{"id":1444,"date":"2020-11-17T07:22:56","date_gmt":"2020-11-17T07:22:56","guid":{"rendered":"http:\/\/matematika.besaba.com\/?p=1444"},"modified":"2021-03-31T19:34:19","modified_gmt":"2021-03-31T19:34:19","slug":"sutazne-ulohy-kategorii-z5-z9-8","status":"publish","type":"post","link":"https:\/\/matematika.besaba.com\/?p=1444","title":{"rendered":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9"},"content":{"rendered":"<p class=\"hh2\"><a name=\"top\"><\/a>70. ro\u010dn\u00edk matematickej olympi\u00e1dy 2020-2021<br \/>\nS\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9<\/p>\n<p><span class=\"povinne\">aktualizovan\u00e9 31.3.2021 19:30<\/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=3450\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt51z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=3478\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt51r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z5-I-1<\/strong><br \/>\nP\u00e1n Krbec s koc\u00farom Koke\u0161om pred\u00e1vali na hrade Kul\u00edkov vstupenky. V sobotu predali 210 detsk\u00fdch vstupeniek po 25 gro\u0161ov a tie\u017e nejak\u00e9 vstupenky pre dospel\u00fdch po 50 gro\u0161ov. Celkom za ten de\u0148 ut\u0155\u017eili 5 950 gro\u0161ov. Ko\u013eko predali vstupeniek pre dospel\u00fdch?<\/p>\n<div class=\"pri\">(Marie Krej\u010dov\u00e1)<\/div>\n<p><strong>Z5-I-2<\/strong><br \/>\nDeti na t\u00e1bore h\u00e1dzali hracou kockou a pod\u013ea v\u00fdsledkov plnili nasleduj\u00face \u00falohy:<\/p>\n<table border=\"1\" cellpadding=\"2\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"width: 20;\">1<\/td>\n<td style=\"width: 100;\">cho\u010fte 1 km na z\u00e1pad<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>cho\u010fte 1 km na v\u00fdchod<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>cho\u010fte 1 km na sever<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>cho\u010fte 1 km na juh<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>stojte na mieste<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>cho\u010fte 3 km na sever<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol>\n<li>Ak\u00fa trasu mohlo Marekovo dru\u017estvo prejs\u0165? Nazna\u010dte aspo\u0148 \u0161tyri mo\u017enosti.<\/li>\n<li>Ak\u00fd mohol by\u0165 celkov\u00fd s\u00fa\u010det v\u0161etk\u00fdch \u010d\u00edsel, ktor\u00e9 tomuto dru\u017estvu padli? Ur\u010dte v\u0161etky mo\u017enosti.<\/li>\n<\/ol>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z5-I-3<\/strong><br \/>\nP\u00e1n re\u017eis\u00e9r Al\u00edk potreboval do telev\u00edznej rozpr\u00e1vky \u0161tyri psy. Dostal ponuku z Gr\u00e9cka, Belgicka, \u00cdrska a z Dolnej Lehoty. Vybral ov\u010diaka, dalmat\u00edna, vlkodava a jazve\u010d\u00edka, ka\u017ed\u00e9ho z inej krajiny, s r\u00f4znym menom a r\u00f4znym vekom.<\/p>\n<ul style=\"list-style-type: circle;\">\n<li>Najstar\u0161\u00ed zo psov bol jazve\u010d\u00edk, mal 5 rokov.<\/li>\n<li>Bucki bol z nich druh\u00fd najmlad\u0161\u00ed.<\/li>\n<li>Vlkodav poch\u00e1dzal z \u00cdrska.<\/li>\n<li>Pes z Dolnej Lehoty sa volal Dun\u010do.<\/li>\n<li>Oddi osl\u00e1vil v\u010dera svoje \u0161tvrt\u00e9 narodeniny.<\/li>\n<li>Ov\u010diak poch\u00e1dzal z Belgicka.<\/li>\n<li>Rubby nebol dalmat\u00edn.<\/li>\n<li>Vlkodav mal tri roky.<\/li>\n<li>Najmlad\u0161\u00ed z vybran\u00fdch psov bol Rubby, mal dva roky.<\/li>\n<\/ul>\n<p>Zistite, ako sa ka\u017ed\u00fd zo \u0161tyroch psov volal, odkia\u013e poch\u00e1dzal, akej bol rasy a ko\u013eko mal rokov.<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z5-I-4<\/strong><br \/>\nMami\u010dka uvarila dom\u00e1cu r\u00edbez\u013eov\u00fa \u0161\u0165avu a nalievala ju do flia\u0161. F\u013ea\u0161e mala dvojak\u00e9: mal\u00e9 s objemom 500 ml a ve\u013ek\u00e9 s objemom 750 ml. Nakoniec jej zv\u00fd\u0161ilo 12 mal\u00fdch flia\u0161 pr\u00e1zdnych, ostatn\u00e9 f\u013ea\u0161e boli \u00faplne naplnen\u00e9. Potom mami\u010dka zistila, \u017ee mohla \u0161\u0165avu nalieva\u0165 tak, aby jej zv\u00fd\u0161ili pr\u00e1zdne iba ve\u013ek\u00e9 f\u013ea\u0161e a v\u0161etky ostatn\u00e9 boli \u00faplne naplnen\u00e9. Ko\u013eko pr\u00e1zdnych flia\u0161 by jej v takom pr\u00edpade zv\u00fd\u0161ilo?<\/p>\n<div class=\"pri\">(Michaela Petrov\u00e1)<\/div>\n<p><strong>Z5-I-5<\/strong><br \/>\nV \u0161tvor\u010dekovej sieti so \u0161tvor\u010dekmi s rozmermi 1 cm\u00d71 cm s\u00fa vyzna\u010den\u00e9 tri mre\u017eov\u00e9 body <em>K<\/em>, <em>O<\/em> a <em>Z<\/em>. Ur\u010dte mre\u017eov\u00fd bod <em>A<\/em> tak, aby obsah \u0161tvoruholn\u00edka <em>KOZA<\/em> bol 4 cm<sup>2<\/sup>.