{"id":1252,"date":"2017-10-20T17:58:23","date_gmt":"2017-10-20T17:58:23","guid":{"rendered":"http:\/\/matematika.besaba.com\/?p=1252"},"modified":"2023-11-02T18:52:02","modified_gmt":"2023-11-02T18:52:02","slug":"sutazne-ulohy-kategorii-z5-z9-6","status":"publish","type":"post","link":"https:\/\/matematika.besaba.com\/?p=1252","title":{"rendered":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9"},"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 Z5-Z9<\/p>\n<p><span class=\"povinne\">aktualizovan\u00e9 24.1.2019 18:30<\/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<table style=\"width: 90%;\" border=\"1\" cellspacing=\"2\" cellpadding=\"5\" align=\"center\">\n<tbody>\n<tr align=\"center\" bgcolor=\"#c0c0c0\">\n<td><span class=\"a\">Z5<\/span><\/td>\n<td><span class=\"a\">Z6<\/span><\/td>\n<td><span class=\"a\">Z7<\/span><\/td>\n<td><span class=\"a\">Z8<\/span><\/td>\n<td><span class=\"a\">Z9<\/span><\/td>\n<\/tr>\n<tr align=\"center\">\n<td><span class=\"a\"><a href=\"#z51\">dom\u00e1ce<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z61\">dom\u00e1ce<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z71\">dom\u00e1ce<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z81\">dom\u00e1ce<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z91\">dom\u00e1ce<\/a><\/span><\/td>\n<\/tr>\n<tr align=\"center\">\n<td><span class=\"a\"><a href=\"#z52\">obvodn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z62\">obvodn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z72\">obvodn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z82\">obvodn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z92\">obvodn\u00e9<\/a><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"4\" bgcolor=\"#c0c0c0\">\u00a0<\/td>\n<td align=\"center\"><span class=\"a\"><a href=\"#z93\">krajsk\u00e9<\/a><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"background-color: #ffff99;\" colspan=\"5\" align=\"center\"><span class=\"rs\">Let\u00e1k a zadania \u00faloh dom\u00e1ceho kola na stiahnutie vo form\u00e1te PDF: <a href=\"https:\/\/skmo.sk\/dokument.php?id=2530\" 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<p><!--more--><\/p>\n<p><a name=\"z51\"><\/a><\/p>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=2521\" target=\"_blank\" rel=\"noopener\"><input name=\"butt52z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2538\" target=\"_blank\" rel=\"noopener\"><input name=\"butt52r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z5-I-1<\/strong><br \/>\nJanko dostal vreckov\u00e9 a chce si za\u0148 k\u00fapi\u0165 nie\u010do dobr\u00e9. Keby si k\u00fapil \u0161tyri kol\u00e1\u010de, zv\u00fd\u0161ilo by mu 0,50 \u20ac. Keby si chcel k\u00fapi\u0165 p\u00e4\u0165 kol\u00e1\u010dov, ch\u00fdbalo by mu 0,60 \u20ac. Keby si k\u00fapil dva kol\u00e1\u010de a tri \u0161i\u0161ky, utratil by cel\u00e9 vreckov\u00e9 bezo zvy\u0161ku. Ko\u013eko stoj\u00ed jedna \u0161i\u0161ka?<\/p>\n<div class=\"pri\">(Lenka Dedkov\u00e1)<\/div>\n<p><strong>Z5-I-2<\/strong><br \/>\nJano mal tri klietky (\u010diernu, strieborn\u00fa, zlat\u00fa) a tri zvierat\u00e1 (mor\u010da, potkana a tchora). V ka\u017edej klietke bolo jedno zviera. Zlat\u00e1 klietka st\u00e1la na\u013eavo od \u010diernej klietky. Strieborn\u00e1 klietka st\u00e1la napravo od klietky s mor\u010da\u0165om. Potkan bol v klietke napravo od striebornej klietky. Ur\u010dte, v ktorej klietke bolo ktor\u00e9 zviera.<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z5-I-3<\/strong><br \/>\nNa obr\u00e1zku je diagram so siedmimi pol\u00ed\u010dkami. Nakreslite do neho hviezdi\u010dky tak, aby boli splnen\u00e9 v\u0161etky nasleduj\u00face podmienky:<\/p>\n<ul>\n<li>Hviezdi\u010diek je celkom 21.<\/li>\n<li>V ka\u017edom pol\u00ed\u010dku je aspo\u0148 jedna hviezdi\u010dka.<\/li>\n<li>V pol\u00ed\u010dkach ozna\u010den\u00fdch A, B, C je dokopy 8 hviezdi\u010diek.<\/li>\n<li>V pol\u00ed\u010dkach ozna\u010den\u00fdch A a B je dokopy menej hviezdi\u010diek ako v pol\u00ed\u010dku ozna\u010denom C.<\/li>\n<li>V pol\u00ed\u010dku ozna\u010denom B je viac hviezdi\u010diek ako v pol\u00ed\u010dku ozna\u010denom A.<\/li>\n<li>V kruhu je celkom 15 hviezdi\u010diek, v trojuholn\u00edku celkom 12 hviezdi\u010diek a v obd\u013a\u017eniku celkom 14 hviezdi\u010diek.<\/li>\n<\/ul>\n<div class=\"pc\"><img decoding=\"async\" src=\"67\/67z5a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z5-I-4<\/strong><br \/>\nEva s Marekom hrali bedminton a Viktor im po\u010d\u00edtal v\u00fdmeny. Po ka\u017ed\u00fdch 10 v\u00fdmen\u00e1ch nakreslil Viktor kr\u00ed\u017eik (X). Potom namiesto ka\u017ed\u00fdch 10 kr\u00ed\u017eikov nakreslil kr\u00fa\u017eok (O) a prisl\u00fachaj\u00facich 10 kr\u00ed\u017eikov<br \/>\nzmazal. Ke\u010f Eva a Marek hru ukon\u010dili, mal Viktor nakreslen\u00e9 toto:<\/p>\n<div class=\"r\">OOOXXXXXXX<\/div>\n<p>Ur\u010dte, ko\u013eko najmenej a ko\u013eko najviac v\u00fdmen Eva s Marekom mohli zohra\u0165.<\/p>\n<div class=\"pri\">(Miroslava Farkas Smitkov\u00e1)<\/div>\n<p><strong>Z5-I-5<\/strong><br \/>\nZostrojte \u013eubovo\u013en\u00fa \u00fase\u010dku <em>AS<\/em>, potom zostrojte kru\u017enicu <em>k<\/em> so stredom v bode <em>S<\/em>, ktor\u00e1 prech\u00e1dza bodom <em>A<\/em>?