{"id":1752,"date":"2024-09-11T20:59:56","date_gmt":"2024-09-11T19:59:56","guid":{"rendered":"https:\/\/matematika.besaba.com\/?p=1752"},"modified":"2025-04-03T21:13:57","modified_gmt":"2025-04-03T20:13:57","slug":"sutazne-ulohy-kategorii-z5-z9-12","status":"publish","type":"post","link":"https:\/\/matematika.besaba.com\/?p=1752","title":{"rendered":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3rii Z5 &#8211; Z9"},"content":{"rendered":"<p class=\"hh2\"><a name=\"top\"><\/a>74. ro\u010dn\u00edk matematickej olympi\u00e1dy 2024-2025<br \/>\nS\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9<\/p>\n<p><span class=\"povinne\">aktualizovan\u00e9 03.11.2024 17:30<\/span><\/p>\n\n<p><!--more--><\/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\">okresn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z62\">okresn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z72\">okresn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z82\">okresn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z92\">okresn\u00e9<\/a><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"4\" bgcolor=\"#c0c0c0\"><\/td>\n<td align=\"center\"><span class=\"a\"><a href=\"#z93\">krajsk\u00e9<\/a><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"z51\"><\/a><\/p>\n<form><a href=\"https:\/\/skmo.sk\/dokument.php?id=5344\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt51z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=5364\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt51r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z5-I-1<\/strong><br \/>\n\u010capkovci maju\u0301 dvoch synov, Karola a o dva roky stars\u030cieho Petra. Nemcovci maju\u0301 dce\u0301ru Boz\u030cenu. Vs\u030cetky tri deti maju\u0301 narodeniny v ten isty\u0301 den\u030c a obe rodiny ich vtedy oslavuju\u0301 spoloc\u030cne. Pri tohtoroc\u030cnej oslave bola Boz\u030cena trikra\u0301t stars\u030cia ako Karol. O tri roky budu\u0301 ma\u0165 Karol a Peter spolu to\u013eko rokov ako Boz\u030cena.<br \/>\nKo\u013eko rokov mali deti pri tohtoroc\u030cnej oslave?<\/p>\n<div class=\"pri\">(Michaela Petrova\u0301<\/div>\n<p><strong>Z5-I-2<\/strong><br \/>\nNa obra\u0301zku je sivy\u0301 s\u030ctvorec so stranou d\u013a\u017eky 10 cm. \u0160tvorec dop\u013an\u030caju\u0301 s\u030ctyri rovnake\u0301 pravouhle\u0301 trojuholn\u00edky do tvaru hviezdy. Su\u0301c\u030cet obsahov ty\u0301chto s\u030ctyroch trojuholn\u00edkov je s\u030ctvorna\u0301sobkom obsahu s\u030ctvorca.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"74\/74z5a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<p>Urc\u030cte d\u013a\u017eku druhej najdlhs\u030cej strany kaz\u030cde\u0301ho z ty\u0301chto s\u030ctyroch trojuholn\u00edkov.<\/p>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z5-I-3<\/strong><br \/>\nV nasleduju\u0301com vy\u0301raze je pa\u0308\u0165kra\u0301t pouz\u030cite\u0301 znamienko + a jeho hodnota 39 je na\u0301sobkom 3:<\/p>\n<div class=\"r\">9 + 8 + 7 + 6 + 5 + 4.<\/div>\n<p>Zmen\u030cte dve zo znamienok \u201e+\u201c na znamienka \u201e\u2212\u201c tak, aby hodnota nove\u0301ho vy\u0301razu bola opa\u0308\u0165 na\u0301sobkom 3. Na\u0301jdite vs\u030cetky moz\u030cnosti.<\/p>\n<div class=\"pri\">(Eva Semera\u0301dova\u0301)<\/div>\n<p><strong>Z5-I-4<\/strong><br \/>\nPinocchio tvrd\u00ed, z\u030ce c\u030c\u00edslo dn\u030ca v da\u0301tume jeho narodenia moz\u030cno bezo zvys\u030cku deli\u0165 3, 4, 5 a 6. Tri z ty\u0301chto s\u030ctyroch informa\u0301ci\u0131\u0301 su\u0301 pravdiv\u00e9, jedna je nepravdiv\u00e1.<br \/>\nKo\u013eky\u0301 den\u030c v mesiaci mo\u0302z\u030ce ma\u0165 Pinocchio narodeniny? Na\u0301jdite vs\u030cetky moz\u030cnosti.<\/p>\n<div class=\"pri\">(Erika Novotna\u0301)<\/div>\n<p><strong>Z5-I-5<\/strong><br \/>\nV sieti chodn\u00edkov vyznac\u030ceny\u0301ch na obra\u0301zku ma\u0301 kaz\u030cdy\u0301 chodn\u00edk medzi susedny\u0301mi kriz\u030covatkami d\u013a\u017eku 1 km.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"74\/74z5a2.png\" alt=\"\" border=\"0\" \/><\/div>\n<p>Ko\u013eko ciest dlhy\u0301ch nanajvy\u0301s\u030c 3 km vedie po chodn\u00edkoch z miesta \ud835\udc34 do miesta \ud835\udc35?