{"id":1306,"date":"2018-09-23T09:47:10","date_gmt":"2018-09-23T09:47:10","guid":{"rendered":"http:\/\/matematika.besaba.com\/?p=1306"},"modified":"2023-11-02T18:52:10","modified_gmt":"2023-11-02T18:52:10","slug":"sutazne-ulohy-kategorii-z5-z9-7","status":"publish","type":"post","link":"https:\/\/matematika.besaba.com\/?p=1306","title":{"rendered":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9"},"content":{"rendered":"<p class=\"hh2\"><a name=\"top\"><\/a>68. ro\u010dn\u00edk matematickej olympi\u00e1dy 2018-2019<br \/>\nS\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9<\/p>\n<p><span class=\"povinne\">aktualizovan\u00e9 30.1.2020 10:25<\/span><\/p>\n<p><!--more--><\/p>\n<p><center><script type=\"text\/javascript\" async=\"true\" charset=\"utf-8\" src=\"http:\/\/sk.search.etargetnet.com\/generic\/advert.php?g=ref:48528,area:600x70,tabl:4,divid:,design_name:green,border_color:ffffff,border_style:none,background_opacity:100,background_color:ffffff,title_color:426000,text_color:000000,url_color:426000,h_title_color:426000,h_text_color:000000,h_url_color:426000,freespace:0,logo_type:3,logo:1,title_underline:0,url_underline:0,h_title_underline:1,h_url_underline:1,nourl:,fsi:11,font:verdana,require_login_pass:,login_pass:,show_in_fb:,show_in_fb_text:,allow_youtube_ads:1\" ><\/script><\/center><\/p>\n<table style=\"width: 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<\/tbody>\n<\/table>\n<p><a name=\"z51\"><\/a><\/p>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=3027\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt51z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=3049\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt51r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z5-I-1<\/strong><br \/>\nMi\u0161ka m\u00e1 p\u00e4\u0165 pasteliek. Vojto ich m\u00e1 menej ako Mi\u0161ka. Vendel\u00edn ich m\u00e1 to\u013eko, ko\u013eko Mi\u0161ka a Vojto spolu. V\u0161etci traja spolu maj\u00fa sedemkr\u00e1t viac pasteliek, ako m\u00e1 Vojto. Ko\u013eko pasteliek m\u00e1 Vendel\u00edn?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z5-I-2<\/strong><br \/>\nTereza dostala \u0161tyri zhodn\u00e9 pravouhl\u00e9 trojuholn\u00edky so stranami d\u013a\u017eok 3 cm, 4 cm a 5 cm. Z t\u00fdchto trojuholn\u00edkov (nie nutne zo v\u0161etk\u00fdch \u0161tyroch) sk\u00fa\u0161ala sklada\u0165 nov\u00e9 \u00fatvary. Postupne sa jej podarilo zlo\u017ei\u0165 \u0161tvoruholn\u00edky s obvodom 14 cm, 18 cm, 22 cm a 26 cm, a to zaka\u017ed\u00fdm dvoma r\u00f4znymi sp\u00f4sobmi (t. j. tak, \u017ee \u017eiadne dva \u0161tvoruholn\u00edky neboli zhodn\u00e9). Nakreslite, ak\u00e9 \u0161tvoruholn\u00edky mohla Tereza zlo\u017ei\u0165.