<\/p>\n<ol type=\"1\">\n<li>Zostrojte na kru\u017enici <em>k<\/em> body <em>E<\/em>, <em>F<\/em>, <em>G<\/em> tak, aby spolu s bodom <em>A<\/em> tvorili obd\u013a\u017enik <em>AEFG<\/em>. N\u00e1jdite aspo\u0148 dve rie\u0161enia.<\/li>\n<li>Zostrojte na kru\u017enici <em>k<\/em> body <em>B<\/em>, <em>C<\/em>, <em>D<\/em> tak, aby spolu s bodom <em>A<\/em> tvorili \u0161tvorec <em>ABCD<\/em>.<\/li>\n<\/ol>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\u00e1)<\/div>\n<p><strong>Z5-I-6<\/strong><br \/>\nNa stole le\u017ealo osem karti\u010diek s \u010d\u00edslami 2, 3, 5, 7, 11, 13, 17, 19. Fero si vybral tri karti\u010dky. S\u010d\u00edtal na nich nap\u00edsan\u00e9 \u010d\u00edsla a zistil, \u017ee ich s\u00fa\u010det je o 1 v\u00e4\u010d\u0161\u00ed ako s\u00fa\u010det \u010d\u00edsel na zvy\u0161n\u00fdch karti\u010dk\u00e1ch. Ktor\u00e9 karti\u010dky mohli zosta\u0165 na stole? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\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=\"z52\"><\/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=2540\" target=\"_blank\" rel=\"noopener\"><input name=\"butt52z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2542\" target=\"_blank\" rel=\"noopener\"><input name=\"butt52r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z5-II-1<\/strong><br \/>\n<img decoding=\"async\" src=\"67\/67z5b1.png\" alt=\"\" align=\"right\" border=\"0\" hspace=\"10\" vspace=\"5\" \/>Na obr\u00e1zku je zn\u00e1zornen\u00fdch p\u00e4\u0165 v\u00fdbehov \u010dasti zoo. Ka\u017ed\u00fd v\u00fdbeh ob\u00fdva jeden z piatich druhov zvierat. Pritom vieme, \u017ee<\/p>\n<ul>\n<li>v\u00fdbeh \u017eir\u00e1f m\u00e1 p\u00e4\u0165 str\u00e1n,<\/li>\n<li>v\u00fdbeh op\u00edc nesused\u00ed ani s v\u00fdbehom nosoro\u017ecov, ani s v\u00fdbehom \u017eir\u00e1f,<\/li>\n<li>v\u00fdbeh levov m\u00e1 rovnak\u00fd po\u010det str\u00e1n ako v\u00fdbeh op\u00edc,<\/li>\n<li>tulene maj\u00fa vo v\u00fdbehu jazierko.<\/li>\n<\/ul>\n<p>Ur\u010dte, ktor\u00e9 zvierat\u00e1 s\u00fa v ktorom v\u00fdbehu.<\/p>\n<div class=\"pri\">(Erika Novotn\u00e1)<\/div>\n<p><strong>Z5-II-2<\/strong><br \/>\nNa stole le\u017ealo p\u00e4\u0165 karti\u010diek s navz\u00e1jom r\u00f4znymi kladn\u00fdmi cel\u00fdmi \u010d\u00edslami. Marek spo\u010d\u00edtal, \u017ee najv\u00e4\u010d\u0161ie z \u010d\u00edsel na karti\u010dk\u00e1ch je o 8 v\u00e4\u010d\u0161ie ako najmen\u0161ie \u010d\u00edslo. Adam vzal zo stola karti\u010dky s najmen\u0161\u00edmi dvoma \u010d\u00edslami a spo\u010d\u00edtal, \u017ee s\u00fa\u010din t\u00fdchto dvoch \u010d\u00edsel je rovn\u00fd 12. Dominik potom spo\u010d\u00edtal, \u017ee s\u00fa\u010det \u010d\u00edsel na zvy\u0161n\u00fdch troch karti\u010dk\u00e1ch je rovn\u00fd 25. Ktor\u00e9 \u010d\u00edsla mohli by\u0165 nap\u00edsan\u00e9 na karti\u010dk\u00e1ch? N\u00e1jdite tri rie\u0161enia.<\/p>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\u00e1)<\/div>\n<p><strong>Z5-II-3<\/strong><br \/>\nZostrojte \u0161tvorec <em>ABCD<\/em> so stranou d\u013a\u017eky 6 cm a priese\u010dn\u00edk jeho uhloprie\u010dok ozna\u010dte <em>S<\/em>. Zostrojte bod <em>K<\/em> tak, aby spolu s bodmi <em>S, B, C<\/em> tvoril \u0161tvorec <em>BKCS<\/em>. Zostrojte bod <em>L<\/em> tak, aby spolu s bodmi <em>S, A , D<\/em> tvoril \u0161tvorec <em>ASDL<\/em>. Zostrojte \u00fase\u010dku <em>KL<\/em>, priese\u010dn\u00edk \u00fase\u010diek <em>KL<\/em> a <em>AD<\/em> ozna\u010dte <em>X<\/em>, priese\u010dn\u00edk \u00fase\u010diek <em>KL<\/em> a <em>BC<\/em> ozna\u010dte <em>Y<\/em>. Zo zadan\u00fdch \u00fadajov vypo\u010d\u00edtajte d\u013a\u017eku lomenej \u010diary <em>KYBAXL<\/em>.<\/p>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\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=\"z61\"><\/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=2522\" target=\"_blank\" rel=\"noopener\"><input name=\"butt61z\" type=\"button\" value=\"Zadanie\" \/><\/a><\/form>\n<p><strong>Z6-I-1<\/strong><br \/>\nAni\u010dka a Blanka si nap\u00edsali ka\u017ed\u00e1 jedno dvojcifern\u00e9 \u010d\u00edslo, ktor\u00e9 za\u010d\u00ednalo sedmi\u010dkou. Diev\u010dat\u00e1 si zvolili r\u00f4zne \u010d\u00edsla. Potom ka\u017ed\u00e1 medzi obe cifry vlo\u017eila nulu, tak\u017ee im vzniklo trojcifern\u00e9 \u010d\u00edslo. Od neho ka\u017ed\u00e1 od\u010d\u00edtala svoje p\u00f4vodn\u00e9 dvojcifern\u00e9 \u010d\u00edslo. V\u00fdsledok ich prekvapil. Ur\u010dte, ako sa ich v\u00fdsledky l\u00ed\u0161ili.<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z6-I-2<\/strong><br \/>\nErika chcela pon\u00faknu\u0165 \u010dokol\u00e1du svojim trom kamar\u00e1tkam. Ke\u010f ju vytiahla z batohu, zistila, \u017ee je pol\u00e1man\u00e1 ako na obr\u00e1zku. (Vyzna\u010den\u00e9 \u0161tvor\u010deky s\u00fa navz\u00e1jom zhodn\u00e9.) Diev\u010dat\u00e1 sa dohodli, \u017ee \u010dokol\u00e1du \u010falej l\u00e1ma\u0165 nebud\u00fa a l\u00f3som ur\u010dia, ak\u00fd ve\u013ek\u00fd k\u00fasok ktor\u00e1 dostane. Zora\u010fte \u0161tyri k\u00fasky \u010dokol\u00e1dy od najmen\u0161ieho po najv\u00e4\u010d\u0161\u00ed.