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"70\/70z5a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z5-I-6<\/strong><br \/>\nMysl\u00edm si p\u00e4\u0165cifern\u00e9 \u010d\u00edslo tvoren\u00e9 p\u00e1rnymi ciframi. Ke\u010f prehod\u00edm cifru na tre\u0165om mieste s akouko\u013evek inou, \u010d\u00edslo sa zmen\u0161\u00ed. \u010ealej prezrad\u00edm, \u017ee prv\u00e1 cifra je dvojn\u00e1sobkom poslednej a druh\u00e1 cifra je dvojn\u00e1sobkom predposlednej. Ak\u00e9 \u010d\u00edslo si mysl\u00edm?<\/p>\n<div class=\"pri\">(Martin Mach)<\/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=3480\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt52z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=3482\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt52r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z5-II-1<\/strong><br \/>\nCesty v na\u0161om parku ved\u00fa po stran\u00e1ch a uhloprie\u010dkach dvoch rovnak\u00fdch obd\u013a\u017enikov:<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"70\/70z5b1.png\" alt=\"\" border=\"0\" \/><\/div>\n<p>Po\u010das v\u00edkendu sme sa parkom pre\u0161li \u0161tyrmi sp\u00f4sobmi. V ka\u017edom pr\u00edpade sme ka\u017edou vyzna\u010denou cestou pre\u0161li pr\u00e1ve raz:<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"70\/70z5b2.png\" alt=\"\" border=\"0\" \/><\/div>\n<p>V prvom pr\u00edpade sme pre\u0161li o 500 metrov menej ako v druhom pr\u00edpade. O ko\u013eko metrov viac sme pre\u0161li v tre\u0165om pr\u00edpade ako vo \u0161tvrtom?<\/p>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z5-II-2<\/strong><br \/>\nSkrinka m\u00e1 tri \u0161upl\u00edky umiestnen\u00e9 nad sebou. V nich s\u00fa tri predmety: gu\u013e\u00f4\u010dka, mu\u0161\u013ea a minca. Ka\u017ed\u00fd \u0161upl\u00edk obsahuje jeden predmet. Vieme, \u017ee:<\/p>\n<ul>\n<li>zelen\u00fd \u0161upl\u00edk je vy\u0161\u0161ie ako modr\u00fd \u0161upl\u00edk,<\/li>\n<li>minca je vy\u0161\u0161ie ako gu\u013e\u00f4\u010dka,<\/li>\n<li>\u010derven\u00fd \u0161upl\u00edk je ni\u017e\u0161ie ako mu\u0161\u013ea,<\/li>\n<li>gu\u013e\u00f4\u010dka je ni\u017e\u0161ie ako \u010derven\u00fd \u0161upl\u00edk.<\/li>\n<\/ul>\n<p>V ktorom \u0161upl\u00edku je minca?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z5-II-3<\/strong><br \/>\nAko po\u010fakovanie za ich pomoc upiekla Maru\u0161ka dvan\u00e1stim mesia\u010dikom dev\u00e4\u0165desiat pern\u00ed\u010dkov. Ke\u010f im ich d\u00e1vala, vzal si ka\u017ed\u00fd to\u013eko pern\u00ed\u010dkov, ak\u00e9 je poradov\u00e9 \u010d\u00edslo jeho mesiaca v roku. Dvaja mesia\u010dikovia s v\u010fakou odmietli a \u017eiadny pern\u00ed\u010dek si nevzali. Maru\u0161ke tak v ko\u0161\u00ed\u010dku zv\u00fd\u0161ilo 21 pern\u00ed\u010dkov. Ur\u010dte v\u0161etky mo\u017en\u00e9 dvojice mesia\u010dikov, ktor\u00ed mohli odmietnu\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=\"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=3451\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt61z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=3599\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt61r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z6-I-1<\/strong><br \/>\nKr\u00e1liky Pe\u010dienka, Fa\u0161\u00edrka, Reze\u0148 a Gul\u00e1\u0161 s\u00fa\u0165a\u017eili v skoku do dia\u013eky. Pe\u010dienka sko\u010dila o 15 cm \u010falej ako Fa\u0161\u00edrka, ktor\u00e1 sko\u010dila o 2 dm menej ako Gul\u00e1\u0161. Reze\u0148 sko\u010dil 2 730 mm, teda o 1 m a 1 dm \u010falej ako Pe\u010dienka. Ur\u010dte poradie a d\u013a\u017eky skokov v\u0161etk\u00fdch kr\u00e1likov.<\/p>\n<div class=\"pri\">(Svetlana Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z6-I-2<\/strong><br \/>\nVzal som klasick\u00fa \u010dierno-bielu \u0161achovnicu, ktor\u00e1 bola tvoren\u00e1 8 \u00d7 8 \u0161tvorcov\u00fdmi pol\u00ed\u010dkami so stranami d\u013a\u017eky 3 cm. Pol\u00ed\u010dka som v danom r\u00e1mci preskupil tak, \u017ee vznikol jeden \u010dierny obd\u013a\u017enik, jeden \u010dierny \u0161tvorec a jeden s\u00favisl\u00fd biely \u00fatvar. Jednotliv\u00e9 pol\u00ed\u010dka sa aj po preskupen\u00ed dot\u00fdkali cel\u00fdmi stranami. \u010dierne \u00fatvary sa nedot\u00fdkali (ani rohom) a ka\u017ed\u00fd z nich mal aspo\u0148 jednu stranu spolo\u010dn\u00fa s okrajom \u0161achovnice. Ur\u010dte najv\u00e4\u010d\u0161\u00ed mo\u017en\u00fd obvod bieleho \u00fatvaru a nakreslite, ako by v takom pr\u00edpade mohol vyzera\u0165.