p><strong>A-I-3<\/strong><br \/>Dan\u00fd je pravouhl\u00fd trojuholn\u00edk <em>ABC<\/em> s preponou <em>AB<\/em> a dlh\u0161ou odvesnou <em>BC<\/em>. Nech <em>D<\/em> je p\u00e4ta v\u00fd\u0161ky z vrcholu <em>C<\/em>. Kru\u017enica <em>k<\/em> so stredom <em>D<\/em> a polomerom <em>CD<\/em> pret\u00edna odvesnu <em>BC<\/em> v bode <em>Q<\/em> a \u010falej priamku <em>AB<\/em> v bodoch <em>E<\/em> a <em>F<\/em> (<em>E<\/em> \u2260 <em>F<\/em> ), pri\u010dom <em>F<\/em> je bodom prepony <em>AB<\/em> . \u00dase\u010dka <em>QE<\/em> pret\u00edna odvesnu <em>AC<\/em> v bode <em>P<\/em>. Dok\u00e1\u017ete, \u017ee |<em>PE<\/em>|=|<em>QF<\/em>|.<\/p>\n<div class=\"pri\">(Jaroslav \u0160vr\u010dek)<\/div>\n<p><strong>A-I-4<\/strong><br \/>Nela s Janou zvolia prirodzen\u00e9 \u010d\u00edslo <em>k<\/em> a n\u00e1sledne hraj\u00fa hru s tabu\u013ekou maj\u00facou rozmery 9 \u00d7 9. Za\u010d\u00ednaj\u00faca Nela v\u017edy vo svojom \u0165ahu vyberie jedno pr\u00e1zdne pol\u00ed\u010dko a vp\u00ed\u0161e do\u0148ho nulu. Jana vo svojom \u0165ahu do nejak\u00e9ho pr\u00e1zdneho pol\u00ed\u010dka nap\u00ed\u0161e jednotku. Navy\u0161e po ka\u017edom \u0165ahu Nely nasleduje <em>k<\/em> \u0165ahov Jany. Ak sa kedyko\u013evek po\u010das hry stane, \u017ee s\u00fa\u010det \u010d\u00edsel v ka\u017edom riadku aj v ka\u017edom st\u013apci je nep\u00e1rny, vyhr\u00e1 Jana. Ak diev\u010dat\u00e1 vyplnia cel\u00fa tabu\u013eku bez toho, aby sa tak stalo, vyhr\u00e1 Nela. N\u00e1jdite najmen\u0161iu hodnotu <em>k<\/em>, pre ktor\u00fa m\u00e1 Jana vyhr\u00e1vaj\u00facu strat\u00e9giu.<\/p>\n<div class=\"pri\">(Michal Rol\u00ednek)<\/div>\n<p><strong>A-I-5<\/strong><br \/> Dan\u00fd je trojuholn\u00edk <em>ABC<\/em> s najkrat\u0161ou stranou <em>BC<\/em>. Na stran\u00e1ch <em>AB<\/em>, <em>AC<\/em> a\u00a0na polpriamkach opa\u010dn\u00fdch k\u00a0polpriamkam <em>BC<\/em>, <em>CB<\/em> zvo\u013eme postupne body <em>X<\/em>, <em>Y<\/em>, <em>K<\/em>, <em>L<\/em> tak, aby platilo |<em>BX<\/em>|=|<em>BK<\/em>|=|<em>BC<\/em>|=|<em>CY<\/em>|=|<em>CL<\/em>|. Priamky <em>KX<\/em> a\u00a0<em>LY<\/em> sa pret\u00ednaj\u00fa v\u00a0bode <em>M<\/em>. Dok\u00e1\u017ete, \u017ee \u0165a\u017eisko trojuholn\u00edka <em>KLM<\/em> je toto\u017en\u00e9 so stredom kru\u017enice vp\u00edsanej do trojuholn\u00edka <em>ABC<\/em>.<\/p>\n<div class=\"pri\">(Tom\u00e1\u0161 Jur\u00edk)<\/div>\n<p><strong>A-I-6<\/strong><br \/>Na tabuli je nap\u00edsan\u00fd s\u00fa\u010din<\/p>\n<div class=\"pc\">1 \u00b7 2 \u00b7 3 \u00b7 \u2026 \u00b7 <em>n<\/em>.<\/div>\n<p>Pre ktor\u00e9 prirodzen\u00e9 \u010d\u00edsla n \u2265 2 je mo\u017en\u00e9 za niektor\u00e9 z \u010dinite\u013eov dop\u00edsa\u0165 v\u00fdkri\u010dn\u00edk a nahradi\u0165 ich tak ich faktori\u00e1lmi, aby v\u00fdsledn\u00fd s\u00fa\u010din bol rovn\u00fd druhej mocnine prirodzen\u00e9ho \u010d\u00edsla?<\/p>\n<div class=\"pri\">(Michal Rol\u00ednek)<\/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=1729\" target=\"_blank\"><input name=\"butta2z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=1730\" target=\"_blank\"><input name=\"butta2r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>A-S-1<\/strong><br \/>Prvo\u010d\u00edslo nazveme pekn\u00e9, ak sa d\u00e1 zap\u00edsa\u0165 ako rozdiel dvoch tret\u00edch mocn\u00edn prirodzen\u00fdch \u010d\u00edsel. Ur\u010dte posledn\u00e9 cifry v\u0161etk\u00fdch pekn\u00fdch prvo\u010d\u00edsel.