<\/p>\n<div class=\"pri\">(Eva Semera\u0301dova\u0301)<\/div>\n<p><strong>Z5-I-6<\/strong><br \/>\nAnic\u030cka navlieka na ni\u0165 bezprostredne za seba kora\u0301liky troch ro\u0302znych tvarov A, B, C. Postupuje tak, z\u030ce tvary strieda sta\u0301le v rovnakom porad\u00ed a postupne zvys\u030cuje poc\u030cty tvarov v skupina\u0301ch, a to takto:<\/p>\n<div class=\"r\">ABCAABBCCAAABBBCCCAAAABBBBCCCC &#8230;<\/div>\n<p>Kora\u0301lik tvaru A zabera\u0301 5 mm nite, kora\u0301lik tvaru B 4 mm, kora\u0301lik tvaru C 3 mm.<br \/>\nKo\u013eko kora\u0301likov potrebuje Anic\u030cka na vy\u0301robu n\u00e1hrdeln\u00edka dlhe\u0301ho aspon\u030c 50 cm?<\/p>\n<div class=\"pri\">(Lenka Dedkova\u0301)<\/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=\"https:\/\/skmo.sk\/dokument.php?id=5380\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt52z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=5382\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt52r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z5-II-1<\/strong><br \/>\nJanka a Danka jedli poc\u030cas ty\u0301z\u030cdn\u030ca ovocie. Janka jedla len hru\u0161ky\u00a0 alebo jablka\u0301, Danka jedla iba c\u030ceres\u030cne. Kaz\u030cdy\u0301 den\u030c zjedla Janka najviac\u00a0jeden kus ovocia a Danka jedla v ten isty\u0301 den\u030c c\u030ceres\u030cne pod\u013ea\u00a0nasleduju\u0301ceho rozpisu:<\/p>\n<ul>\n<li>Ke\u010f Janka zjedla hrus\u030cku, zjedla Danka dve c\u030ceres\u030cne.<\/li>\n<li>Ke\u010f Janka zjedla jablko, zjedla Danka tri c\u030ceres\u030cne.<\/li>\n<li>Ke\u010f Janka nezjedla z\u030ciadne ovocie, zjedla Danka s\u030ces\u0165 c\u030ceres\u030cn\u0131\u0301.<\/li>\n<\/ul>\n<p>Od pondelka do nedele zjedla Danka dokopy 19 c\u030ceres\u030cn\u0131\u0301.<br \/>\nKo\u013eko ktore\u0301ho ovocia mohla zjes\u0165 za ten isty\u0301 ty\u0301z\u030cden\u030c Janka? Na\u0301jdite obe moz\u030cnosti.<\/p>\n<div class=\"pri\">(E. Novotn\u00e1)<\/div>\n<p><strong>Z5-II-2<\/strong><br \/>\nOkruz\u030cna\u0301 cesta spa\u0301ja tri dediny tak ako na obra\u0301zku. Vo vyznac\u030cenom smere to je z Pa\u0301voviec do Krtkovian 10 km, z Mn\u030caukova do Pa\u0301voviec 15 km a z Krtkovian do Mn\u030caukova 16 km.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"74\/74z5b1.png\" alt=\"\" border=\"0\" \/><\/div>\n<p>Aka\u0301 dlha\u0301 je cela\u0301 okruz\u030cna\u0301 cesta?<\/p>\n<div class=\"pri\">(E. Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z5-II-3<\/strong><br \/>\nZ 2025 rovnaky\u0301ch s\u030ctvorcov je zloz\u030ceny\u0301 u\u0301tvar pod\u013ea pravidla naznac\u030cene\u0301ho na obra\u0301zku. Strana s\u030ctvorca je 1 cm.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"74\/74z5b2.png\" alt=\"\" border=\"0\" \/><\/div>\n<p>Urc\u030cte obvod u\u0301tvaru.<\/p>\n<div class=\"pri\">(K. 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=\"z61\"><\/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=\"https:\/\/skmo.sk\/dokument.php?id=5343\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt61z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=5391\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt61r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z6-I-1<\/strong><br \/>\nP\u00e1n Vafli\u010dka vypr\u00e1\u017ea a pred\u00e1va \u0161i\u0161ky, p\u00e1n \u0160i\u0161ka pe\u010die a pred\u00e1va vafle. Obaja cukr\u00e1ri maj\u00fa ka\u017ed\u00fd t\u00fd\u017ede\u0148 otvoren\u00e9 od pondelka do piatka. Lenka u nich kupuje ka\u017ed\u00fd pondelok 2 vafle a 1 \u0161i\u0161ku, ka\u017ed\u00fd utorok 3 \u0161i\u0161ky a 1 vaf\u013eu, ka\u017ed\u00fa stredu 4 \u0161i\u0161ky, ka\u017ed\u00fd \u0161tvrtok 3 vafle a ka\u017ed\u00fd piatok 2 \u0161i\u0161ky a 2 vafle. P\u00e1n \u0160i\u0161ka si raz v\u0161imol, \u017ee od prv\u00e9ho pondelka tohto mesiaca predal Lenke dokopy 30 vafl\u00ed. Ko\u013eko \u0161i\u0161iek predal Lenke za rovnak\u00e9 obdobie p\u00e1n Vafli\u010dka?