<\/p>\n<div class=\"pri\">(Martin Mach)<\/div>\n<p><strong>Z6-I-3<\/strong><br \/>\nMami\u010dka dala do misy 56 jah\u00f4d a 39 mal\u00edn a zaniesla ich Eme, ktor\u00e1 si \u010d\u00edtala. Ema si \u010d\u00edtanie spr\u00edjemnila ma\u0161krten\u00edm, a to tak, \u017ee si postupne brala po dvoch n\u00e1hodn\u00fdch kusoch ovocia:<\/p>\n<ul>\n<li>Ke\u010f vytiahla dve maliny, vymenila ich u mami\u010dky za jednu jahodu a t\u00fa vr\u00e1tila do misy.<\/li>\n<li>Ke\u010f vytiahla dve jahody, jednu zjedla a druh\u00fa vr\u00e1tila do misy.<\/li>\n<li>Ke\u010f vytiahla jednu jahodu a jednu malinu, zjedla jahodu a malinu vr\u00e1tila do misy.<\/li>\n<\/ul>\n<p>Takto nejak\u00fa chv\u00ed\u013eu ma\u0161krtila, a\u017e v mise zostal jedin\u00fd kus ovocia. Rozhodnite (a vysvetlite), \u010di to bola jahoda, alebo malina.<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z6-I-4<\/strong><br \/>\nCtibor naprogramoval dva spolupracuj\u00face rysovacie roboty Mikiho a Nikiho. Miki vie zostrojova\u0165 \u0161tvorce, pravideln\u00e9 p\u00e4\u0165uholn\u00edky a pravideln\u00e9 \u0161es\u0165uholn\u00edky. Po\u010das jedn\u00e9ho d\u0148a v\u0161ak rysuje iba navz\u00e1jom zhodn\u00e9 mnohouholn\u00edky. Niki do v\u0161etk\u00fdch Mikiho mnohouholn\u00edkov dop\u013a\u0148a v\u0161etky uhloprie\u010dky.<\/p>\n<ol>\n<li>V pondelok zostrojil Miki rovnak\u00fd po\u010det \u00fase\u010diek ako Niki. Ak\u00e9 mnohouholn\u00edky rysovali?<\/li>\n<li>V utorok zostrojil Miki 18 \u00fase\u010diek. Ko\u013eko ich zostrojil Niki?<\/li>\n<li>V stredu zostrojili Miki a Niki dokopy 70 \u00fase\u010diek. Ko\u013eko mnohouholn\u00edkov im dal Ctibor rysova\u0165?<\/li>\n<\/ol>\n<div class=\"pri\">(Michaela Petrov\u00e1)<\/div>\n<p><strong>Z6-I-5<\/strong><br \/>\nPetra vpisovala do kr\u00fa\u017ekov \u010d\u00edsla 1, 2, 3, 4, 5, 6, 7, 8 tak, \u017ee ka\u017ed\u00e9 bolo pou\u017eit\u00e9 pr\u00e1ve raz a \u017ee s\u00fa\u010det \u010d\u00edsel na ka\u017edej strane trojuholn\u00edka bol rovnak\u00fd. Ak\u00fd najv\u00e4\u010d\u0161\u00ed s\u00fa\u010det mohla takto dosta\u0165? Uve\u010fte pr\u00edklad mo\u017en\u00e9ho vyplnenia.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"70\/70z6a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Al\u017ebeta Bohinikov\u00e1)<\/div>\n<p><strong>Z6-I-6<\/strong><br \/>\nAni\u010dka a Marienka maj\u00fa ka\u017ed\u00e1 svoje ob\u013e\u00faben\u00e9 prirodzen\u00e9 \u010d\u00edslo. Ak vyn\u00e1sob\u00edme Ani\u010dkino \u010d\u00edslo samo so sebou, vyjde n\u00e1m stokr\u00e1t v\u00e4\u010d\u0161ie \u010d\u00edslo, ako ke\u010f vyn\u00e1sob\u00edme Marienkino \u010d\u00edslo samo so sebou. Ak s\u010d\u00edtame Ani\u010dkino a Marienkino ob\u013e\u00faben\u00e9 \u010d\u00edslo, z\u00edskame \u010d\u00edslo o 18 v\u00e4\u010d\u0161ie, ako je polovica Ani\u010dkinho \u010d\u00edsla. Ur\u010dte Ani\u010dkino a Marienkino ob\u013e\u00faben\u00e9 \u010d\u00edslo.<\/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 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=3606\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt62z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=3609\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt62r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z6-II-1<\/strong><br \/>\n\u0160tyri veveri\u010dky zjedli dokopy 2020 orie\u0161kov, ka\u017ed\u00e1 najmenej 103 orie\u0161kov. Prv\u00e1 veveri\u010dka zjedla viac orie\u0161kov ako ktor\u00e1ko\u013evek z ostatn\u00fdch troch veveri\u010diek. Druh\u00e1 a tretia veveri\u010dka zjedli dokopy 1277 orie\u0161kov. Ko\u013eko orie\u0161kov zjedla prv\u00e1 veveri\u010dka?<\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z6-II-2<\/strong><br \/>\nV s\u00fa\u010dinovej pyram\u00edde je v ka\u017edom pol\u00ed\u010dku jedno kladn\u00e9 cel\u00e9 \u010d\u00edslo, ktor\u00e9 je s\u00fa\u010dinom \u010d\u00edsel z dvoch susediacich pol\u00ed\u010dok z ni\u017e\u0161ej vrstvy. Vo vrchole trojvrstvovej s\u00fa\u010dinovej pyram\u00eddy je \u010d\u00edslo 90. Ak\u00e9 \u010d\u00edslo m\u00f4\u017ee by\u0165 vo vyzna\u010denom pol\u00ed\u010dku? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"70\/70z6b1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">()<\/div>\n<p><strong>Z6-II-3<\/strong><br \/>\nDvierka kr\u00e1lik\u00e1rne s\u00fa vyroben\u00e9 z dreven\u00e9ho r\u00e1mu a dr\u00f4ten\u00e9ho pletiva so \u0161tvorcov\u00fdmi okami. Latky r\u00e1mu s\u00fa \u0161irok\u00e9 5 cm. Niektor\u00e9 mre\u017eov\u00e9 body pletiva sa nach\u00e1dzaj\u00fa presne na vn\u00fatorn\u00fdch hran\u00e1ch r\u00e1mu ako na obr\u00e1zku. Vn\u00fatorn\u00e1 (pletivov\u00e1) \u010das\u0165 dvierok m\u00e1 obsah 432 cm<sup>2<\/sup>. Ur\u010dte vonkaj\u0161ie rozmery (t. j. \u0161\u00edrku a v\u00fd\u0161ku) cel\u00fdch dvierok.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"70\/70z6b2.