<\/p>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\u00e1)<\/div>\n<p><strong>Z5-I-3<\/strong><br \/>\n\u0160tefka rada oslavuje, tak\u017ee okrem naroden\u00edn vymyslela e\u0161te antinarodeniny: d\u00e1tum antinaroden\u00edn vznikne tak, \u017ee sa vymen\u00ed \u010d\u00edslo d\u0148a a \u010d\u00edslo mesiaca v d\u00e1tume narodenia. Sama sa narodila 8.11., tak\u017ee antinarodeniny m\u00e1 11.8. Jej mami\u010dka antinarodeniny oslavova\u0165 nem\u00f4\u017ee: narodila sa 23.7., jej antinarodeniny by mali by\u0165 7.23., \u010do ale nie je d\u00e1tum \u017eiadneho d\u0148a v roku. Jej brat s\u00edce antinarodeniny oslavova\u0165 m\u00f4\u017ee, ale m\u00e1 ich v ten ist\u00fd de\u0148 ako narodeniny: narodil sa 3.3. Ko\u013eko dn\u00ed v roku je tak\u00fdch, \u017ee \u010dlovek, ktor\u00fd sa toho d\u0148a narodil, m\u00f4\u017ee oslavova\u0165 svoje antinarodeniny, a to v in\u00fd de\u0148 ako svoje narodeniny?<\/p>\n<div class=\"pri\">(Veronika Huc\u00edkov\u00e1)<\/div>\n<p><strong>Z5-I-4<\/strong><br \/>\nV novej klubovni boli iba stoli\u010dky a st\u00f4l. Ka\u017ed\u00e1 stoli\u010dka mala \u0161tyri nohy, st\u00f4l bol trojnoh\u00fd. Do klubovne pri\u0161li skauti. Ka\u017ed\u00fd si sadol na svoju stoli\u010dku, dve stoli\u010dky zostali neobsaden\u00e9 a po\u010det n\u00f4h v miestnosti bol 101. Ur\u010dte, ko\u013eko stoli\u010diek bolo v klubovni.<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z5-I-5<\/strong><br \/>\nTom\u00e1\u0161 dostal dev\u00e4\u0165 karti\u010diek, na ktor\u00fdch boli nasleduj\u00face \u010d\u00edsla a matematick\u00e9 symboly:<\/p>\n<div class=\"pc\">18, 19, 20, 20, +, -, \u00d7, (, )<\/div>\n<p>Karti\u010dky ukladal tak, \u017ee ved\u013ea seba nikdy nele\u017eali dve karti\u010dky s \u010d\u00edslami, t. j. striedali sa karti\u010dky s \u010d\u00edslami a karti\u010dky so symbolmi. Takto vzniknut\u00e9 \u00falohy vypo\u010d\u00edtal a v\u00fdsledok si zap\u00edsal. Ur\u010dte, ak\u00fd najv\u00e4\u010d\u0161\u00ed v\u00fdsledok mohol Tom\u00e1\u0161 z\u00edska\u0165.<\/p>\n<div class=\"pri\">(Karel Pazourek)<\/div>\n<p><strong>Z5-I-6<\/strong><br \/>\nNa obr\u00e1zku je hrac\u00ed pl\u00e1nik a cesta, ktor\u00fa Juro zam\u00fd\u0161\u013eal prejs\u0165 z prav\u00e9ho doln\u00e9ho rohu do \u013eav\u00e9ho horn\u00e9ho. Potom zistil, \u017ee m\u00e1 pl\u00e1nik chybne pooto\u010den\u00fd, teda \u017ee by neza\u010d\u00ednal v pravom dolnom rohu. Tvar zam\u00fd\u0161\u013eanej cesty u\u017e ale nemohol zmeni\u0165 a musel ju prejs\u0165 pri spr\u00e1vnom nato\u010den\u00ed pl\u00e1nika. Pre ka\u017ed\u00e9 z troch mo\u017en\u00fdch nato\u010den\u00ed prekreslite uveden\u00fa cestu a ur\u010dte, ko\u013ek\u00fdmi siv\u00fdmi pol\u00ed\u010dkami t\u00e1to cesta prech\u00e1dza.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"68\/68z5a1.