<\/p>\n<div class=\"pri\">(Michaela Petrov\u00e1)<\/div>\n<p><strong>Z6-I-2<\/strong><br \/>\nV obd\u013a\u017eniku so stranami d\u013a\u017eok 4 cm a 8 cm le\u017eia dve r\u00f4zne polkru\u017enice, z ktor\u00fdch ka\u017ed\u00e1 m\u00e1 krajn\u00e9 body v jeho dvoch susedn\u00fdch vrcholoch a dot\u00fdka sa proti\u013eahlej strany. Zostrojte \u0161tvorec tak\u00fd, \u017ee jeho dva vrcholy le\u017eia na jednej polkru\u017enici, zvy\u0161n\u00e9 dva na druhej a jeho strany s\u00fa rovnobe\u017en\u00e9 so stranami obd\u013a\u017enika.<\/p>\n<div class=\"pri\">(Karel Pazourek)<\/div>\n<p><strong>Z6-I-3<\/strong><br \/>\nP\u00e4\u0165cifern\u00fd palindro\u0301m je tak\u00e9 p\u00e4\u0165cifern\u00e9 \u010d\u00edslo, ktor\u00e9 m\u00e1 na mieste jednotiek rovnak\u00fa cifru ako na mieste desa\u0165tis\u00edcok a na mieste desiatok rovnak\u00fa cifru ako na mieste tis\u00edcok. N\u00e1jdite najmen\u0161\u00ed p\u00e4\u0165cifern\u00fd palindro\u0301m delite\u013en\u00fd \u010d\u00edslom 36.<\/p>\n<div class=\"pri\">(Iveta Jan\u010digov\u00e1)<\/div>\n<p><strong>Z6-I-4<\/strong><br \/>\n\u0160\u00e1rka s \u013dubo\u0161om spolo\u010dne zasadili 70 tulip\u00e1nov r\u00f4znych farieb. \u0160\u00e1rka nesadila \u017elt\u00e9 tulip\u00e1ny a p\u00e4\u0165 dev\u00e4t\u00edn tulip\u00e1nov, ktor\u00e9 zasadila, boli \u010derven\u00e9. \u013dubo\u0161 nesadil \u010derven\u00e9 tulip\u00e1ny a dve sedemn\u00e1stiny tulip\u00e1nov, ktor\u00e9 zasadil, boli \u017elt\u00e9. Ko\u013eko zasaden\u00fdch tulip\u00e1nov malo in\u00fa farbu ako \u010derven\u00fa alebo \u017elt\u00fa?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z6-I-5<\/strong><br \/>\nTri kamar\u00e1tky sa po rokoch zi\u0161li a rozpr\u00e1vali sa o tom, kde ktor\u00e1 z nich b\u00fdva:<br \/>\nPrv\u00e1: \u201eJa b\u00fdvam v Hru\u0161t\u00edne.\u201c.<br \/>\nDruh\u00e1: \u201eJa neb\u00fdvam v O\u010dovej.\u201c.<br \/>\nTretia druhej: \u201eTy neb\u00fdva\u0161 v Jasenove.\u201c.<br \/>\nKamar\u00e1tky naozaj b\u00fdvaj\u00fa v spom\u00ednan\u00fdch dedin\u00e1ch, ka\u017ed\u00e1 v inej. Jedna z kamar\u00e1tok nepovedala pravdu a nebola to t\u00e1 z O\u010dovej.<br \/>\nRozhodnite, kde ktor\u00e1 z kamar\u00e1tok b\u00fdva.<\/p>\n<div class=\"pri\">(Michaela Petrov\u00e1)<\/div>\n<p><strong>Z6-I-6<\/strong><br \/>\nV \u0161tvorcovej sieti s\u00fa tri kruhy a tri trojuholn\u00edky, ka\u017ed\u00fd v inom pol\u00ed\u010dku. Ka\u017ed\u00fd \u00fatvar m\u00e1 aspo\u0148 jedn\u00e9ho suseda, pri\u010dom susedia s\u00fa pol\u00ed\u010dka so spolo\u010dnou stranou. Obsaden\u00e9 pol\u00ed\u010dka tvoria s\u00favisl\u00fa oblas\u0165, t. j. od ka\u017ed\u00e9ho ku ka\u017ed\u00e9mu sa d\u00e1 dosta\u0165 cez susedov. Ka\u017ed\u00fa noc sa ka\u017ed\u00fd \u00fatvar m\u00f4\u017ee zmeni\u0165 pod\u013ea toho, ako cez de\u0148 vyzerali jeho susedia:<\/p>\n<ul style=\"list-style-type: disc;\">\n<li>Ak je \u00fatvar kruh a medzi jeho susedmi bolo viac trojuholn\u00edkov ako kruhov, tak sa zmen\u00ed na trojuholn\u00edk.<\/li>\n<li>Ak je \u00fatvar trojuholn\u00edk a medzi jeho susedmi bolo viac kruhov ako trojuholn\u00edkov, tak sa zmen\u00ed na kruh.<\/li>\n<li>V ostatn\u00fdch pr\u00edpadoch sa \u00fa tvar nezmen\u00ed.<\/li>\n<\/ul>\n<p>Pr\u00edklad \u0161tvorcovej siete a premeny jej \u00fatvarov po jednej noci je na obr\u00e1zku:<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"74\/74z6a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<p>N\u00e1jdite aspo\u0148 jedno rozmiestnenie troch kruhov a troch trojuholn\u00edkov do siete tak\u00e9,<br \/>\na) aby sa v noci nezmenili;<br \/>\nb) aby sa ka\u017ed\u00fd \u00fatvar ka\u017ed\u00fa noc zmenil;<br \/>\nc) aby po nieko\u013ek\u00fdch nociach boli v\u0161etky \u00fatvary rovnak\u00e9.<\/p>\n<div class=\"pri\">(Iveta Jan\u010digov\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=\"https:\/\/skmo.sk\/dokument.php?id=5406\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt62z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=5408\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt62r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z6-II-1<\/strong><br \/>\nPrirodzene\u0301 c\u030c\u0131\u0301slo nazveme pekn\u00e9, ak je va\u0308c\u030cs\u030cie ako 7000 a su\u0301c\u030cin jeho cifier je 252.