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"67\/67z6a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Katar\u00edna Jasen\u010d\u00e1kov\u00e1)<\/div>\n<p><strong>Z6-I-3<\/strong><br \/>\nJano mal 100 rovnak\u00fdch zav\u00e1rac\u00edch flia\u0161, z ktor\u00fdch si staval trojbok\u00e9 pyram\u00eddy. Najvy\u0161\u0161ie poschodie pyram\u00eddy m\u00e1 v\u017edy jednu f\u013ea\u0161u, druh\u00e9 poschodie zhora predstavuje rovnostrann\u00fd trojuholn\u00edk, ktor\u00e9ho strana pozost\u00e1va z dvoch flia\u0161, at\u010f. Pr\u00edklad kon\u0161trukcie trojposchodovej pyram\u00eddy je na obr\u00e1zku.<\/p>\n<table border=\"0\" align=\"center\">\n<tbody>\n<tr align=\"center\">\n<td width=\"150\"><img decoding=\"async\" src=\"67\/67z6a2.png\" alt=\"\" border=\"0\" \/><\/td>\n<td width=\"150\"><img decoding=\"async\" src=\"67\/67z6a3.png\" alt=\"\" border=\"0\" \/><\/td>\n<td width=\"150\"><img decoding=\"async\" src=\"67\/67z6a4.png\" alt=\"\" border=\"0\" \/><\/td>\n<\/tr>\n<tr align=\"center\">\n<td>1. poschodie<\/td>\n<td>1. a 2. poschodie<\/td>\n<td>trojposchodov\u00e1 pyram\u00edda<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Ko\u013eko flia\u0161 Jano potreboval na p\u00e4\u0165poschodov\u00fa pyram\u00eddu?<br \/>\nKo\u013eko poschod\u00ed mala pyram\u00edda, na ktor\u00fa bolo pou\u017eit\u00fdch \u010do najviac Janov\u00fdch flia\u0161?<\/p>\n<div class=\"pri\">(Katar\u00edna Jasen\u010d\u00e1kov\u00e1)<\/div>\n<p><strong>Z6-I-4<\/strong><br \/>\nVeronika m\u00e1 klasick\u00fa \u0161achovnicu s 8 x 8 pol\u00ed\u010dkami. Riadky s\u00fa ozna\u010den\u00e9 ciframi 1 a\u017e 8, st\u013apce p\u00edsmenami A a\u017e H. Veronika polo\u017eila na pol\u00ed\u010dko B1 jazdca, s ktor\u00fdm mo\u017eno pohybova\u0165 iba tak ako v \u0161achu.<\/p>\n<ol type=\"1\">\n<li>Je mo\u017en\u00e9 premiestni\u0165 jazdca \u0161tyrmi \u0165ahmi na pol\u00ed\u010dko H1?<\/li>\n<li>Je mo\u017en\u00e9 premiestni\u0165 jazdca piatimi \u0165ahmi na pol\u00ed\u010dko E6?<\/li>\n<\/ol>\n<p><em>Ak \u00e1no, pop\u00ed\u0161te v\u0161etky mo\u017en\u00e9 postupnosti \u0165ahov. Ak nie, zd\u00f4vodnite, pre\u010do to mo\u017en\u00e9 nie je.<\/em><\/p>\n<div class=\"pri\">(Katar\u00edna Jasen\u010d\u00e1kov\u00e1)<\/div>\n<p><strong>Z6-I-5<\/strong><br \/>\nV plechovke boli \u010derven\u00e9 a zelen\u00e9 cukr\u00edky. Cyril zjedol 2\/5 v\u0161etk\u00fdch \u010derven\u00fdch cukr\u00edkov a Zuzka zjedla 3\/5 v\u0161etk\u00fdch zelen\u00fdch cukr\u00edkov. Teraz tvoria \u010derven\u00e9 cukr\u00edky 3\/8 v\u0161etk\u00fdch cukr\u00edkov v plechovke. Ko\u013eko najmenej cukr\u00edkov mohlo by\u0165 p\u00f4vodne v plechovke?<\/p>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\u00e1)<\/div>\n<p><strong>Z6-I-6<\/strong><br \/>\nZostrojte \u013eubovo\u013en\u00fa \u00fase\u010dku DS, potom zostrojte kru\u017enicu k so stredom v bode S, ktor\u00e1 prech\u00e1dza bodom D.<\/p>\n<ol type=\"1\">\n<li>Zostrojte rovnostrann\u00fd trojuholn\u00edk <em>DAS<\/em>, ktor\u00e9ho vrchol <em>A<\/em> le\u017e\u00ed na kru\u017enici <em>k<\/em>.<\/li>\n<li>Zostrojte rovnostrann\u00fd trojuholn\u00edk <em>ABC<\/em>, ktor\u00e9ho vrcholy <em>B<\/em> a <em>C<\/em> tie\u017e le\u017eia na kru\u017enici <em>k<\/em>.<\/li>\n<\/ol>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\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=\"z62\"><\/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=2758\" target=\"_blank\" rel=\"noopener\"><input name=\"butt62z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2761\" target=\"_blank\" rel=\"noopener\"><input name=\"butt62r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z6-II-1<\/strong><br \/>\nJanko dostal na Vianoce knihu, ktor\u00fa hne\u010f na \u0160tedr\u00fd de\u0148 24. decembra za\u010dal \u010d\u00edta\u0165. \u010c\u00edtal denne rovnak\u00fd po\u010det str\u00e1n, a to a\u017e do 31. janu\u00e1ra v novom roku. Toho d\u0148a zistil, \u017ee zatia\u013e pre\u010d\u00edtal 78 str\u00e1n, \u010do bola pr\u00e1ve tretina knihy. S\u00fa\u010dasne zistil, \u017ee ak by chcel knihu do\u010d\u00edta\u0165 pr\u00e1ve v de\u0148 svojich naroden\u00edn, musel by po\u010d\u00ednaj\u00fac zajtraj\u0161kom ka\u017ed\u00fd de\u0148 pre\u010d\u00edta\u0165 o \u0161tyri strany viac ako doposia\u013e. Ur\u010dte, kedy m\u00e1 Janko narodeniny.<\/p>\n<div class=\"pri\">(Miroslava Farkas Smitkov\u00e1)<\/div>\n<p><strong>Z6-II-2<\/strong><br \/>\nEvi\u010dka mala stavebnicu s deviatimi dielikmi, ktor\u00e9 boli ozna\u010den\u00e9 1, 2, 3, 4, 5, 6, 7, 8 a 9. \u010casom sa jej podarilo v\u0161etky dieliky strati\u0165, a to nasleduj\u00facim sp\u00f4sobom:<\/p>\n<ul>\n<li>najsk\u00f4r stratila \u0161tyri dieliky ozna\u010den\u00e9 nep\u00e1rnymi \u010d\u00edslami,<\/li>\n<li>potom stratila dieliky so s\u00fa\u010dinom \u010d\u00edsel 24,<\/li>\n<li>nakoniec stratila posledn\u00e9 dva dieliky, na ktor\u00fdch boli p\u00e1rne \u010d\u00edsla.<\/li>\n<\/ul>\n<p>Zistite, ktor\u00e9 \u010d\u00edsla mohli by\u0165 nap\u00edsan\u00e9 na posledn\u00fdch dvoch dielikoch. N\u00e1jdite dve rie\u0161enia.