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 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=3051\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt52z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=3053\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt52r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z5-II-1<\/strong><br \/>\nPretekov chrtov sa z\u00fa\u010dastnilo 36 psov. Po\u010det psov, ktor\u00e9 do cie\u013ea dobehli pred Dun\u010dom, bol \u0161tyrikr\u00e1t men\u0161\u00ed ako po\u010det t\u00fdch, ktor\u00e9 dobehli za n\u00edm. Ko\u013ek\u00fd bol Dun\u010do?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z5-II-2<\/strong><br \/>\nAg\u00e1ta nap\u00edsala dvadsa\u0165cifern\u00e9 \u010d\u00edslo 12345678901234567890. Filip si z Ag\u00e1tinho \u010d\u00edsla vybral \u0161tyri cifry, ktor\u00e9 boli zap\u00edsan\u00e9 bezprostredne za sebou. Tieto \u0161tyri cifry sk\u00fa\u0161al zap\u00edsa\u0165 v r\u00f4znom porad\u00ed a nakoniec sa mu z nich podarilo zostavi\u0165 dvojicu dvojcifern\u00fdch \u010d\u00edsel, z ktor\u00fdch prv\u00e9 bolo o 1 v\u00e4\u010d\u0161ie ako druh\u00e9. Ktor\u00e9 dvojice \u010d\u00edsel mohol Filip zostavi\u0165? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\u00e1)<\/div>\n<p><strong>Z5-II-3<\/strong><br \/>\nAni\u010dka vyrazila z hotela na prech\u00e1dzku, i\u0161la 5 km na sever, potom 2 km na v\u00fdchod, 3 km na juh a nakoniec 4 km na z\u00e1pad. Tak do\u0161la k rybn\u00edku, kde sa ok\u00fapala. Vojto vy\u0161iel z kempu, i\u0161iel 3 km na juh, 4 km na z\u00e1pad a 1 km na sever. Tak do\u0161iel ku skale, ktor\u00e1 bola 5 km z\u00e1padne od rybn\u00edka, v ktorom sa k\u00fapala Ani\u010dka. Inokedy vy\u0161la Ani\u010dka zo svojho hotela a Vojto zo svojho kempu, obaja chceli d\u00f4js\u0165 k rybn\u00edku, v ktorom sa predt\u00fdm k\u00fapala Ani\u010dka, a obaja postupovali iba v smeroch \u0161tyroch svetov\u00fdch str\u00e1n. Ur\u010dte, ko\u013eko najmenej kilometrov musela prejs\u0165 Ani\u010dka a ko\u013eko Vojto.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"68\/68z5b1.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 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=3026\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt61z\" type=\"button\" value=\"Zadanie\" \/><\/a><\/form>\n<p><strong>Z6-I-1<\/strong><br \/>\nIvan a Mirka sa delili o hru\u0161ky v mise. Ivan si v\u017edy berie dve hru\u0161ky a Mirka polovicu toho, \u010do v mise ost\u00e1va. Takto postupne odoberali Ivan, Mirka, Ivan, Mirka a nakoniec Ivan, ktor\u00fd vzal posledn\u00e9 dve hru\u0161ky. Ur\u010dte, kto mal nakoniec viac hru\u0161iek a o ko\u013eko.<\/p>\n<div class=\"pri\">(Monika Dillingerov\u00e1)<\/div>\n<p><strong>Z6-I-2<\/strong><br \/>\nErnest si zo \u0161tvor\u010dekov\u00e9ho papiera vystrihol \u0161tvorec 4 \u00d7 4. Kristi\u00e1n v \u0148om vystrihol dve diery, pozri dva \u010dierne \u0161tvor\u010deky na obr\u00e1zku. Tento \u00fatvar sk\u00fa\u0161al Ernest rozstrihn\u00fa\u0165 pod\u013ea vyzna\u010den\u00fdch \u010diar na dve zhodn\u00e9 \u010dasti. N\u00e1jdite aspo\u0148 \u0161tyri r\u00f4zne sp\u00f4soby, ako to mohol Ernest spravi\u0165. (<em>Pritom dve strihania pova\u017eujte za r\u00f4zne, ak \u010dasti vzniknut\u00e9 jedn\u00fdm strihan\u00edm nie s\u00fa zhodn\u00e9 s \u010das\u0165ami vzniknut\u00fdmi druh\u00fdm strihan\u00edm.