<\/p>\n<div class=\"pri\">(Patrik Bak, Michal Rol\u00ednek)<\/div>\n<p><strong>A-S-2<\/strong><br \/>Kladn\u00e9 re\u00e1lne \u010d\u00edsla <em>a, b, c, d<\/em> sp\u013a\u0148aj\u00fa rovnosti<\/p>\n<div class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"65\/65ab1.png\" alt=\"\" width=\"67\" height=\"29\" align=\"middle\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0a\u00a0\u00a0\u00a0\u00a0\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"65\/65ab2.png\" alt=\"\" width=\"68\" height=\"28\" align=\"middle\" \/><\/div>\n<p>Dok\u00e1\u017ete nerovnos\u0165 <em>ab<\/em> \u2265 4 a n\u00e1jdite najmen\u0161iu mo\u017en\u00fa hodnotu v\u00fdrazu <em>ab + cd<\/em>.<\/p>\n<div class=\"pri\">(Jarom\u00edr \u0160im\u0161a)<\/div>\n<p><strong>A-S-3<\/strong><br \/>Dan\u00fd je lichobe\u017en\u00edk <em>ABCD<\/em> (AB \u2551 CD), v ktorom plat\u00ed |<em>BC<\/em>|=|<em>AB<\/em>|+|<em>CD<\/em>|. Dok\u00e1\u017ete, \u017ee<\/p>\n<p>\u00a0\u00a0\u00a0\u00a0\u00a0a) na ramene <em>AD<\/em> le\u017e\u00ed nejak\u00fd bod kru\u017enice maj\u00facej priemer <em>BC<\/em>,<\/p>\n<p>\u00a0\u00a0\u00a0\u00a0\u00a0b) na ramene <em>BC<\/em> le\u017e\u00ed nejak\u00fd bod kru\u017enice maj\u00facej priemer <em>AD<\/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<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=1731\" target=\"_blank\"><input name=\"butta3z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=1732\" target=\"_blank\"><input name=\"butta3r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>A-II-1<\/strong><br \/>Na tabuli s\u00fa nap\u00edsan\u00e9 r\u00f4zne kladn\u00e9 cel\u00e9 \u010d\u00edsla. Ich aritmetick\u00fd priemer je desatinn\u00e9 \u010d\u00edslo, ktor\u00e9ho desatinn\u00e1 \u010das\u0165 je presne 0,2016. Ak\u00fa najmen\u0161iu hodnotu m\u00f4\u017ee tento priemer ma\u0165?<\/p>\n<div class=\"pri\">(Patrik Bak)<\/div>\n<p><strong>A-II-2<\/strong><br \/>Dan\u00fd je \u0161tvorec <em>ABCD<\/em> so stranou d\u013a\u017eky 1. Na jeho strane <em>CD<\/em> zvol\u00edme bod <em>E<\/em> tak, aby platilo |\u2220<em>BAE<\/em>|= 60\u00b0. \u010ealej zvo\u013eme \u013eubovo\u013en\u00fd vn\u00fatorn\u00fd bod \u00fase\u010dky <em>AE<\/em> a ozna\u010dme ho <em>X<\/em>. Bodom <em>X<\/em> potom ve\u010fme kolmicu na priamku <em>BX<\/em> a jej priese\u010dn\u00edk s priamkou <em>BC<\/em> ozna\u010dme <em>Y<\/em>. Ak\u00e1 je najmen\u0161ia mo\u017en\u00e1 d\u013a\u017eka \u00fase\u010dky <em>BY<\/em>?<\/p>\n<div class=\"pri\">(Michal Rol\u00ednek)<\/div>\n<p><strong>A-II-3<\/strong><br \/>Ko\u013ek\u00fdmi sp\u00f4sobmi sa d\u00e1 rozdeli\u0165 mno\u017eina {1,2, &#8230;, 12} na \u0161es\u0165 disjunktn\u00fdch dvojprvkov\u00fdch podmno\u017e\u00edn tak\u00fdch, \u017ee ka\u017ed\u00e1 z nich obsahuje navz\u00e1jom nes\u00fadelite\u013en\u00e9 \u010d\u00edsla (teda tak\u00e9, ktor\u00e9 nemaj\u00fa spolo\u010dn\u00e9ho delite\u013ea v\u00e4\u010d\u0161ieho ako 1)?<\/p>\n<div class=\"pri\">(Martin Pan\u00e1k)<\/div>\n<p><strong>A-II-4<\/strong><br \/>Ur\u010dte najmen\u0161ie re\u00e1lne \u010d\u00edslo <em>m<\/em>, pre ktor\u00e9 mo\u017eno n\u00e1js\u0165 re\u00e1lne \u010d\u00edsla <em>a<\/em> a <em>b<\/em> tak, aby nerovnos\u0165<\/p>\n<div class=\"r\">|x<sup>2<\/sup> + <em>a<\/em>x + <em>b<\/em>|\u00a0\u2264 <em>m<\/em> (x<sup>2<\/sup> + 1)<\/div>\n<p>platila pre ka\u017ed\u00e9 <em>x<\/em>\u00a0\u2208 \u2329-1, 1\u232a.<\/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 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>65. ro\u010dn\u00edk matematickej olympi\u00e1dy 2015-2016S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3rie A, B a C aktualizovan\u00e9 12.8.2016 15:35<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","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":[22],"tags":[],"class_list":["post-1119","post","type-post","status-publish","format-standard","hentry","category-65-rocnik"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.6 - 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