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=\"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=3452\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt71z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=3600\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt71r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z7-I-1<\/strong><br \/>\nUr\u010dte, ktor\u00e1 cifra je na 1000. mieste za desatinnou \u010diarkou v desatinnom rozvoji \u010d\u00edsla\u00a0<img decoding=\"async\" style=\"vertical-align: middle;\" src=\"70\/70z7a1.png\" alt=\"\" \/>.<\/p>\n<div class=\"pri\">(Marie Krej\u010dov\u00e1)<\/div>\n<p><strong>Z7-I-2<\/strong><br \/>\nKubo sa dohodol s ba\u010dom, \u017ee sa mu bude stara\u0165 o ovce. Ba\u010da Kubovi s\u013e\u00fabil, \u017ee po roku slu\u017eby dostane dvadsa\u0165 zlat\u00fdch a k tomu jednu ovcu. Len\u017ee Kubo dal v\u00fdpove\u010f pr\u00e1ve vtedy, ke\u010f uplynul siedmy mesiac slu\u017eby. Aj tak ho Ba\u010da spravodlivo odmenil a zaplatil mu p\u00e4\u0165 zlat\u00fdch a jednu ovcu. Na ko\u013eko zlat\u00fdch si ba\u010da cenil jednu ovcu?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z7-I-3<\/strong><br \/>\nPre skupinu det\u00ed plat\u00ed, \u017ee v ka\u017edej trojici det\u00ed zo skupiny je chlapec menom Adam a v ka\u017edej \u0161tvorici je diev\u010da menom Be\u00e1ta. Ko\u013eko nanajv\u00fd\u0161 det\u00ed m\u00f4\u017ee by\u0165 v takej skupine a ak\u00e9 s\u00fa v tom pr\u00edpade ich men\u00e1?<\/p>\n<div class=\"pri\">(Jaroslav Zhouf)<\/div>\n<p><strong>Z7-I-4<\/strong><br \/>\nMedzi pr\u00edstavmi Mumraj a Zm\u00e4tok pendluj\u00fa po rovnakej trase dve lode. V pr\u00edstavoch tr\u00e1via zanedbate\u013en\u00fd \u010das, hne\u010f sa ot\u00e1\u010daj\u00fa a pokra\u010duj\u00fa v plavbe. R\u00e1no v rovnakom okamihu vypl\u00e1va modr\u00e1 lo\u010f z pr\u00edstavu Mumraj a zelen\u00e1 lo\u010f z pr\u00edstavu Zm\u00e4tok. Prv\u00fdkr\u00e1t sa lode m\u00ed\u0148aj\u00fa 20 km od pr\u00edstavu Mumraj a po nejakom \u010dase sa stretn\u00fa priamo v tomto pr\u00edstave. To u\u017e modr\u00e1 lo\u010f stihla upl\u00e1va\u00bb trasu medzi pr\u00edstavmi \u0161tyrikr\u00e1t, zatia\u013e \u010do zelen\u00e1 lo\u010f iba trikr\u00e1t. Ak\u00e1 dlh\u00e1 je trasa medzi pr\u00edstavmi Mumraj a Zm\u00e4tok?<\/p>\n<div class=\"pri\">(Franti\u0161ek Steinhauser)<\/div>\n<p><strong>Z7-I-5<\/strong><br \/>\nOd\u010d\u00edtacia pyram\u00edda je pyram\u00edda tvoren\u00e1 nez\u00e1porn\u00fdmi cel\u00fdmi \u010d\u00edslami, z ktor\u00fdch ka\u017ed\u00e9 je rozdielom dvoch najbli\u017e\u0161\u00edch \u010d\u00edsel z predch\u00e1dzaj\u00faceho riadka (\u010d\u00edtan\u00e9 odspodu nahor). Tu je pr\u00edklad od\u010d\u00edtacej pyram\u00eddy:<\/p>\n<div class=\"pc\">\n<table border=\"0\" align=\"center\">\n<tbody>\n<tr align=\"center\">\n<td style=\"width: 5;\"><\/td>\n<td style=\"width: 5;\"><\/td>\n<td style=\"width: 5;\"><\/td>\n<td style=\"width: 5;\">1<\/td>\n<td style=\"width: 5;\"><\/td>\n<td style=\"width: 5;\"><\/td>\n<td style=\"width: 5;\"><\/td>\n<\/tr>\n<tr align=\"center\">\n<td><\/td>\n<td><\/td>\n<td>2<\/td>\n<td><\/td>\n<td>1<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr align=\"center\">\n<td><\/td>\n<td>2<\/td>\n<td><\/td>\n<td>4<\/td>\n<td><\/td>\n<td>5<\/td>\n<td><\/td>\n<\/tr>\n<tr align=\"center\">\n<td>5<\/td>\n<td><\/td>\n<td>7<\/td>\n<td><\/td>\n<td>3<\/td>\n<td><\/td>\n<td>8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>V\u00fdzna\u010dn\u00e9 \u010d\u00edslo je najv\u00e4\u010d\u0161ie \u010d\u00edslo od\u010d\u00edtacej pyram\u00eddy. V\u00fdborn\u00e1 pyram\u00edda je od\u010d\u00edtacia pyram\u00edda, ktor\u00e1 m\u00e1 vo vrchole 0 a aspo\u0148 jeden riadok tvoren\u00fd navz\u00e1jom r\u00f4znymi \u010d\u00edslami.<\/p>\n<ol>\n<li>Ko\u013eko najmenej riadkov mus\u00ed ma\u0165 v\u00fdborn\u00e1 pyram\u00edda?<\/li>\n<li>Ktor\u00e9 najmen\u0161ie v\u00fdzna\u010dn\u00e9 \u010d\u00edslo m\u00f4\u017ee by\u0165 obsiahnut\u00e9 vo v\u00fdbornej pyram\u00edde s najmen\u0161\u00edm po\u010dtom riadkov?<\/li>\n<\/ol>\n<div class=\"pri\">(Katar\u00edna Jasen\u010d\u00e1kov\u00e1)<\/div>\n<p><strong>Z7-I-6<\/strong><br \/>\nV trojuholn\u00edku <em>ABC<\/em> le\u017e\u00ed na strane <em>AC<\/em> bod <em>D<\/em> a na strane <em>BC<\/em> bod <em>E<\/em>. Ve\u013ekosti uhlov <em>ABD<\/em>, <em>BAE<\/em>, <em>CAE<\/em> a <em>CBD<\/em> s\u00fa postupne 30\u00b0, 60\u00b0, 20\u00b0a 30\u00b0. Ur\u010dte ve\u013ekos\u0165 uhla <em>AED<\/em>.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"70\/70z7a2.png\" alt=\"\" border=\"0\" \/><\/div>\n<p>Pozn\u00e1mka: obr\u00e1zok je iba ilustra\u010dn\u00fd.