<br \/>\nNa\u0301jdite dve najmens\u030cie pekne\u0301 c\u030c\u0131\u0301sla.<\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z6-II-2<\/strong><br \/>\nCestic\u030cky medzi u\u0301krytmi cvrc\u030ckov Adama, Borisa, Cyrila, Daniela a Erika tvoria sie\u0165 ako na obra\u0301zku a ich u\u0301 kryty su\u0301 oznac\u030cene\u0301 prvy\u0301mi p\u0131\u0301smenami ich mien. Pritom plat\u0131\u0301:<\/p>\n<ul style=\"list-style-type: disc;\">\n<li>Cestic\u030cky medzi u\u0301 krytmi Adama, Borisa a Erika tvoria rovnostranny\u0301 trojuholn\u0131\u0301k.<\/li>\n<li>Cestic\u030cky medzi u\u0301krytmi Borisa, Cyrila, Daniela a Erika tvoria obd\u013az\u030cnik s obsahom 360 dm<sup>2<\/sup>.<\/li>\n<li>Precha\u0301dzka po cestic\u030cka\u0301ch od Adama k Borisovi, Cyrilovi, Danielovi, Erikovi a Adamovi je o 24 dm dlhs\u030cia ako precha\u0301dzka od Adama k Borisovi, Erikovi a Adamovi.<\/li>\n<\/ul>\n<p>Aka\u0301 dlha\u0301 je cestic\u030cka medzi u\u0301krytmi Adama a Erika?<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"74\/74z6b1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">()<\/div>\n<p><strong>Z6-II-3<\/strong><br \/>\nJakub prec\u030c\u0131\u0301tal knihu poc\u030cas troch dn\u0131\u0301. V utorok prec\u030c\u0131\u0301tal tretinu vs\u030cetky\u0301ch stra\u0301n, v stredu prec\u030c\u0131\u0301tal tri sedminy zvys\u030cny\u0301ch stra\u0301n a posledny\u0301ch 32 stra\u0301n prec\u030c\u0131\u0301tal vo s\u030ctvrtok.<br \/>\nKo\u013eko stra\u0301n mala kniha?<\/p>\n<div class=\"pri\">()<\/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\n<form><a href=\"https:\/\/skmo.sk\/dokument.php?id=5342\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt71z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=5392\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt71r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z7-I-1<\/strong><br \/>\nAlenka a Zuzka jedli slivky. Prv\u00fd de\u0148 zjedla Alenka tri \u0161tvrtiny toho, \u010do v ten ist\u00fd de\u0148 zjedla Zuzka. Druh\u00fd de\u0148 zjedla Zuzka tri polovice toho, \u010do v ten ist\u00fd de\u0148 zjedla Alenka. Dokopy za oba dni zjedli 31 sliviek a ka\u017ed\u00e9 diev\u010da ka\u017ed\u00fd de\u0148 zjedlo celo\u010d\u00edseln\u00fd po\u010det sliviek. Ko\u013eko sliviek zjedla za oba dni Alenka?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z7-I-2<\/strong><br \/>\nMikul\u00e1\u0161 postavil pyram\u00eddu zo 6 rovnak\u00fdch kociek s hranami d\u013a\u017eky 7 cm. Spodn\u00e9 poschodie tvorili 3 kocky, prostredn\u00e9 poschodie 2 kocky a horn\u00e9 poschodie 1 kocka. Susedn\u00e9 kocky v ka\u017edom poschod\u00ed mali spolo\u010dn\u00fa stenu, poschodia navz\u00e1jom nepre\u010dnievali. V\u00ed\u0165azoslav posunul kocky tak, \u017ee ka\u017ed\u00e1 kocka v horn\u00fdch dvoch poschodiach st\u00e1la na dvoch spodn\u00fdch kock\u00e1ch a medzi susedn\u00fdmi kockami v spodn\u00fdch dvoch poschodiach boli medzery \u0161irok\u00e9 tretinu hrany kocky. A\u017e na tieto medzery poschodia navz\u00e1jom nepre\u010dnievali. O ko\u013eko cm<sup>2<\/sup> sa l\u00ed\u0161ia povrchy p\u00f4vodnej a upravenej pyram\u00eddy?<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"74\/74z7a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Vladim\u00edr Dedek)<\/div>\n<p><strong>Z7-I-3<\/strong><br \/>\nPankr\u00e1c, Serv\u00e1c a Bonif\u00e1c sa ubytovali v hoteli. \u010c\u00edsla izieb boli trojcifern\u00e9 a cifra na mieste stoviek ur\u010dovala poschodie, na ktorom sa izba nach\u00e1dzala. Na ra\u0148ajk\u00e1ch si pod\u013ea pr\u00edveskov na k\u013e\u00fa\u010doch od izieb v\u0161imli, \u017ee:<\/p>\n<ul style=\"list-style-type: disc;\">\n<li>v \u010d\u00edslach ich izieb s\u00fa pou\u017eit\u00e9 v\u0161etky cifry od 1 do 9,<\/li>\n<li>Pankr\u00e1covo \u010d\u00edslo je delite\u013en\u00e9 9, Serv\u00e1covo \u010d\u00edslo je delite\u013en\u00e9 8, Bonif\u00e1covo \u010d\u00edslo je delite\u013en\u00e9 7,<\/li>\n<li>Bonif\u00e1covo \u010d\u00edslo je 4 kr\u00e1 t v\u00e4\u010d\u0161ie ako Pankr\u00e1covo \u010d\u00edslo,<\/li>\n<li>Serv\u00e1c b\u00fdva na poschod\u00ed medzi Pankr\u00e1com a Bonif\u00e1com.