<\/p>\n<div class=\"pri\">(Erika Novotn\u00e1)<\/div>\n<p><strong>Z6-II-3<\/strong><br \/>\nZostrojte \u0161tvorec <em>ABCD<\/em> so stranou d\u013a\u017eky 6 cm. Zostrojte priamku <em>p<\/em> rovnobe\u017en\u00fa s uhloprie\u010dkou <em>AC<\/em> a prech\u00e1dzaj\u00facu bodom <em>D<\/em>. Zostrojte obd\u013a\u017enik <em>ACEF<\/em> tak, aby vrcholy <em>E<\/em> a <em>F<\/em> le\u017eali na priamke <em>p<\/em>. Zo zadan\u00fdch \u00fadajov vypo\u010d\u00edtajte obsah obd\u013a\u017enika <em>ACEF<\/em>.<\/p>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\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=\"z71\"><\/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=2523\" target=\"_blank\" rel=\"noopener\"><input name=\"butt71z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2756\" target=\"_blank\" rel=\"noopener\"><input name=\"butt71r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z7-I-1<\/strong><br \/>\nPeter povedal Pavlovi: &#8222;<em>Nap\u00ed\u0161 dvojcifern\u00e9 prirodzen\u00e9 \u010d\u00edslo, ktor\u00e9 m\u00e1 t\u00fa vlastnos\u0165, \u017ee ke\u010f od neho od\u010d\u00edta\u0161 dvojcifern\u00e9 prirodzen\u00e9 \u010d\u00edslo nap\u00edsan\u00e9 naopak, dostane\u0161 rozdiel 63.<\/em>&#8220; Ktor\u00e9 \u010d\u00edslo mohol Pavol nap\u00edsa\u0165? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z7-I-2<\/strong><br \/>\nDan\u00e9 s\u00fa dve dvojice rovnobe\u017en\u00fdch priamok <em>AB<\/em>||<em>CD<\/em> a <em>AC<\/em>||<em>BD<\/em>. Bod <em>E<\/em> le\u017e\u00ed na priamke <em>BD<\/em>, bod <em>F<\/em> je stredom \u00fase\u010dky <em>BD<\/em>, bod <em>G<\/em> je stredom \u00fase\u010dky <em>CD<\/em> a obsah trojuholn\u00edka <em>ACE<\/em> je 20 cm<sup>2<\/sup>. Ur\u010dte obsah trojuholn\u00edka <em>DFG<\/em>.<\/p>\n<div class=\"pri\">(Vladim\u00edra Semer\u00e1kov\u00e1)<\/div>\n<p><strong>Z7-I-3<\/strong><br \/>\nZoologick\u00e1 z\u00e1hrada pon\u00fakala \u0161kolsk\u00fdm skupin\u00e1m v\u00fdhodn\u00e9 vstupn\u00e9: ka\u017ed\u00fd piaty \u017eiak dost\u00e1va vstupenku zdarma. P\u00e1n u\u010dite\u013e 6.A spo\u010d\u00edtal, \u017ee ak k\u00fapi vstupn\u00e9 de\u0165om zo svojej triedy, u\u0161etr\u00ed za \u0161tyri vstupenky a zaplat\u00ed 19,95 \u20ac. Pani u\u010dite\u013eka 6.B mu navrhla, nech k\u00fapi vstupenky de\u0165om oboch tried naraz, a tak bud\u00fa plati\u0165 44,10 \u20ac. Ko\u013eko det\u00ed z 6.A a ko\u013eko det\u00ed z 6.B i\u0161lo do zoo? (Cena vstupenky v centoch je celo\u010d\u00edseln\u00e1.)<\/p>\n<div class=\"pri\">(Libor \u0160im\u016fnek)<\/div>\n<p><strong>Z7-I-4<\/strong><br \/>\nNa stole le\u017ealo \u0161es\u0165 karti\u010diek s ciframi 1, 2, 3, 4, 5, 6. Ane\u017eka z t\u00fdchto karti\u010diek<br \/>\nzlo\u017eila \u0161es\u0165cifern\u00e9 \u010d\u00edslo, ktor\u00e9 bolo delite\u013en\u00e9 \u0161iestimi. Potom postupne odoberala karti\u010dky sprava. Ke\u010f odobrala prv\u00fa karti\u010dku, zostalo na stole p\u00e4\u0165cifern\u00e9 \u010d\u00edslo delite\u013en\u00e9 piatimi. Ke\u010f odobrala \u010fal\u0161iu karti\u010dku, zostalo \u0161tvorcifern\u00e9 \u010d\u00edslo delite\u013en\u00e9 \u0161tyrmi. Ke\u010f odoberala \u010falej, z\u00edskala postupne trojcifern\u00e9 \u010d\u00edslo delite\u013en\u00e9 tromi a dvojcifern\u00e9 \u010d\u00edslo delite\u013en\u00e9 dvoma. Ktor\u00e9 \u0161es\u0165cifern\u00e9 \u010d\u00edslo mohla Ane\u017eka p\u00f4vodne zlo\u017ei\u0165? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\u00e1)<\/div>\n<p><strong>Z7-I-5<\/strong><br \/>\nProkop zostrojil trojuholn\u00edk <em>ABC<\/em>, ktor\u00e9ho vn\u00fatorn\u00fd uhol pri vrchole <em>A<\/em> bol v\u00e4\u010d\u0161\u00ed ako 60\u00b0 a vn\u00fatorn\u00fd uhol pri vrchole <em>B<\/em> bol men\u0161\u00ed ako 60\u00b0. Juraj narysoval v polrovine ur\u010denej priamkou <em>AB<\/em> a bodom <em>C<\/em> bod <em>D<\/em>, a to tak, \u017ee trojuholn\u00edk <em>ABD<\/em> bol rovnostrann\u00fd. Potom chlapci zistili, \u017ee trojuholn\u00edky <em>ACD<\/em> a <em>BCD<\/em> s\u00fa rovnoramenn\u00e9 s hlavn\u00fdm vrcholom <em>D<\/em>. Ur\u010dte ve\u013ekos\u0165 uhla <em>ACB<\/em>.<\/p>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z7-I-6<\/strong><br \/>\nVodn\u00edk Chaluha nalieval hmlu do rozmanit\u00fdch, r\u00f4zne ve\u013ek\u00fdch n\u00e1dob, ktor\u00e9 si starostlivo zoradil na polici. Pri nalievan\u00ed postupoval postupne z jednej strany, \u017eiadnu n\u00e1dobu nepreskakoval. Do ka\u017edej n\u00e1doby sa vojde aspo\u0148 deciliter hmly. Keby nalieval hmlu sedemlitrovou odmerkou, hmla z prvej odmerky by naplnila presne 11 n\u00e1dob, hmla z druhej odmerky by naplnila presne \u010fal\u0161\u00edch 12 n\u00e1dob a hmla z tretej odmerky by naplnila presne 7 n\u00e1dob. Ak by pou\u017eil p\u00e4\u0165litrov\u00fa odmerku, tak hmla z prvej odmerky by naplnila presne 8 n\u00e1dob, z druhej presne 10 n\u00e1dob, z tretej presne 7 n\u00e1dob a zo \u0161tvrtej odmerky presne 4 n\u00e1doby. Rozhodnite, \u010di je tridsiata n\u00e1doba v porad\u00ed v\u00e4\u010d\u0161ia ako dvadsiatapiata.