<\/p>\n<div class=\"pri\">(Al\u017ebeta Bohinikov\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=3607\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt72z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=3610\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt72r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z7-II-1<\/strong><br \/>\nNa rozpr\u00e1vkovom ostrove \u017eij\u00fa draci a kyklopi. V\u0161etci draci s\u00fa \u010derven\u00ed, trojhlav\u00ed a dvojnoh\u00ed. V\u0161etci kyklopi s\u00fa hned\u00ed, jednohlav\u00ed a dvojnoh\u00ed. Kyklopi maj\u00fa jedno oko uprostred \u010dela, draci maj\u00fa na ka\u017edej hlave dve o\u010di. Dokopy maj\u00fa kyklopi a draci 42 o\u010d\u00ed a 34 n\u00f4h. Ko\u013eko drakov a ko\u013eko kyklopov \u017eije na ostrove?<\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z7-II-2<\/strong><br \/>\nPravideln\u00fd \u0161es\u0165uholn\u00edk je \u0161tyrmi svojimi uhloprie\u010dkami rozdelen\u00fd na \u0161es\u0165 trojuholn\u00edkov a jeden \u0161tvoruholn\u00edk ako na obr\u00e1zku. Obsah \u0161tvoruholn\u00edka F je 1,8 cm<sup>2<\/sup>. Ur\u010dte obsahy trojuholn\u00edkov A, B, C, D, E a G.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"70\/70z7b1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">()<\/div>\n<p><strong>Z7-II-3<\/strong><br \/>\nBludi\u010dka Jozef\u00edna tancuje pri mo\u010diari, pri\u010dom pou\u017e\u00edva kroky dvojakej d\u013a\u017eky &#8211; kr\u00e1tke meraj\u00fa 45 cm, dlh\u00e9 60 cm. \u010casom si vy\u0161liapala ov\u00e1lny chodn\u00edk, po ktorom za dlh\u00fdch noc\u00ed tancuje st\u00e1le dokola. Ak opakuje tri dlh\u00e9 kroky dopredu a jeden kr\u00e1tky vzad, tak dev\u00e4\u0165desiatym krokom dotancuje presne tam, kde za\u010d\u00ednala. Ak opakuje tri kr\u00e1tke kroky dopredu a jeden dlh\u00fd vzad, tak jej tie\u017e vych\u00e1dza krok presne tam, kde za\u010d\u00ednala. Ko\u013ek\u00fdm krokom dotancuje Jozef\u00edna na p\u00f4vodn\u00e9 miesto v druhom pr\u00edpade?<\/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=\"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=3453\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt81z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=3601\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt81r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z8-I-1<\/strong><br \/>\nMysl\u00edm si p\u00e4\u0165cifern\u00e9 \u010d\u00edslo, ktor\u00e9 nie je delite\u013en\u00e9 tromi ani \u0161tyrmi. Ak ka\u017ed\u00fa cifru zv\u00e4\u010d\u0161\u00edm o jedna, z\u00edskam p\u00e4\u0165cifern\u00e9 \u010d\u00edslo, ktor\u00e9 je delite\u013en\u00e9 tromi. Ak ka\u017ed\u00fa cifru o jedna zmen\u0161\u00edm, z\u00edskam p\u00e4\u0165cifern\u00e9 \u010d\u00edslo delite\u013en\u00e9 \u0161tyrmi. Ak prehod\u00edm \u013eubovo\u013en\u00e9 dve cifry, \u010d\u00edslo sa zmen\u0161\u00ed. Ak\u00e9 \u010d\u00edslo si m\u00f4\u017eem myslie\u0165? N\u00e1jdite v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Martin Mach)<\/div>\n<p><strong>Z8-I-2<\/strong><br \/>\nNa z\u00e1hrade st\u00e1li tri debny s jablkami. Spolu bolo jab\u013ak viac ako 150, av\u0161ak menej ako 190. Potom Marienka premiestnila z prvej debny do dvoch \u010fal\u0161\u00edch debien jablk\u00e1 tak, \u017ee sa ich po\u010det v ka\u017edej z t\u00fdchto dvoch debien oproti predo\u0161l\u00e9mu stavu zdvojn\u00e1sobil. Obdobn\u00fdm sp\u00f4sobom Marta premiestnila jablk\u00e1 z druhej debny do prvej a tretej. Nakoniec \u0160tefka pod\u013ea rovnak\u00fdch pravidiel premiestnila jablk\u00e1 z tretej debny do prvej a druhej. Ke\u010f pri\u0161iel na z\u00e1hradu Vojto, za\u010dudoval sa, \u017ee v ka\u017edej debne bol rovnak\u00fd po\u010det jab\u013ak. Ko\u013eko jab\u013ak bolo v jednotliv\u00fdch debn\u00e1ch p\u00f4vodne?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z8-I-3<\/strong><br \/>\nV trojuholn\u00edku <em>ABC<\/em> je bod <em>S<\/em> stredom vp\u00edsanej kru\u017enice. Obsah \u0161tvoruholn\u00edka <em>ABCS<\/em> je rovn\u00fd \u0161tyrom p\u00e4tin\u00e1m obsahu trojuholn\u00edka <em>ABC<\/em>. D\u013a\u017eky str\u00e1n trojuholn\u00edka <em>ABC<\/em> vyjadren\u00e9 v centimetroch s\u00fa v\u0161etky celo\u010d\u00edseln\u00e9 a obvod trojuholn\u00edka <em>ABC<\/em> je 15 cm. Ur\u010dte d\u013a\u017eky str\u00e1n trojuholn\u00edka <em>ABC<\/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 \/>\nJarka bola na brig\u00e1de s nemennou dennou mzdou. Za tri dni si zarobila to\u013eko eur, \u017ee si k\u00fapila stolov\u00fa hru a e\u0161te jej 49\u20ac zv\u00fd\u0161ilo. Keby str\u00e1vila na brig\u00e1de p\u00e4\u0165 dn\u00ed, mohla by si k\u00fapi\u0165 dve tak\u00e9 stolov\u00e9 hry a e\u0161te by jej zv\u00fd\u0161ilo 54\u20ac. Ko\u013eko eur st\u00e1la stolov\u00e1 hra?