<\/em>)<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"68\/68z6a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Al\u017ebeta Bohinikov\u00e1)<\/div>\n<p><strong>Z6-I-3<\/strong><br \/>\nNa obr\u00e1zku s\u00fa nazna\u010den\u00e9 dva rady \u0161es\u0165uholn\u00edkov\u00fdch pol\u00ed\u010dok, ktor\u00e9 doprava pokra\u010duj\u00fa bez obmedzenia. Do ka\u017ed\u00e9ho pol\u00ed\u010dka doplnte jedno kladn\u00e9 cel\u00e9 \u010d\u00edslo tak, aby s\u00fa\u010din \u010d\u00edsel v \u013eubovo\u013en\u00fdch troch susediacich pol\u00ed\u010dkach bol 2018. Ur\u010dte \u010d\u00edslo, ktor\u00e9 bude v 2019. pol\u00ed\u010dku v hornom rade.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"68\/68z6a2.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\u00e1)<\/div>\n<p><strong>Z6-I-4<\/strong><br \/>\nP\u00e1n Tich\u00e1\u010dek mal na z\u00e1hrade troch sadrov\u00fdch trpasl\u00edkov: najv\u00e4\u010d\u0161ieho volal Ma\u0161ko, prostredn\u00e9ho Jarko a najmen\u0161ieho Fanko. Ke\u010f\u017ee sa s nimi r\u00e1d hr\u00e1val, \u010dasom zistil, \u017ee ke\u010f postav\u00ed Fanka na Jarka, s\u00fa rovnako vysok\u00ed ako Ma\u0161ko. Ke\u010f naopak postav\u00ed Fanka na Ma\u0161ka, meraj\u00fa spolu o 34 cm viac ako Jarko. A ke\u010f postav\u00ed na Ma\u0161ka Jarka, s\u00fa o 72 cm vy\u0161\u0161\u00ed ako Fanko. Ako vysok\u00ed s\u00fa trpasl\u00edci p\u00e1na Tich\u00e1\u010dka?<\/p>\n<div class=\"pri\">(Michaela Petrov\u00e1)<\/div>\n<p><strong>Z6-I-5<\/strong><br \/>\nV nasleduj\u00facom pr\u00edklade na s4\u00edtanie predstavuj\u00fa rovnak\u00e9 p\u00edsmen\u00e1 rovnak\u00e9 cifry, r\u00f4zne p\u00edsmen\u00e1 r\u00f4zne cifry:<\/p>\n<div class=\"pc\">\n<table style=\"border-collapse: collapse;\" border=\"0\" cellspacing=\"0\" cellpadding=\"0\" align=\"center\">\n<tbody>\n<tr align=\"center\">\n<td width=\"20\">R<\/td>\n<td width=\"20\">A<\/td>\n<td width=\"20\">T<\/td>\n<td width=\"20\">A<\/td>\n<td width=\"20\">M<\/td>\n<\/tr>\n<tr align=\"center\">\n<td style=\"border-bottom: 1pt solid black;\">\u00a0<\/td>\n<td style=\"border-bottom: 1pt solid black;\">\u00a0<\/td>\n<td style=\"border-bottom: 1pt solid black;\">R<\/td>\n<td style=\"border-bottom: 1pt solid black;\">A<\/td>\n<td style=\"border-bottom: 1pt solid black;\">D<\/td>\n<\/tr>\n<tr align=\"center\">\n<td>U<\/td>\n<td>L<\/td>\n<td>O<\/td>\n<td>H<\/td>\n<td>Y<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Nahra\u010fte p\u00edsmen\u00e1 ciframi tak, aby bol pr\u00edklad spr\u00e1vne. N\u00e1jdite dve r\u00f4zne nahradenia.