<\/p>\n<div class=\"pri\">(Karel Pazourek)<\/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=\"z72\"><\/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=2759\" target=\"_blank\" rel=\"noopener\"><input name=\"butt72z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2762\" target=\"_blank\" rel=\"noopener\"><input name=\"butt72r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z7-II-1<\/strong><br \/>\nFlori\u00e1n prem\u00fd\u0161\u013eal, ak\u00fa kyticu nech\u00e1 mami\u010dke uviaza\u0165 na De\u0148 matiek. V kvetin\u00e1rstve pod\u013ea cenn\u00edka spo\u010d\u00edtal, \u017ee \u010di k\u00fapi 5 klasick\u00fdch gerber alebo 7 minigerber, kytica bude st\u00e1\u0165 po doplnen\u00ed ozdobnej stuhy rovnako, a to 29,50 eur. Ak by v\u0161ak k\u00fapil iba 2 minigerbery a 1 klasick\u00fa gerberu bez \u010fal\u0161\u00edch doplnkov, zaplatil by 10,20 eur. Ko\u013eko stoj\u00ed 1 minigerbera?<\/p>\n<div class=\"pri\">(Libor \u0160im\u016fnek)<\/div>\n<p><strong>Z7-II-2<\/strong><br \/>\nNa stole le\u017ealo \u0161es\u0165 karti\u010diek s ciframi 1, 2, 3, 4, 5 a 6. Kamila z troch karti\u010diek zlo\u017eila trojcifern\u00e9 \u010d\u00edslo, ktor\u00e9 bolo v\u00e4\u010d\u0161ie ako 500 a bolo delite\u013en\u00e9 \u0161tyrmi. Filip zo zvy\u0161n\u00fdch troch karti\u010diek zlo\u017eil trojcifern\u00e9 \u010d\u00edslo delite\u013en\u00e9 tromi aj piatimi. Kamila potom obe trojcifern\u00e9 \u010d\u00edsla s\u010d\u00edtala a Filip si v\u0161imol, \u017ee tento s\u00fa\u010det je trojcifern\u00e9 \u010d\u00edslo, ktor\u00e9 sa \u010d\u00edta rovnako z\u013eava ako sprava. Ktor\u00e9 \u010d\u00edsla mohli Kamila s Filipom zlo\u017ei\u0165? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\u00e1)<\/div>\n<p><strong>Z7-II-3<\/strong><br \/>\nZostrojte kru\u017enicu so stredom <em>S<\/em> a polomerom 3 cm. Zostrojte dva navz\u00e1jom kolm\u00e9 priemery <em>AC<\/em> a <em>BD<\/em> tejto kru\u017enice. Zostrojte rovnoramenn\u00e9 trojuholn\u00edky <em>ABK<\/em>, <em>BCL<\/em>, <em>CDM<\/em>, <em>DAN<\/em> tak, aby:<\/p>\n<ul>\n<li>z\u00e1klad\u0148ou ka\u017ed\u00e9ho trojuholn\u00edka bola strana \u0161tvoruholn\u00edka <em>ABCD<\/em>,<\/li>\n<li>z\u00e1klad\u0148a ka\u017ed\u00e9ho trojuholn\u00edka bola zhodn\u00e1 s v\u00fd\u0161kou na t\u00fato stranu,<\/li>\n<li>\u017eiadny trojuholn\u00edk neprekr\u00fdval \u0161tvoruholn\u00edk <em>ABCD<\/em>.<\/li>\n<\/ul>\n<p>Zo zadan\u00fdch \u00fadajov vypo\u010d\u00edtajte obsah mnohouholn\u00edka <em>AKBLCMDN<\/em>.<\/p>\n<div class=\"pri\">(Marie Krej\u010dov\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=\"z81\"><\/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=2524\" target=\"_blank\" rel=\"noopener\"><input name=\"butt81z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2757\" target=\"_blank\" rel=\"noopener\"><input name=\"butt81r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z8-I-1<\/strong><br \/>\nVyjadrite \u010d\u00edslo mili\u00f3n pomocou \u010d\u00edsel obsahuj\u00facich iba cifry 9 a algebrick\u00fdch oper\u00e1ci\u00ed plus, m\u00ednus, kr\u00e1t, delen\u00e9, mocnina a odmocnina. Ur\u010dte aspo\u0148 tri r\u00f4zne rie\u0161enia.<\/p>\n<div class=\"pri\">(Lenka Dedkov\u00e1)<\/div>\n<p><strong>Z8-I-2<\/strong><br \/>\nV ostrouhlom trojuholn\u00edku <em>KLM<\/em> m\u00e1 uhol <em>KLM<\/em> ve\u013ekos\u0165 68\u25e6. Bod <em>V<\/em> je priese\u010dn\u00edkom v\u00fd\u0161ok a <em>P je p\u00e4tou v\u00fd\u0161ky na stranu <em>LM<\/em>. Os uhla <em>PVM<\/em> je rovnobe\u017en\u00e1 so stranou <em>KM<\/em>. Porovnajte ve\u013ekosti uhlov <em>MKL<\/em> a <em>LMK<\/em>.<\/em><\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z8-I-3<\/strong><br \/>\nAdelka mala na papieri nap\u00edsan\u00e9 dve \u010d\u00edsla. Ke\u010f k nim prip\u00edsala e\u0161te ich najv\u00e4\u010d\u0161\u00ed spolo\u010dn\u00fd delite\u013e a najmen\u0161\u00ed spolo\u010dn\u00fd n\u00e1sobok, dostala \u0161tyri r\u00f4zne \u010d\u00edsla men\u0161ie ako 100. S \u00fa\u017easom zistila, \u017ee ke\u010f vydel\u00ed najv\u00e4\u010d\u0161ie z t\u00fdchto \u0161tyroch \u010d\u00edsel najmen\u0161\u00edm, dostane najv\u00e4\u010d\u0161\u00ed spolo\u010dn\u00fd delite\u013e v\u0161etk\u00fdch \u0161tyroch \u010d\u00edsel. Ktor\u00e9 \u010d\u00edsla mala Adelka nap\u00edsan\u00e9 na papieri?<\/p>\n<div class=\"pri\">(Michaela Petrov\u00e1)<\/div>\n<p><strong>Z8-I-4<\/strong><br \/>\nRoboti R\u00f3bert a Hubert skladaj\u00fa a rozoberaj\u00fa mlyn\u010deky na k\u00e1vu. Pritom ka\u017ed\u00fd z nich mlyn\u010dek zlo\u017e\u00ed \u0161tyrikr\u00e1t r\u00fdchlej\u0161ie, ako ho ten druh\u00fd rozoberie. Ke\u010f r\u00e1no pri\u0161li do dielne, nieko\u013eko mlyn\u010dekov u\u017e tam bolo zlo\u017een\u00fdch. O 9:00 za\u010dal Hubert stava\u0165 a R\u00f3bert rozobera\u0165, presne o 12:00 Hubert dokon\u010dil skladanie mlyn\u010deka a R\u00f3bert rozoberanie in\u00e9ho. Spolu za t\u00fato zmenu pribudlo 27 mlyn\u010dekov. O 13:00 za\u010dal R\u00f3bert stava\u0165 a Hubert rozobera\u0165, presne o 19:00 dokon\u010dil R\u00f3bert skladanie posledn\u00e9ho mlyn\u010deka a Hubert rozoberanie in\u00e9ho. Spolu za t\u00fato zmenu pribudlo 120 mlyn\u010dekov. Za ako dlho zlo\u017e\u00ed mlyn\u010dek Hubert? Za ako dlho ho zlo\u017e\u00ed R\u00f3bert?