<\/li>\n<\/ul>\n<p>Ur\u010dte \u010d\u00edsla izieb Pankr\u00e1ca, Serv\u00e1ca a Bonif\u00e1ca.<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1, Erika Novotn\u00e1)<\/div>\n<p><strong>Z7-I-4<\/strong><br \/>\nV jednej z piatich n\u00e1dob o\u010d\u00edslovan\u00fdch 1, 2, 3, 4, 5 je minca. Sprievodn\u00e9 n\u00e1pisy oznamuj\u00fa:<\/p>\n<ul style=\"list-style-type: disc;\">\n<li>&#8222;Minca je v n\u00e1dobe s nep\u00e1rnym \u010d\u00edslom.&#8220;,<\/li>\n<li>&#8222;Minca je v n\u00e1dobe s \u010d\u00edslom v\u00e4\u010d\u0161\u00edm ako 3.&#8220;,<\/li>\n<li>&#8222;Minca je v n\u00e1dobe s \u010d\u00edslom men\u0161\u00edm ako 4.&#8220;.<\/li>\n<\/ul>\n<p>\u010cestn\u00fd str\u00e1\u017eca s bezchybn\u00fdm \u00fasudkom dod\u00e1va: &#8222;Jeden z n\u00e1pisov nie je pravdiv\u00fd, zvy\u0161n\u00e9 dva pravdiv\u00e9 s\u00fa. Hoci viem, ktor\u00fd n\u00e1pis pravdiv\u00fd nie je, neviem ur\u010di\u0165, v ktorej n\u00e1dobe je minca.&#8220;. Rozhodnite, ktor\u00fd z n\u00e1pisov nie je pravdiv\u00fd.<\/p>\n<div class=\"pri\">(Karel Pazourek)<\/div>\n<p><strong>Z7-I-5<\/strong><br \/>\nNech \ud835\udc34\ud835\udc35\ud835\udc36 je trojuholn\u00edk tak\u00fd, \u017ee |\ud835\udc34\ud835\udc35| = 6 cm, |\ud835\udc35\ud835\udc36| = 8 cm a |\ud835\udc34\ud835\udc36| = 12 cm.<br \/>\nZostrojte polkru\u017enicu tak\u00fa, \u017ee sa dot\u00fdka str\u00e1n \ud835\udc34\ud835\udc35 a \ud835\udc35\ud835\udc36 a jej krajn\u00e9 body le\u017eia na strane \ud835\udc34\ud835\udc36.<\/p>\n<div class=\"pri\">(Karel Pazourek)<\/div>\n<p><strong>Z7-I-6<\/strong><br \/>\nKatka a \u0160tevo pe\u010d\u00fa ka\u017ed\u00fd na svojej panvici bez prest\u00e1vok jednu palacinku za druhou a hotov\u00e9 palacinky d\u00e1vaj\u00fa na spolo\u010dn\u00fd tanier. Obaja za\u010dali piec\u0165 s\u00fa\u010dasne. Katke trv\u00e1 ka\u017ed\u00e1 palacinka 3 min\u00faty, \u0160tevovi trv\u00e1 ka\u017ed\u00e1 palacinka 4 min\u00faty. Ka\u017ed\u00fdch 5 min\u00fat od za\u010diatku pe\u010denia sa objav\u00ed ma\u0161krtn\u00fd koc\u00far Lucifer. Ak sa Katka i \u0160tevo venuj\u00fa pe\u010deniu, tak im Lucifer jednu hotov\u00fa palacinku ukradne. Ak niekto z nich pr\u00e1ve d\u00e1va palacinku z panvice na tanier, tak sa schov\u00e1 a palacinku neukradne.<br \/>\nKo\u013eko palaciniek musia Katka so \u0160tevom upiec\u0165, aby im ich ostalo 150? Ako dlho im to bude trva\u0165?<\/p>\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=\"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=\"https:\/\/skmo.sk\/dokument.php?id=5409\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt72z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=5411\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt72r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z7-II-1<\/strong><br \/>\nMamic\u030cka si nachystala pern\u0131\u0301c\u030cky na zdobenie. Kaz\u030cdy\u0301 pern\u0131\u0301c\u030cek zdob\u0131\u0301 rovnako dlho. Keby pri zdoben\u0131\u0301 kaz\u030cde\u0301ho pern\u0131\u0301c\u030cka bola o jednu minu\u0301tu ry\u0301chlejs\u030cia, mohla by skonc\u030ci\u0165 o 48 minu\u0301t sko\u0302r alebo by v takto us\u030cetrenom c\u030case mohla ozdobi\u0165 (v novom zry\u0301chlenom tempe) presne 12 \u010fals\u030c\u0131\u0301ch pern\u0131\u0301c\u030ckov.<br \/>\nKo\u013eko pern\u0131\u0301c\u030ckov si mamic\u030cka nachystala a ako dlho jej bude trva\u0165 ich zdobenie (v po\u0302vodnom nezry\u0301chlenom tempe)?<\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z7-II-2<\/strong><br \/>\nV u\u0301tulku je 60 zvierat, a to vy\u0301hradne mac\u030cky a psy. Tretina mac\u030ciek a tri osminy psov nie su\u0301 ani rok stare\u0301, 39 zvierat ma\u0301 rok alebo viac.<br \/>\nKo\u013eko je v u\u0301tulku mac\u030ciek a ko\u013eko psov?