<\/p>\n<div class=\"pri\">(Erika Novotn\u00e1)<\/div>\n<p><strong>Z6-I-6<\/strong><br \/>\nV dvan\u00e1s\u0165uholn\u00edku <em>ABCDEFGHIJKL<\/em> s\u00fa ka\u017ed\u00e9 dve susedn\u00e9 strany navz\u00e1jom kolm\u00e9 a v\u0161etky strany s v\u00fdnimkou str\u00e1n <em>AL<\/em> a <em>GF<\/em> s\u00fa navz\u00e1jom zhodn\u00e9. Strany <em>AL<\/em> a <em>GF<\/em> s\u00fa oproti ostatn\u00fdm stran\u00e1m dvojn\u00e1sobne dlh\u00e9. \u00dase\u010dky <em>BG<\/em> a <em>EL<\/em> sa pret\u00ednaj\u00fa v bode <em>M<\/em> a rozde\u013euj\u00fa dvan\u00e1s\u0165uholn\u00edk na \u0161es\u0165 \u00fatvarov (tri trojuholn\u00edky, dva \u0161tvoruholn\u00edky a jeden p\u00e4\u0165uholn\u00edk). \u0160tvoruholn\u00edk <em>EFGM<\/em> m\u00e1 obsah 7 cm<sup>2<\/sup>. Ur\u010dte obsahy ostatn\u00fdch piatich \u00fatvarov.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"68\/68z6a4.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 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=3130\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt62z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=3127\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt62r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z6-II-1<\/strong><br \/>\nJano, Dano, Anna a Hana maj\u00fa ka\u017ed\u00fd svoje ob\u013e\u00faben\u00e9 \u010d\u00edslo. Vieme, \u017ee<\/p>\n<ul>\n<li>ke\u010f s\u010d\u00edtame Danovo \u010d\u00edslo a trojn\u00e1sobok Janovho \u010d\u00edsla, vyjde nep\u00e1rne \u010d\u00edslo,<\/li>\n<li>ke\u010f od\u010d\u00edtame od seba Annino a Hanino \u010d\u00edslo a tento v\u00fdsledok vyn\u00e1sob\u00edme piatimi, vyjde nep\u00e1rne \u010d\u00edslo,<\/li>\n<li>ke\u010f vyn\u00e1sob\u00edme Danovo \u010d\u00edslo Hanin\u00fdm \u010d\u00edslom a k v\u00fdsledku pripo\u010d\u00edtame 17, vyjde p\u00e1rne \u010d\u00edslo.<\/li>\n<\/ul>\n<p>Ur\u010dte, \u010die ob\u013e\u00faben\u00e9 \u010d\u00edsla s\u00fa nep\u00e1rne a \u010die p\u00e1rne.<\/p>\n<div class=\"pri\"> (Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z6-II-2<\/strong><br \/>\nAni\u010dka m\u00e1 v kasi\u010dke na\u0161poren\u00fdch 290 minc\u00ed, a to jednoeurovky a dvojeurovky. Ke\u010f pou\u017eije \u0161tvrtinu v\u0161etk\u00fdch dvojeuroviek, zlo\u017e\u00ed rovnak\u00fa sumu, ako ke\u010f pou\u017eije tretinu v\u0161etk\u00fdch jednoeuroviek. Ak\u00fa sumu m\u00e1 Ani\u010dka na\u0161poren\u00fa?<\/p>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\u00e1)<\/div>\n<p><strong>Z6-II-3<\/strong><br \/>\n\u00datvar na obr\u00e1zku je zlo\u017een\u00fd z piatich zhodn\u00fdch \u0161tvorcov a je rozdelen\u00fd \u00fase\u010dkami na tri farebne odl\u00ed\u0161en\u00e9 \u010dasti. Obsah sivej \u010dasti je o 0,6 cm<sup>2<\/sup> v\u00e4\u010d\u0161\u00ed ako obsah bielej \u010dasti. Ur\u010dte obsah cel\u00e9ho \u00fatvaru.