<\/p>\n<div class=\"pri\">(Karel Pazourek)<\/div>\n<p><strong>Z8-I-5<\/strong><br \/>\nP\u00e1n Strieborn\u00fd usporiadal v\u00fdstavu. Vystavoval 120 prste\u0148ov, ktor\u00e9 le\u017eali na stoloch pozd\u013a\u017e stien s\u00e1ly a tvorili tak jednu ve\u013ek\u00fa kru\u017enicu. Prehliadka za\u010d\u00ednala pri vchodov\u00fdch dver\u00e1ch v ozna\u010denom smere. Ka\u017ed\u00fd tret\u00ed prste\u0148 v rade bol zlat\u00fd, ka\u017ed\u00fd \u0161tvrt\u00fd prste\u0148 bol staro\u017eitn\u00fd a ka\u017ed\u00fd desiaty prste\u0148 mal diamant. Prste\u0148, ktor\u00fd nemal \u017eiadnu z t\u00fdchto troch vlastnost\u00ed, bol falzifik\u00e1t. Ko\u013eko bolo na v\u00fdstave zlat\u00fdch prste\u0148ov, ktor\u00e9 boli staro\u017eitn\u00e9 a z\u00e1rove\u0148 mali diamant? Ko\u013eko vystavil p\u00e1n Strieborn\u00fd falzifik\u00e1tov?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z8-I-6<\/strong><br \/>\nBody A, B, C, D a E s\u00fa vrcholmi nepravidelnej p\u00e4\u0165c\u00edpej hviezdy, pozri obr\u00e1zok. Ur\u010dte s\u00fa\u010det vyzna\u010den\u00fdch uhlov.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"70\/70z8a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<p><em>Pozn\u00e1mka: obr\u00e1zok je iba ilustra\u010dn\u00fd.<\/em><\/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=3608\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt82z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=3611\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt82r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z8-II-1<\/strong><br \/>\nDeti na t\u00e1bore z\u00edskavali za splnen\u00e9 \u00falohy body. Tie bolo mo\u017en\u00e9 \u010falej meni\u0165: p\u00e4\u0165 bodov za samolepku, \u0161es\u0165 bodov za pe\u010diatku. Jaro si prepo\u010d\u00edtal, \u017ee keby chcel len samolepky, zv\u00fd\u0161il by mu jeden bod nevyu\u017eit\u00fd. Keby si vybral len pe\u010diatky, nevyu\u017eil by tri body. Nakoniec dok\u00e1zal uplatni\u0165 v\u0161etky svoje body, pri\u010dom z\u00edskal dvojn\u00e1sobn\u00e9 mno\u017estvo pe\u010diatok ako samolepiek. Ko\u013eko najmenej bodov mohol ma\u0165 Jaro pred menen\u00edm?<\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z8-II-2<\/strong><br \/>\nEli\u0161ka umiest\u0148ovala kol\u00e1\u010de do \u0161kat\u00fa\u013e, z ktor\u00fdch potom postavila pyram\u00eddu ako na obr\u00e1zku. Pritom ka\u017ed\u00e1 \u0161katu\u013ea vo vy\u0161\u0161om rade obsahovala to\u013eko kol\u00e1\u010dov ako dve susediace \u0161katule pod \u0148ou dokopy. V troch \u0161katuliach ozna\u010den\u00fdch hviezdi\u010dkami bolo tri, p\u00e4\u0165 a \u0161es\u0165 kol\u00e1\u010dov. Eli\u0161ka si v\u0161imla, \u017ee keby ozna\u010den\u00e9 \u0161katule akoko\u013evek zamenila a pod\u013ea predch\u00e1dzaj\u00faceho pravidla upravila po\u010dty kol\u00e1\u010dov v ostatn\u00fdch \u0161katuliach, celkov\u00fd po\u010det kol\u00e1\u010dov by sa nezmen\u0161il. Ko\u013eko kol\u00e1\u010dov bolo v ozna\u010denej \u0161katuli v druhom rade zdola?<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"70\/70z8b1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">()<\/div>\n<p><strong>Z8-II-3<\/strong><br \/>\nPre v\u0161eobecn\u00fd trojuholn\u00edk ABC s\u00fa dan\u00e9 body D, E, F:<\/p>\n<ul>\n<li>bod D je v tretine \u00fase\u010dky AB, bli\u017e\u0161ie k bodu A,<\/li>\n<li>bod E je v tretine \u00fase\u010dky BC, bli\u017e\u0161ie k bodu B,<\/li>\n<li>bod F je v tretine \u00fase\u010dky CD, bli\u017e\u0161ie k bodu C.<\/li>\n<\/ul>\n<p>Ur\u010dte pomer obsahov trojuholn\u00edkov ABC a DEF.<\/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=3454\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt91z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=3479\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt91r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-I-1<\/strong><br \/>\nSl\u00e1vka si nap\u00edsala farebn\u00fdmi fixkami \u0161tyri r\u00f4zne prirodzen\u00e9 \u010d\u00edsla: \u010derven\u00e9, modr\u00e9, zelen\u00e9 a \u017elt\u00e9. Ke\u010f \u010derven\u00e9 \u010d\u00edslo vydel\u00ed modr\u00fdm, dostane ako ne\u00fapln\u00fd podiel zelen\u00e9 \u010d\u00edslo a \u017elt\u00e9 predstavuje zvy\u0161ok po tomto delen\u00ed. Ke\u010f vydel\u00ed modr\u00e9 \u010d\u00edslo zelen\u00fdm, vyjde jej delenie bezo zvy\u0161ku a podielom je \u010d\u00edslo \u017elt\u00e9. Sl\u00e1vka prezradila, \u017ee dve z jej \u0161tyroch \u010d\u00edsel s\u00fa 97 a 101. Ur\u010dte ostatn\u00e9 Sl\u00e1vkine \u010d\u00edsla a prira\u010fte jednotliv\u00fdm \u010d\u00edslam farby. N\u00e1jdite v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Michaela Petrov\u00e1)<\/div>\n<p><strong>Z9-I-2<\/strong><br \/>\nN\u00e1jdite v\u0161etky dvojice nez\u00e1porn\u00fdch cel\u00fdch \u010d\u00edsel <strong><em>x<\/em><\/strong> a jednocifern\u00fdch prirodzen\u00fdch \u010d\u00edsel <strong><em>y<\/em><\/strong>, pre ktor\u00e9 plat\u00ed:<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"70\/70z9a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<p>Z\u00e1pis na pravej strane rovnosti ozna\u010duje periodick\u00e9 \u010d\u00edslo.