<\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z7-II-3<\/strong><br \/>\nKosos\u030ctvorec \ud835\udc34\ud835\udc35\ud835\udc36\ud835\udc37 je zloz\u030ceny\u0301 z rovnobez\u030cn\u0131\u0301kov s navza\u0301jom rovnaky\u0301mi obsahmi. Vyznac\u030cena\u0301 spoloc\u030cna\u0301 strana \ud835\udc47\ud835\udc48 dvoch rovnobez\u030cn\u0131\u0301kov ma\u0301 d\u013az\u030cku 2 cm.<br \/>\nUrc\u030cte obvod kosos\u030ctvorca \ud835\udc34\ud835\udc35\ud835\udc36\ud835\udc37.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"74\/74z7b1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">()<\/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<form><a href=\"https:\/\/skmo.sk\/dokument.php?id=5341\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt81z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=5393\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt81r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z8-I-1<\/strong><br \/>\nIvan, Jaro, Karol a \u013dubo\u0161 maj\u00fa dokopy 90 zn\u00e1mok. Keby mal Ivan o dve zn\u00e1mky menej, Jaro o dve viac, Karol dvojn\u00e1sobok a \u013dubo\u0161 polovicu toho, \u010do teraz, mali by v\u0161etci rovnako. Ko\u013eko zn\u00e1mok m\u00e1 ka\u017ed\u00fd z chlapcov?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z8-I-2<\/strong><br \/>\nN\u00e1jdite aspo\u0148 jedno rozdelenie rovnoramenn\u00e9ho trojuholn\u00edka so z\u00e1klad\u0148ou d\u013a\u017eky 12 cm a v\u00fd\u0161kou na z\u00e1klad\u0148u d\u013a\u017eky 18 cm na tri lichobe\u017en\u00edky s rovnak\u00fdm obsahom.<\/p>\n<div class=\"pri\">(Lenka Dedkov\u00e1)<\/div>\n<p><strong>Z8-I-3<\/strong><br \/>\n\u010c\u00edsla <em>a, b, c, d<\/em> s\u00fa tak\u00e9, \u017ee plat\u00ed:<\/p>\n<ul style=\"list-style-type: disc;\">\n<li>\u010c\u00edslo <em>a<\/em> d\u00e1va po delen\u00ed 3 zvy\u0161ok 1.<\/li>\n<li>\u010c\u00edslo <em>b<\/em> d\u00e1va po delen\u00ed 6 zvy\u0161ok 2.<\/li>\n<li>Plat\u00ed <em>a \u2212 b = d \u2212 c<\/em>.<\/li>\n<li>\u010c\u00edslo <em>d<\/em> je delite\u013en\u00e9 3.<\/li>\n<\/ul>\n<p>Ak\u00fd zvy\u0161ok po delen\u00ed 9 m\u00f4\u017ee d\u00e1va\u0165 \u010d\u00edslo <em>c<\/em>? N\u00e1jdite v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z8-I-4<\/strong><br \/>\nNech \ud835\udc34\ud835\udc35\ud835\udc36\ud835\udc37\ud835\udc38 je pravideln\u00fd p\u00e4\u0165uholn\u00edk. Rovnobe\u017eka s priamkou \ud835\udc34\ud835\udc35 prech\u00e1dzaj\u00faca bodom \ud835\udc36 pret\u00edna priamku \ud835\udc35\ud835\udc37 v bode \ud835\udc39. Kolmica na priamku \ud835\udc36\ud835\udc39 prech\u00e1dzaj\u00faca bodom \ud835\udc36 pret\u00edna priamku \ud835\udc35\ud835\udc37 v bode \ud835\udc3a.<br \/>\nUr\u010dte ve\u013ekos\u0165 uhla \ud835\udc34\ud835\udc3a\ud835\udc39.<\/p>\n<div class=\"pri\">(Patrik Bak)<\/div>\n<p><strong>Z8-I-5<\/strong><br \/>\nUr\u010dte v\u0161etky mo\u017en\u00e9 dvojice \u010d\u00edsel (<em>a, b<\/em>) tak\u00e9, \u017ee podiel najmen\u0161ieho spolo\u010dn\u00e9ho n\u00e1sobku a najv\u00e4\u010d\u0161ieho spolo\u010dn\u00e9ho delite\u013ea \u010d\u00edsel <em>a<\/em> a <em>b<\/em> je 75 a s\u00fa\u010det \u010d\u00edsel <em>a<\/em> a <em>b<\/em> je v\u00e4\u010d\u0161\u00ed ako 100 a men\u0161\u00ed ako 200.<\/p>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z8-I-6<\/strong><br \/>\nRyb\u00e1r \u0160\u0165uka chytil nieko\u013eko r\u00fdb. Ke\u010f predal tri naj\u0165a\u017e\u0161ie ryby majite\u013eovi miestnej re\u0161taur\u00e1cie, zn\u00ed\u017eil celkov\u00fa hmotnos\u0165 svojho \u00falovku o 35 %. Ke\u010f dal tri naj\u013eah\u0161ie ryby svojmu psovi, zn\u00ed\u017eil hmotnos\u0165 zost\u00e1vaj\u00facich uloven\u00fdch r\u00fdb o p\u00e4\u0165 trin\u00e1stin.<br \/>\nKo\u013eko r\u00fdb chytil p\u00e1n \u0160\u0165uka?<\/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=\"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=\"https:\/\/skmo.sk\/dokument.php?id=5412\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt82z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=5414\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt82r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z8-II-1<\/strong><br \/>\nKu\u0301zeln\u0131\u0301kov povraz je dlhs\u030c\u0131\u0301 ako 10 m. Nech by rozdelil povraz na dve c\u030casti v hociktorom z troch pomerov 3 \u2236 5, 7 \u2236 11, 13 \u2236 17, boli by d\u013az\u030cky oboch c\u030cast\u0131\u0301 vyjadrene\u0301 v centimetroch zakaz\u030cdy\u0301m prirodzeny\u0301m c\u030c\u0131\u0301slom.