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"68\/68z6b1.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 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=3025\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt71z\" type=\"button\" value=\"Zadanie\" \/><\/a><input name=\"butt71r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z7-I-1<\/strong><br \/>\nNa ka\u017edej z troch karti\u010diek je nap\u00edsan\u00e1 jedna cifra r\u00f4zna od nuly (na r\u00f4znych karti\u010dk\u00e1ch nie s\u00fa nutne r\u00f4zne cifry). Vieme, \u00bee ak\u00e9ko\u013evek trojcifern\u00e9 \u010d\u00edslo zlo\u017een\u00e9 z t\u00fdchto karti\u010diek je delite\u013en\u00e9 \u0161iestimi. Navy\u0161e mo\u017eno z t\u00fdchto karti\u010diek zlo\u017ei\u0165 trojcifern\u00e9 \u010d\u00edslo delite\u013en\u00e9 jeden\u00e1stimi. Ak\u00e9 cifry m\u00f4\u017eu by\u0165 na karti\u010dk\u00e1ch? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Veronika Huc\u00edkov\u00e1)<\/div>\n<p><strong>Z7-I-2<\/strong><br \/>\nV dvan\u00e1s\u0165uholn\u00edku <em>ABCDEFGHIJKL<\/em> s\u00fa ka\u017ed\u00e9 dve susedn\u00e9 strany navz\u00e1jom kolm\u00e9 a v\u0161etky strany s v\u00fdnimkou str\u00e1n <em>AL<\/em> a <em>GF<\/em> s\u00fa navz\u00e1jom zhodn\u00e9. Strany <em>AL<\/em> a <em>GF<\/em> s\u00fa oproti ostatn\u00fdm stran\u00e1m dvojn\u00e1sobne dlh\u00e9. \u00dase\u010dky <em>BG<\/em> a <em>EL<\/em> sa pret\u00ednaj\u00fa v bode <em>M<\/em>. \u0160tvoruholn\u00edk <em>ABMJ<\/em> m\u00e1 obsah 1,8 cm<sup>2<\/sup>. Ur\u010dte obsah \u0161tvoruholn\u00edka <em>EFGM<\/em>.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"68\/68z7a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z7-I-3<\/strong><br \/>\nDedo pripravil pre svojich \u0161es\u0165 vn\u00fa\u010dat k\u00f4pku lieskov\u00fdch orie\u0161kov s t\u00fdm, nech si ich nejako rozober\u00fa. Prv\u00fd pri\u0161iel Adam, odpo\u010d\u00edtal si polovicu, pribral si e\u0161te jeden orie\u0161ok a odi\u0161iel. Rovnako sa zachoval druh\u00fd Bob, tret\u00ed Cyril, \u0161tvrt\u00fd Dano aj piaty Edo. Iba Ferko smutne h\u013eadel na pr\u00e1zdny st\u00f4l, u\u017e pre neho \u017eiadny orie\u0161ok nezv\u00fd\u0161il. Ko\u013eko orie\u0161kov bolo p\u00f4vodne na k\u00f4pke?<\/p>\n<div class=\"pri\">(Marta Volfov\u00e1)<\/div>\n<p><strong>Z7-I-4<\/strong><br \/>\nBetka sa hrala s ozuben\u00fdmi kolesami, ktor\u00e9 ukladala tak, ako je nazna\u010den\u00e9 na obr\u00e1zku. Ke\u010f potom zato\u010dila jedn\u00fdm okolo, to\u010dili sa v\u0161etky ostatn\u00e9. Nakoniec bola spokojn\u00e1 so s\u00fakoles\u00edm, pri\u010dom prv\u00e9 koleso malo 32 a druh\u00e9 24 zubov. Ke\u010f sa tretie koleso oto\u010dilo presne osemkr\u00e1t, druh\u00e9 koleso urobilo p\u00e4\u0165 ot\u00e1\u010dok a \u010das\u0165 \u0161iestej a prv\u00e9 koleso urobilo \u0161tyri ot\u00e1\u010dky a \u010das\u0165 piatej. Zistite, ko\u013eko zubov malo tretie koleso.