<\/p>\n<div class=\"pri\">(Karel Pazourek)<\/div>\n<p><strong>Z9-I-3<\/strong><br \/>\nV rovnostrannom trojuholn\u00edku <em>ABC<\/em> je bod <em>T<\/em> jeho \u0165a\u017eiskom, bod <em>R<\/em> je obrazom bodu <em>T<\/em> v osovej s\u00famernosti pod\u013ea priamky <em>AB<\/em> a bod <em>N<\/em> je obrazom bodu <em>T<\/em> v osovej s\u00famernosti pod\u013ea priamky <em>BC<\/em>. Ur\u010dte pomer obsahov trojuholn\u00edkov <em>ABC<\/em> a <em>TRN<\/em>.<\/p>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z9-I-4<\/strong><br \/>\nNa stene bolo nap\u00edsan\u00e9 dvakr\u00e1t to ist\u00e9 p\u00e4\u0165cifern\u00e9 \u010d\u00edslo. Pat pred jeden z\u00e1pis \u010d\u00edsla prip\u00edsal jednotku, Mat prip\u00edsal jednotku za ten druh\u00fd z\u00e1pis \u010d\u00edsla. T\u00fdm dostali dve \u0161es\u0165cifern\u00e9 \u010d\u00edsla, z ktor\u00fdch jedno bolo trikr\u00e1t v\u00e4\u010d\u0161ie ako druh\u00e9. Ktor\u00e9 p\u00e4\u0165cifern\u00e9 \u010d\u00edslo bolo p\u00f4vodne nap\u00edsan\u00e9 na stene?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z9-I-5<\/strong><br \/>\nNa ihrisku s\u00fa nakreslen\u00e9 tri rovnako ve\u013ek\u00e9 kruhy. Rozostavte 16 kolkov tak, aby v ka\u017edom kruhu st\u00e1lo 9 kolkov. N\u00e1jdite aspo\u0148 osem podstatne odli\u0161n\u00fdch rozostaven\u00ed, t.j. tak\u00fdch rozostaven\u00ed, pri ktor\u00fdch sa nerozli\u0161uj\u00fa kolky ani kruhy.<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z9-I-6<\/strong><br \/>\nJozef a M\u00e1ria objavili na dovolenke pravideln\u00fd ihlan, ktor\u00e9ho podstavou bol \u0161tvorec so stranou 230 m a ktor\u00e9ho v\u00fd\u0161ka bola rovn\u00e1 polomeru kruhu s rovnak\u00fdm obvodom ako m\u00e1 podstavn\u00fd \u0161tvorec. M\u00e1ria ozna\u010dila vrcholy \u0161tvorca <em>ABCD<\/em> Jozef vyzna\u010dil na priamke sp\u00e1jaj\u00facej bod <em>B<\/em> s vrcholom ihlana tak\u00fd bod <em>E<\/em>, \u017ee d\u013a\u017eka lomenej \u010diary <em>AEC<\/em> bola najkrat\u0161ia mo\u017en\u00e1. Ur\u010dte d\u013a\u017eku lomenej \u010diary <em>AEC<\/em> zaokr\u00fahlen\u00fa na cel\u00e9 centimetre.<\/p>\n<div class=\"pri\">(Marie Krej\u010dov\u00e1, Franti\u0161ek Steinhauser)<\/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=3481\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt92z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=3483\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt92r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-II-1<\/strong><br \/>\nBabi\u010dka mala \u0161tvorcov\u00fa z\u00e1hradu. Dok\u00fapila nieko\u013eko susedn\u00fdch pozemkov, \u010d\u00edm z\u00edskala zasa \u0161tvorcov\u00fd pozemok, ktor\u00e9ho strana bola o tri metre dlh\u0161ia ako strana p\u00f4vodnej z\u00e1hrady. Takto bola v\u00fdmera pozemku o dev\u00e4\u0165 \u0161tvorcov\u00fdch metrov v\u00e4\u010d\u0161ia ako dvojn\u00e1sobok p\u00f4vodnej v\u00fdmery. Ak\u00e1 dlh\u00e1 bola strana p\u00f4vodnej z\u00e1hrady?<\/p>\n<div class=\"pri\">(Katar\u00edna Buz\u00e1kov\u00e1)<\/div>\n<p><strong>Z9-II-2<\/strong><br \/>\nBabi\u010dka e\u0161te nem\u00e1 100 rokov, vnu\u010dka m\u00e1 viac ako 10 rokov a vek babi\u010dky je n\u00e1sobkom veku vnu\u010dky. Ke\u010f vnu\u010dka nap\u00edsala vek babi\u010dky a za\u0148 vek svoj, dostala \u0161tvorcifern\u00e9 \u010d\u00edslo. Ke\u010f babi\u010dka nap\u00edsala vek vnu\u010dky a za\u0148 vek svoj, dostala in\u00e9 \u0161tvorcifern\u00e9 \u010d\u00edslo. Rozdiel t\u00fdchto dvoch \u0161tvorcifern\u00fdch \u010d\u00edsel je 7128. Ko\u013eko rokov m\u00f4\u017ee ma\u0165 babi\u010dka a ko\u013eko vnu\u010dka? Uve\u010fte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z9-II-3<\/strong><br \/>\nKarol, Miro a \u013dudo porovn\u00e1vali svoje zbierky zn\u00e1mok. Ke\u010f kontrolovali po\u010dty, zistili, \u017ee Karol a Miro maj\u00fa dokopy 101 zn\u00e1mok, Karol a \u013dudo 115 zn\u00e1mok, Miro a \u013dudo 110. Ke\u010f overovali, \u010do by mohli meni\u0165, zistili, \u017ee \u017eiadnu zn\u00e1mku nemaj\u00fa v\u0161etci rovnak\u00fa, ale \u017ee Karol a Miro maj\u00fa 5 zn\u00e1mok rovnak\u00fdch, Karol a \u013dudo 12 rovnak\u00fdch, Miro a \u013dudo 7. Ko\u013eko zn\u00e1mok m\u00e1 \u013dudo in\u00fdch ako ostatn\u00ed chlapci?