<br \/>\nAka\u0301 je najmens\u030cia moz\u030cna\u0301 d\u013az\u030cka ku\u0301zeln\u0131\u0301kovho povrazu?<\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z8-II-2<\/strong><br \/>\nMonika si vybrala dve c\u030c\u0131\u0301sla, aby presku\u0301s\u030cala schopnosti robota Popletu. Najprv mu dala sc\u030c\u0131\u0301ta\u0165 obe c\u030c\u0131\u0301sla. Popletov vy\u0301sledok bol o 4,1 mens\u030c\u0131\u0301 ako vy\u0301sledok pri spra\u0301vnom sc\u030c\u0131\u0301tan\u0131\u0301. Potom mu dala sc\u030c\u0131\u0301ta\u0165 trojna\u0301sobok prve\u0301ho c\u030c\u0131\u0301sla s druhy\u0301m c\u030c\u0131\u0301slom. Teraz bol Popletov vy\u0301sledok o 8,4 mens\u030c\u0131\u0301 ako vy\u0301sledok pri spra\u0301vnom sc\u030c\u0131\u0301tan\u0131\u0301. \u010coskoro zistila, z\u030ce Popleta nepoc\u030c\u0131\u0301ta su\u0301c\u030cet dvoch zadany\u0301ch c\u030c\u0131\u0301sel, ale spra\u0301vne vypoc\u030c\u0131\u0301ta ich aritmeticky\u0301 priemer.<br \/>\nKtore\u0301 c\u030c\u0131\u0301sla si Monika vybrala?<\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z8-II-3<\/strong><br \/>\nV trojuholn\u0131\u0301ku \ud835\udc34\ud835\udc35\ud835\udc36 lez\u030c\u0131\u0301 bod \ud835\udc37 na strane \ud835\udc35\ud835\udc36 a bod \ud835\udc38 na strane \ud835\udc34\ud835\udc36 tak, z\u030ce plat\u0131\u0301<\/p>\n<div class=\"r\">|AB| = |BE| = |EC| = |CD|,<br \/>|BD| = |DE|.<\/div>\n<p>Urc\u030cte ve\u013ekosti uhlov \ud835\udc34\ud835\udc36\ud835\udc35 a \ud835\udc35\ud835\udc34\ud835\udc37.<\/p>\n<div class=\"pri\">()<\/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<form><a href=\"https:\/\/skmo.sk\/dokument.php?id=5340\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt91z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=5365\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt91r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-I-1<\/strong><br \/>\nNa\u0301jdite vs\u030cetky dvojice cely\u0301ch \u010d\u00edsel (\ud835\udc65, \ud835\udc66) tak\u00e9, z\u030ce \ud835\udc65 + \ud835\udc66 je prvo\u010d\u00edslo a 3\ud835\udc65 + 5\ud835\udc66 = 16.<\/p>\n<div class=\"pri\">(Patrik Bak)<\/div>\n<p><strong>Z9-I-2<\/strong><br \/>\nPravidelny\u0301 s\u030ctvorboky\u0301 hranol ma\u0301 objem 864 cm<sup>3<\/sup> a obsah jeho pla\u0301s\u030c\u0165a je dvojna\u0301sobkom obsahu jeho podstavy. Urc\u030cte ve\u013ekos\u0165 jeho telesovej uhlopriec\u030cky.<\/p>\n<div class=\"pri\">(Vladim\u0131\u0301r Dedek)<\/div>\n<p><strong>Z9-I-3<\/strong><br \/>\nNa\u0301jdite najva\u0308c\u030cs\u030cie moz\u030cne\u0301 \ud835\udc5b, pre ktore\u0301 je moz\u030cne\u0301 mnoz\u030cinu {1, 2, \u2026 , \ud835\udc5b} rozdeli\u0165 do 5 nepra\u0301zdnych podmno\u017e\u00edn tak, aby \u010d\u00edsla v kaz\u030cdej podmnoz\u030cine boli po dvoch nes\u00fadelite\u013en\u00e9.<\/p>\n<div class=\"pri\">(Toma\u0301s\u030c Ba\u0301rta)<\/div>\n<p><strong>Z9-I-4<\/strong><br \/>\nRozhodnite, c\u030ci je moz\u030cne\u0301 k \u010d\u00edslu s ciferny\u0301m su\u0301c\u030ctom 2024 pripo\u010d\u00edta\u0165 jednociferne\u0301 \u010d\u00edslo tak, aby vy\u0301sledne\u0301 \u010d\u00edslo malo ciferny\u0301 su\u0301c\u030cet 74.<\/p>\n<div class=\"pri\">(Toma\u0301s\u030c Ba\u0301rta)<\/div>\n<p><strong>Z9-I-5<\/strong><br \/>\nNech \ud835\udc34\ud835\udc35\ud835\udc36 je trojuholn\u00edk tak\u00fd, z\u030ce strana \ud835\udc34\ud835\udc35 je dvakra\u0301t dlhs\u030cia ako strana \ud835\udc34\ud835\udc36 a os uhla \ud835\udc35\ud835\udc34\ud835\udc36 pret\u00edna stranu \ud835\udc35\ud835\udc36 v bode \ud835\udc37. Nech rovnobez\u030cka so stranou \ud835\udc34\ud835\udc35 precha\u0301dzaju\u0301ca bodom \ud835\udc37 pret\u00edna stranu \ud835\udc34\ud835\udc36 v bode \ud835\udc38 a rovnobez\u030cka s u\u0301sec\u030ckou \ud835\udc34\ud835\udc37 precha\u0301dzaju\u0301ca bodom \ud835\udc38 pret\u00edna stranu \ud835\udc35\ud835\udc36 v bode \ud835\udc39.