<\/p>\n<div class=\"pri\">(Karel Pazourek)<\/div>\n<p><strong>Z8-I-5<\/strong><br \/>\nZhodn\u00e9 obd\u013a\u017eniky <em>ABCD<\/em> a <em>EFGH<\/em> s\u00fa umiestnen\u00e9 tak, \u017ee ich zhodn\u00e9 strany s\u00fa rovnobe\u017en\u00e9. Body <em>I, J, K, L, M<\/em> a <em>N<\/em> s\u00fa priese\u010dn\u00edky pred\u013a\u017een\u00fdch str\u00e1n ako na obr\u00e1zku. Obsah obd\u013a\u017enika <em>BNHM<\/em> je 12 cm<sup>2<\/sup>, obsah obd\u013a\u017enika <em>MBCK<\/em> je 63 cm<sup>2<\/sup> a obsah obd\u013a\u017enika <em>MLGH<\/em> je 28 cm<sup>2<\/sup>. Ur\u010dte obsah obd\u013a\u017enika <em>IFJD<\/em>.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"67\/67z8a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z8-I-6<\/strong><br \/>\nPriamka predstavuje \u010d\u00edseln\u00fa os a vyzna\u010den\u00e9 body zodpovedaj\u00fa \u010d\u00edslam a, -a, a + 1, av\u0161ak v neur\u010denom porad\u00ed. Zostrojte body, ktor\u00e9 zodpovedaj\u00fa \u010d\u00edslam 0 a 1. Preberte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"67\/67z8a2.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Michaela Petrov\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=\"z82\"><\/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=2760\" target=\"_blank\" rel=\"noopener\"><input name=\"butt82z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2763\" target=\"_blank\" rel=\"noopener\"><input name=\"butt82r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z8-II-1<\/strong><br \/>\nJo\u017eko si do zo\u0161ita nap\u00edsal nasleduj\u00facu \u00falohu:<\/p>\n<div class=\"r\">M + A + M + R+A + D + M + A + T + E + M + A + T + I + K + U =<\/div>\n<p>Potom nahradzoval p\u00edsmen\u00e1 ciframi od 1 do 9, a to tak, \u017ee r\u00f4zne p\u00edsmen\u00e1 nahradzoval r\u00f4znymi ciframi a rovnak\u00e9 rovnak\u00fdmi. Ak\u00fd najv\u00e4\u010d\u0161\u00ed s\u00fa\u010det mohol Jo\u017eko dosta\u0165? A mohol Jo\u017eko dosta\u0165 s\u00fa\u010det 50?<br \/>\n\u010casom sa Jo\u017ekovi podarilo vytvori\u0165 s\u00fa\u010det 59. Ktor\u00e1 cifra mohla v takom pr\u00edpade zodpoveda\u0165 p\u00edsmenu T? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Erika Novotn\u00e1)<\/div>\n<p><strong>Z8-II-2<\/strong><br \/>\nKocka s hranou 12 cm bola rozdelen\u00e1 na men\u0161ie navz\u00e1jom zhodn\u00e9 koc\u00f4\u010dky tak, \u017ee s\u00fa\u010det v\u0161etk\u00fdch ich povrchov bol osemkr\u00e1t v\u00e4\u010d\u0161\u00ed ako povrch p\u00f4vodnej kocky. Ur\u010dte, ko\u013eko bolo mal\u00fdch koc\u00f4\u010dok a ak\u00e9 dlh\u00e9 boli ich hrany.<\/p>\n<div class=\"pri\">(Marta Volfov\u00e1)<\/div>\n<p><strong>Z8-II-3<\/strong><br \/>\nV rovnostrannom trojuholn\u00edku <em>ABC<\/em> so stranou d\u013a\u017eky 8 cm je bod <em>D<\/em> stred strany <em>BC<\/em> a bod <em>E<\/em> je stred strany <em>AC<\/em>. Bod <em>F<\/em> le\u017e\u00ed na \u00fase\u010dke <em>BC<\/em> tak, \u017ee obsah trojuholn\u00edka <em>ABF<\/em> je rovnak\u00fd ako obsah \u0161tvoruholn\u00edka <em>ABDE<\/em>. Vypo\u010d\u00edtajte d\u013a\u017eku \u00fase\u010dky <em>BF<\/em>.<\/p>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\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=\"z91\"><\/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=2525\" target=\"_blank\" rel=\"noopener\"><input name=\"butt91z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2539\" target=\"_blank\" rel=\"noopener\"><input name=\"butt91r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-I-1<\/strong><br \/>\nVekov\u00fd priemer v\u0161etk\u00fdch \u013eud\u00ed, ktor\u00ed sa zi\u0161li na rodinnej oslave, bol rovn\u00fd po\u010dtu pr\u00edtomn\u00fdch. Teta Beta, ktor\u00e1 mala 29 rokov, sa vz\u00e1p\u00e4t\u00ed ospravedlnila a odi\u0161la. Aj po odchode tety Bety bol vekov\u00fd priemer v\u0161etk\u00fdch pr\u00edtomn\u00fdch \u013eud\u00ed rovn\u00fd ich po\u010dtu. Ko\u013eko \u013eud\u00ed bolo p\u00f4vodne na oslave?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z9-I-2<\/strong><br \/>\nV lichobe\u017en\u00edku <em>VODY<\/em> plat\u00ed, \u017ee <em>VO<\/em> je dlh\u0161ou z\u00e1klad\u0148ou, priese\u010dn\u00edk uhloprie\u010dok <em>K<\/em> del\u00ed \u00fase\u010dku <em>VD<\/em> v pomere 3 : 2 a obsah trojuholn\u00edka <em>KOV<\/em> je rovn\u00fd 13,5 cm<sup>2<\/sup>. Ur\u010dte obsah cel\u00e9ho lichobe\u017en\u00edka.<\/p>\n<div class=\"pri\">(Michaela Petrov\u00e1)<\/div>\n<p><strong>Z9-I-3<\/strong><br \/>\nRoboti R\u00f3bert a Hubert skladaj\u00fa a rozoberaj\u00fa mlyn\u010deky na k\u00e1vu. Pritom ka\u017ed\u00fd z nich mlyn\u010dek zlo\u017e\u00ed \u0161tyrikr\u00e1t r\u00fdchlej\u0161ie, ako ho s\u00e1m rozoberie. Ke\u010f r\u00e1no pri\u0161li do dielne, nieko\u013eko mlyn\u010dekov u\u017e tam bolo zlo\u017een\u00fdch. O 7:00 za\u010dal Hubert stava\u0165 a R\u00f3bert rozobera\u0165, presne o 12:00 Hubert dokon\u010dil skladanie mlyn\u010deka a R\u00f3bert rozoberanie in\u00e9ho. Celkom za t\u00fato zmenu pribudlo 70 mlyn\u010dekov. O 13:00 za\u010dal R\u00f3bert stava\u0165 a Hubert rozobera\u0165, presne o 22:00 dokon\u010dil R\u00f3bert skladanie posledn\u00e9ho mlyn\u010deka a Hubert rozoberanie in\u00e9ho. Celkom za t\u00fato zmenu pribudlo 36 mlyn\u010dekov. Za ako dlho by zlo\u017eili 360 mlyn\u010dekov, keby R\u00f3bert aj Hubert skladali spolo\u010dne?