<\/p>\n<div class=\"pri\">(Miroslava Farkas Smitkov\u00e1)<\/div>\n<p><strong>Z9-II-4<\/strong><br \/>\nP\u00e1n u\u010dite\u013e chcel po Adamovi a Eve, aby vypo\u010d\u00edtali obvod lichobe\u017en\u00edka, ktor\u00e9ho dlh\u0161ia z\u00e1klad\u0148a merala 30 cm, v\u00fd\u0161ka 24 cm a ramen\u00e1 25 cm a 30 cm. Adamovi vy\u0161iel in\u00fd obvod ako Eve, aj tak v\u0161ak p\u00e1n u\u010dite\u013e oboch pochv\u00e1lil za spr\u00e1vne rie\u0161enia. Ur\u010dte v\u00fdsledky Adama a Evy.<\/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=\"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=\"https:\/\/skmo.sk\/dokument.php?id=3573\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt93z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=3574\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt93r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-III-1<\/strong><br \/>\nPre tri nezn\u00e1me prirodzen\u00e9 \u010d\u00edsla plat\u00ed, \u017ee<\/p>\n<ul>\n<li>najv\u00e4\u010d\u0161\u00ed spolo\u010dn\u00fd delite\u013e prv\u00e9ho a druh\u00e9ho je 8,<\/li>\n<li>najv\u00e4\u010d\u0161\u00ed spolo\u010dn\u00fd delite\u013e druh\u00e9ho a tretieho je 2,<\/li>\n<li>najv\u00e4\u010d\u0161\u00ed spolo\u010dn\u00fd delite\u013e prv\u00e9ho a tretieho je 6,<\/li>\n<li>najmen\u0161\u00ed spolo\u010dn\u00fd n\u00e1sobok v\u0161etk\u00fdch troch \u010d\u00edsel je 1680,<\/li>\n<li>najv\u00e4\u010d\u0161ie z \u010d\u00edsel je v\u00e4\u010d\u0161ie ako 100, ale nie je v\u00e4\u010d\u0161ie ako 200,<\/li>\n<li>jedno z \u010d\u00edsel je \u0161tvrtou mocninou cel\u00e9ho \u010d\u00edsla.<\/li>\n<\/ul>\n<p>O ktor\u00e9 \u010d\u00edsla ide? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z9-III-2<\/strong><br \/>\nTrojuholn\u00edk <em>ACH<\/em> je ur\u010den\u00fd tromi vrcholmi kocky <em>ABCDEFGH<\/em> (pozri obr\u00e1zok). V\u00fd\u0161ka tohto trojuholn\u00edka na stranu <em>CH<\/em> m\u00e1 ve\u013ekos\u0165 12 cm. Vypo\u010d\u00edtajte obsah trojuholn\u00edka <em>ACH<\/em> a ve\u013ekos\u0165 hrany danej kocky.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"70\/70z9c1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Marie Krej\u010dov\u00e1)<\/div>\n<p><strong>Z9-III-3<\/strong><br \/>\nDan\u00e1 je postupnos\u0165 siedmich \u010d\u00edsel <em>a, b, c, d, e, f, g<\/em>. Ka\u017ed\u00e9 z \u010d\u00edsel <em>b, c, d, e, f<\/em> je aritmetick\u00fdm priemerom susedn\u00fdch dvoch \u010d\u00edsel. Uk\u00e1\u017ete, \u017ee \u010d\u00edslo <em>d<\/em> je aritmetick\u00fdm priemerom \u010d\u00edsel <em>a<\/em> a <em>g<\/em>. <\/p>\n<div class=\"pri\">(Karel Pazourek)<\/div>\n<p><strong>Z9-III-4<\/strong><br \/>\nDan\u00e9 s\u00fa rovnobe\u017en\u00edky ABGH a DEGH, ktor\u00fdch vrcholy A, B, D a E le\u017eia na jednej priamke. Bod <em>C<\/em> je priese\u010dn\u00edkom \u00fase\u010diek <em>BG<\/em> a <em>DH<\/em>, bod <em>I<\/em> le\u017e\u00ed na \u00fase\u010dke <em>AH<\/em> a bod <em>F<\/em> le\u017e\u00ed na \u00fase\u010dke <em>EG<\/em>. Mnohouholn\u00edk <em>ABCDEGH<\/em> pozost\u00e1va zo siedmich trojuholn\u00edkov, pri\u010dom medzi trojuholn\u00edkmi <em>ABI<\/em>, <em>BCI<\/em>, <em>CHI<\/em>, <em>DEF<\/em>, <em>CDF<\/em> a <em>CFG<\/em> je jeden s obsahom 3 cm<sup>2<\/sup>, jeden s obsahom 5 cm<sup>2<\/sup>, dva s obsahom 7 cm<sup>2<\/sup> a jeden s obsahom 10 cm<sup>2<\/sup>. Okrem trojuholn\u00edkov s obsahmi 7 cm<sup>2<\/sup> nem\u00e1 \u017eiadna \u010fal\u0161ia dvojica z uveden\u00fdch siedmich trojuholn\u00edkov rovnak\u00fd obsah. Rozhodnite, \u010di mo\u017eno s istotou ur\u010di\u0165 trojuholn\u00edky s obsahmi 7 cm<sup>2<\/sup>. \u010ealej ur\u010dte obsah mnohouholn\u00edka <em>ABCDEGH<\/em>; n\u00e1jdite v\u0161etky mo\u017enosti.<br \/> <br \/>\n<em>Pozn\u00e1mka: Obr\u00e1zok je len ilustra\u010dn\u00fd. <\/em><\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"70\/70z9c2.png\" alt=\"\" border=\"0\" \/><\/div>\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; 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>70. ro\u010dn\u00edk matematickej olympi\u00e1dy 2020-2021 S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9 aktualizovan\u00e9 31.3.2021 19:30<\/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":[35],"tags":[],"class_list":["post-1444","post","type-post","status-publish","format-standard","hentry","category-70-rocnik-2020-21"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.6 - 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