<br \/>\nUrc\u030cte pomer d\u013a\u017eok u\u0301sec\u030ciek \ud835\udc34\ud835\udc37 a \ud835\udc38\ud835\udc39.<\/p>\n<div class=\"pri\">(Ma\u0301ria Doma\u0301nyova\u0301)<\/div>\n<p><strong>Z9-I-6<\/strong><br \/>\nPlavci Pstruh a Kapor si chceli zmera\u0165 svoje sily. Z proti\u013eahly\u0301ch stra\u0301n baze\u0301na skoc\u030cili su\u0301c\u030casne do susedny\u0301ch dra\u0301h a pla\u0301vali proti sebe, kaz\u030cdy\u0301 svojou kons\u030ctantnou ry\u0301chlos\u0165ou. Prvy\u0301kra\u0301t sa plavci minuli vo vzdialenosti 8 metrov od Pstruhovej s\u030ctartovacej strany, na konci dra\u0301hy sa ry\u0301chlo otoc\u030cili a pla\u0301vali naspa\u0308\u0165. Druhy\u0301kra\u0301t sa plavci minuli vo vzdialenosti 5 metrov od Kaprovej s\u030ctartovacej strany, dopla\u0301vali na koniec dra\u0301hy, a ty\u0301m sa preteky skonc\u030cili.<br \/>\nUrc\u030cte, kto vyhral a aka\u0301 bola d\u013a\u017eka baze\u0301na.<\/p>\n<div class=\"pri\">(Libus\u030ce Hozova\u0301)<\/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=\"https:\/\/skmo.sk\/dokument.php?id=5383\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt92z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=5385\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt92r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-II-1<\/strong><br \/>\nNa\u0301jdite vs\u030cetky dvojciferne\u0301 prirodzene\u0301 c\u030c\u0131\u0301sla, ktore\u0301 maju\u0301 nasleduju\u0301cu vlastnos\u0165: Ke\u010f pred c\u030c\u0131\u0301slo prip\u0131\u0301s\u030ceme su\u0301c\u030cin jeho prvej cifry a jeho prvej cifry zva\u0308c\u030cs\u030cenej o 1, dostaneme druhu\u0301 mocninu<br \/>\npo\u0302vodne\u0301ho c\u030c\u0131\u0301sla.<\/p>\n<div class=\"pri\">(K. Pazourek)<\/div>\n<p><strong>Z9-II-2<\/strong><br \/>\nNa\u0301jdite vs\u030cetky trojice navza\u0301jom ro\u0302znych c\u030c\u0131\u0301sel taky\u0301ch, z\u030ce:<\/p>\n<ul>\n<li>jedno z nich je aritmeticky\u0301m priemerom zvys\u030cny\u0301ch dvoch,<\/li>\n<li>su\u0301c\u030cet aritmeticke\u0301ho priemeru najva\u0308c\u030cs\u030cieho a stredne\u0301ho c\u030c\u0131\u0301sla a aritmeticke\u0301ho priemeru stredne\u0301ho a najmens\u030cieho c\u030c\u0131\u0301sla je 628 a rozdiel ty\u0301chto dvoch aritmeticky\u0301ch priemerov je 83.<\/li>\n<\/ul>\n<p>Na porad\u0131\u0301 c\u030c\u0131\u0301sel v trojici neza\u0301lez\u030c\u0131\u0301.<\/p>\n<div class=\"pri\">(K. Pazourek)<\/div>\n<p><strong>Z9-II-3<\/strong><br \/>\nVc\u030cera vydojili na farme Doj dvakra\u0301t viac mlieka ako na farme Hoj a na farme Loj dvakra\u0301t viac mlieka ako na farme Doj. Kaz\u030cda\u0301 farma poslala c\u030cas\u0165 vydojene\u0301ho mlieka na vy\u0301robu masla. Farma Doj poslala na vy\u0301robu masla 7\/8 svojho mlieka a farma Hoj 3\/4 svojho mlieka. Z mlieka vydojene\u0301ho na vs\u030cetky\u0301ch troch farma\u0301ch is\u030clo na vy\u0301robu masla dokopy 90 %.<br \/>\nAku\u0301 c\u030cas\u0165 svojho mlieka poslala na vy\u0301robu masla farma Loj?<\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z9-II-4<\/strong><br \/>\nObd\u013az\u030cnik <em>ABCD<\/em> ma\u0301 obsah 82 cm<sup>2<\/sup>. Bod <em>E<\/em> je stredom strany <em>CD<\/em> a bod <em>P<\/em> je priesec\u030cn\u0131\u0301kom u\u0301sec\u030ciek <em>AC<\/em> a <em>BE<\/em>.<br \/>\nUrc\u030cte obsah trojuholn\u0131\u0301ka <em>ABP<\/em>.<\/p>\n<div class=\"pri\">(E. Novotn\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><input name=\"butt93z\" type=\"button\" value=\"Zadanie\" \/><input name=\"butt93r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z9-III-1<\/strong><\/p>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z9-III-2<\/strong><\/p>\n<div class=\"pri\">(Marie Krej\u010dov\u00e1)<\/div>\n<p><strong>Z9-III-3<\/strong><br \/>\n(Karel Pazourek)<\/p>\n<p><strong>Z9-III-4<\/strong><\/p>\n<div class=\"pri\">(Eva Semer\u00e1dov\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><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; 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