<\/p>\n<div class=\"pri\">(Karel Pazourek)<\/div>\n<p><strong>Z9-I-4<\/strong><br \/>\n\u010c\u00edsla 1, 2, 3, 4, 5, 6, 7, 8 a 9 sa chystali na cestu vlakom s tromi vag\u00f3nmi. Chceli sa rozsadi\u0165 tak, aby v ka\u017edom vag\u00f3ne sedeli tri \u010d\u00edsla a najv\u00e4\u010d\u0161ie z ka\u017edej trojice bolo rovn\u00e9 s\u00fa\u010dtu zvy\u0161n\u00fdch dvoch. Sprievodca tvrdil, \u017ee to nie je probl\u00e9m, a sna\u017eil sa \u010d\u00edslam pom\u00f4c\u0165. Naopak v\u00fdpravca tvrdil, \u017ee to nie je mo\u017en\u00e9. Rozhodnite, kto z nich mal pravdu.<\/p>\n<div class=\"pri\">(Erika Novotn\u00e1)<\/div>\n<p><strong>Z9-I-5<\/strong><br \/>\nVn\u00fatri obd\u013a\u017enika <em>ABCD<\/em> le\u017eia body <em>E<\/em> a <em>F<\/em> tak, \u017ee \u00fase\u010dky <em>EA<\/em>, <em>ED<\/em>, <em>EF<\/em>, <em>FB<\/em>, <em>FC<\/em> s\u00fa navz\u00e1jom zhodn\u00e9. Strana <em>AB<\/em> je dlh\u00e1 22 cm a kru\u017enica op\u00edsan\u00e1 trojuholn\u00edku <em>AFD<\/em> m\u00e1 polomer 10 cm. Ur\u010dte d\u013a\u017eku strany <em>BC<\/em>.<\/p>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\u00e1)<\/div>\n<p><strong>Z9-I-6<\/strong><br \/>\nNa priamke predstavuj\u00facej \u010d\u00edseln\u00fa os uv\u00e1\u017ete navz\u00e1jom r\u00f4zne body zodpovedaj\u00face \u010d\u00edslam <em>a<\/em>, <em>2a<\/em>, <em>3a + 1<\/em> vo v\u0161etk\u00fdch mo\u017en\u00fdch poradiach. Pri ka\u017edej mo\u017enosti rozhodnite, \u010di je tak\u00e9 usporiadanie mo\u017en\u00e9. Ak \u00e1no, uve\u010fte konkr\u00e9tny pr\u00edklad, ak nie, zd\u00f4vodnite pre\u010do. (Michaela Petrov\u00e1)<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"67\/67z9a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Michaela Petrov\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=\"z92\"><\/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=2541\" target=\"_blank\" rel=\"noopener\"><input name=\"butt92z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2543\" target=\"_blank\" rel=\"noopener\"><input name=\"butt92r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-II-1<\/strong><br \/>\n\u0160tefka a Terezka dostali bonboni\u00e9ru, v ktorej bolo 35 \u010dokol\u00e1dov\u00fdch cukr\u00edkov. Prv\u00fd de\u0148 zjedla Terezka <em>2\/5<\/em> toho, \u010do zjedla v tento de\u0148 \u0160tefka. Druh\u00fd de\u0148 zjedla \u0160tefka <em>3\/4<\/em> toho, \u010do zjedla v tento de\u0148 Terezka. Na konci druh\u00e9ho d\u0148a bola bonboni\u00e9ra pr\u00e1zdna. Ko\u013eko cukr\u00edkov celkom zjedla Terezka, ke\u010f vieme, \u017ee po\u010det cukr\u00edkov zjeden\u00fdch Terezkou a po\u010det cukr\u00edkov zjeden\u00fdch \u0160tefkou sa l\u00ed\u0161ia o najmen\u0161iu mo\u017en\u00fa hodnotu? Ka\u017ed\u00fd de\u0148 zjedlo ka\u017ed\u00e9 z diev\u010dat aspo\u0148 jeden cukr\u00edk a \u017eiadny cukr\u00edk nebol delen\u00fd na \u010dasti.<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z9-II-2<\/strong><br \/>\nVo \u0161tvoruholn\u00edku <em>ABCD<\/em> s\u00fa strany <em>AB<\/em> a <em>CD<\/em> rovnobe\u017en\u00e9, pri\u010dom strana <em>AB<\/em> je dvakr\u00e1t dlh\u0161ia ako strana <em>CD<\/em>. Bod <em>S<\/em> je priese\u010dn\u00edkom uhloprie\u010dok \u0161tvoruholn\u00edka <em>ABCD<\/em> a trojuholn\u00edky <em>ABS<\/em> a <em>CDS<\/em> s\u00fa oba rovnostrann\u00e9. Bod <em>E<\/em> je tak\u00fd bod \u00fase\u010dky <em>BS<\/em>, \u017ee ve\u013ekos\u0165 uhla <em>ACE<\/em> je 30\u00b0. Ur\u010dte pomer obsahov \u0161tvoruholn\u00edka <em>ABCD<\/em> a trojuholn\u00edka <em>EBC<\/em>.<\/p>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z9-II-3<\/strong><br \/>\n\u0160tyri kamar\u00e1tky na\u0161li v u\u010debnici nasleduj\u00facu pozn\u00e1mku:<\/p>\n<p class=\"pc\"><em>Vieme, \u017ee <img decoding=\"async\" src=\"67\/67z9b5.png\" alt=\"\" border=\"0\" \/> a \u017ee <img decoding=\"async\" src=\"67\/67z9b3.png\" alt=\"\" border=\"0\" \/> je prirodzen\u00e9 \u010d\u00edslo.<\/em><\/p>\n<p>Teraz prem\u00fd\u0161\u013eaj\u00fa a dohaduj\u00fa sa, \u010do m\u00f4\u017eu poveda\u0165 o \u010d\u00edsle c:<\/p>\n<p style=\"margin-left: 50px;\">Anna: &#8222;Ur\u010dite to nem\u00f4\u017ee by\u0165 <img decoding=\"async\" src=\"67\/67z9b4.png\" alt=\"\" border=\"0\" \/>.&#8220;<br \/>\nDana: &#8222;M\u00f4\u017ee to by\u0165 napr. 98.&#8220;<br \/>\nHana: &#8222;M\u00f4\u017ee to by\u0165 ak\u00e9ko\u013evek p\u00e1rne \u010d\u00edslo.&#8220;<br \/>\nJana: &#8222;Tak\u00e9 \u010d\u00edslo je ur\u010dite len jedno.&#8220;<\/p>\n<p>Rozhodnite, ktor\u00e9 z diev\u010dat m\u00e1 (maj\u00fa) pravdu, a vysvetlite pre\u010do.<\/p>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z9-II-4<\/strong><br \/>\nKv\u00e1der s rozmermi 20 cm \u00d7 30 cm \u00d7 40 cm je polo\u017een\u00fd tak, \u017ee hrana d\u013a\u017eky 20 cm le\u017e\u00ed na stole a hrana d\u013a\u017eky 40 cm zviera so stolom uhol 30\u00b0. Kv\u00e1der je \u010diasto\u010dne naplnen\u00fd vodou, ktor\u00e1 om\u00e1\u010da horn\u00fa stenu s rozmermi 20 cm \u00d7 40 cm z jednej \u0161tvrtiny. Ur\u010dte objem vody v kv\u00e1dri.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"67\/67z9b2.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Marie Krej\u010dov\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=\"z93\"><\/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=2728\" target=\"_blank\" rel=\"noopener\"><input name=\"butt93z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2729\" target=\"_blank\" rel=\"noopener\"><input name=\"butt93r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-III-1<\/strong><br \/>\nVo fin\u00e1le s\u00fa\u0165a\u017ee spolo\u010densk\u00fdch tancov mal ka\u017ed\u00fd z dvadsiatich piatich porotcov ohodnoti\u0165 p\u00e4\u0165 tane\u010dn\u00fdch p\u00e1rov zn\u00e1mkami 1 a\u017e 5 ako v \u0161kole, pri\u010dom ka\u017ed\u00fa zn\u00e1mku musel pou\u017ei\u0165 pr\u00e1ve raz. Po s\u00fa\u0165a\u017ei bola zverejnen\u00e1 tabu\u013eka s priemermi zn\u00e1mok pre ka\u017ed\u00fd p\u00e1r:<\/p>\n<p><center><\/p>\n<table class=\"aligncenter\" style=\"width: 400px; border-collapse: collapse;\" border=\"2\" cellspacing=\"2\" cellpadding=\"2\">\n<tr class=\"r\">\n<td width=\"80\">p\u00e1r A<\/td>\n<td width=\"80\">p\u00e1r B<\/td>\n<td width=\"80\">p\u00e1r C<\/td>\n<td width=\"80\">p\u00e1r D<\/td>\n<td width=\"80\">p\u00e1r E<\/td>\n<\/tr>\n<tr class=\"r\">\n<td>4,68<\/td>\n<td>3,86<\/td>\n<td>3,36<\/td>\n<td>1,44<\/td>\n<td>1,60<\/td>\n<\/tr>\n<\/table>\n<p><\/center><\/p>\n<p>Vz\u00e1p\u00e4t\u00ed sa zistilo, \u017ee pr\u00e1ve jeden z priemerov je v tabu\u013eke uveden\u00fd zle. Zistite, ktor\u00fd priemer je chybn\u00fd, a opravte ho. Jedn\u00fdm z porotcov bol str\u00fdko tane\u010dn\u00edka p\u00e1ru E. P\u00e1ry v\u0161ak hodnotil \u010destne a bez zaujatosti nasleduj\u00facim sp\u00f4sobom: 1 pre D, 2 pre E, 3 pre B, 4 pre C a 5 pre A. Napriek tomu mu po vyhl\u00e1sen\u00ed opraven\u00fdch v\u00fdsledkov v\u0155talo v hlave, \u010di by b\u00fdval mohol nepoctiv\u00fdm ozn\u00e1mkovan\u00edm posun\u00fa\u0165 p\u00e1r E na prv\u00e9 miesto. Zistite, \u010di to mohol dok\u00e1za\u0165.<\/p>\n<div class=\"pri\">(Libor \u0160im\u016fnek)<\/div>\n<p><strong>Z9-III-2<\/strong><br \/>\nKa\u017ed\u00e9 z \u010d\u00edsel <strong><em>a<\/em><\/strong> a <em><strong>b<\/strong><\/em> sa d\u00e1 vyjadri\u0165 ako s\u00fa\u010din troch prvo\u010d\u00edsel men\u0161\u00edch ako 10. Ka\u017ed\u00e9 prvo\u010d\u00edslo men\u0161ie ako 10 je pr\u00edtomn\u00e9 v rozklade aspo\u0148 jedn\u00e9ho z \u010d\u00edsel <em><strong>a<\/strong><\/em> a <em><strong>b<\/strong><\/em>. Najv\u00e4\u010d\u0161\u00ed spolo\u010dn\u00fd delite\u013e \u010d\u00edsel <em><strong>a<\/strong><\/em> a <em><strong>b<\/strong><\/em> je rovn\u00fd najv\u00e4\u010d\u0161iemu spolo\u010dn\u00e9mu delite\u013eovi \u010d\u00edsel <em><strong>a<\/strong><\/em> a <em><strong>b<\/strong><\/em> a s\u00fa\u010dasne dvojn\u00e1sobku najv\u00e4\u010d\u0161ieho spolo\u010dn\u00e9ho delite\u013ea \u010d\u00edsel <em><strong>a<\/strong><\/em> a <em><strong>b<\/strong><\/em>. Ur\u010dte \u010d\u00edsla <em><strong>a<\/strong><\/em> a <em><strong>b<\/strong><\/em>.<\/p>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z9-III-3<\/strong><br \/>\nNa tajomnom ostrove \u017eij\u00fa dva druhy domorodcov: jedni hovoria v\u00fdlu\u010dne pravdu (poctivci), druh\u00ed st\u00e1le klam\u00fa (klam\u00e1ri). V\u00fdskumn\u00edci tam stretli nieko\u013eko skup\u00edn domorodcov a v\u017edy sa ka\u017ed\u00e9ho zo skupiny sp\u00fdtali, ko\u013eko je v ich skupine poctivcov.<\/p>\n<ul>\n<li>Ako odpovede od jednej \u0161tvor\u010dlennej skupiny dostali v\u0161etky \u010d\u00edsla rovnak\u00e9.<\/li>\n<li>Ako odpovede od druhej skupiny dostali \u010d\u00edsla 0, 1, 3, 3, 3, 4.<\/li>\n<\/ul>\n<p>Ko\u013eko poctivcov mohlo by\u0165 v jednej a ko\u013eko v druhej skupine? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Marta Volfov\u00e1)<\/div>\n<p><strong>Z9-III-4<\/strong><br \/>\nV koso\u0161tvorci <em>ABCD<\/em> so stranou d\u013a\u017eky 4 cm a uhloprie\u010dkou <em>AC<\/em> d\u013a\u017eky 4 cm je bod <em>K<\/em> stredom strany <em>BC<\/em>. \u010ealej s\u00fa zostrojen\u00e9 body <em>L<\/em> a <em>M<\/em>, ktor\u00e9 tvoria spolu s bodmi <em>D<\/em> a <em>K<\/em> \u0161tvorec <em>KLMD<\/em>. Vypo\u010d\u00edtajte obsah \u0161tvorca <em>KLMD<\/em>.<\/p>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\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<p><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><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 Z5-Z9 aktualizovan\u00e9 24.1.2019 18:30 Z5 Z6 Z7 Z8 Z9 dom\u00e1ce dom\u00e1ce dom\u00e1ce dom\u00e1ce dom\u00e1ce obvodn\u00e9 obvodn\u00e9 obvodn\u00e9 obvodn\u00e9 obvodn\u00e9 \u00a0 krajsk\u00e9 Let\u00e1k a zadania \u00faloh dom\u00e1ceho kola na stiahnutie vo form\u00e1te PDF: (ulo\u017ei\u0165 ako &#8230;)<\/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-1252","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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