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"68\/68z7a2.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Erika Novotn\u00e1)<\/div>\n<p><strong>Z7-I-5<\/strong><br \/>\nV z\u00e1hradn\u00edctve Rose si jedna predaj\u0148a objednala celkom 120 ru\u017e\u00ed vo farbe \u010dervenej a \u017eltej, druh\u00e1 predaj\u0148a celkom 105 ru\u017e\u00ed vo farbe \u010dervenej a bielej a tretia predaj\u0148a celkom 45 ru\u017e\u00ed vo farbe \u017eltej a bielej. Z\u00e1hradn\u00edctvo z\u00e1kazku splnilo, a to tak, \u017ee ru\u017e\u00ed rovnakej farby dodalo do ka\u017ed\u00e9ho obchodu rovnako. Ko\u013eko celkom \u010derven\u00fdch, ko\u013eko bielych a ko\u013eko \u017elt\u00fdch ru\u017e\u00ed dodalo z\u00e1hradn\u00edctvo do t\u00fdchto troch predajn\u00ed?<\/p>\n<div class=\"pri\">(Marta Volfov\u00e1)<\/div>\n<p><strong>Z7-I-6<\/strong><br \/>\nDan\u00fd je rovnoramenn\u00fd pravouhl\u00fd trojuholn\u00edk <em>ABS<\/em> so z\u00e1klad\u0148ou <em>AB<\/em>. Na kru\u017enici, ktor\u00e1 m\u00e1 stred v bode <em>S<\/em> a prech\u00e1dza bodmi <em>A<\/em> a <em>B<\/em>, le\u017e\u00ed bod <em>C<\/em> tak, \u017ee trojuholn\u00edk <em>ABC<\/em> je rovnoramenn\u00fd. Ur\u010dte, ko\u013eko bodov <em>C<\/em> vyhovuje uveden\u00fdm podmienkam, a v\u0161etky tak\u00e9 body zostrojte.<\/p>\n<div class=\"pri\">(Karol 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=3131\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt72z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=3128\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt72r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z7-II-1<\/strong><br \/>\nMajka, Va\u0161o a Zuzka po\u010d\u00edtali cez v\u00edkend \u00falohy. Majka a Va\u0161o vypo\u010d\u00edtali dokopy 25 \u00faloh. Zuzka a Va\u0161o vypo\u010d\u00edtali dokopy 32 \u00faloh. Pritom Zuzka vypo\u010d\u00edtala dvakr\u00e1t viac \u00faloh ako Majka. Ko\u013eko \u00faloh vypo\u010d\u00edtal Va\u0161o?<\/p>\n<div class=\"pri\"> (Monika Dillingerov\u00e1)<\/div>\n<p><strong>Z7-II-2<\/strong><br \/>\nMedzi cifry \u010d\u00edsla 2019 vlo\u017ete dve cifry tak, aby vzniknut\u00e9 \u0161es\u0165cifern\u00e9 \u010d\u00edslo<\/p>\n<ul>\n<li>za\u010d\u00ednalo 2 a kon\u010dilo 9,<\/li>\n<li>bolo zlo\u017een\u00e9 zo \u0161iestich r\u00f4znych cifier,<\/li>\n<li>bolo delite\u013en\u00e9 tromi,<\/li>\n<li>jeho prv\u00e9 troj\u010d\u00edslie bolo delite\u013en\u00e9 tromi,<\/li>\n<li>jeho prv\u00e9 \u0161tvor\u010d\u00edslie bolo delite\u013en\u00e9 \u0161tyrmi,<\/li>\n<li>s\u00fa\u010det vlo\u017een\u00fdch cifier bol nep\u00e1rny.<\/li>\n<\/ul>\n<p>Ur\u010dte rozdiel najv\u00e4\u010d\u0161ieho a najmen\u0161ieho \u0161es\u0165cifern\u00e9ho \u010d\u00edsla s uveden\u00fdmi vlastnos\u0165ami.<\/p>\n<div class=\"pri\"> (Lucie R\u016f\u017ei\u010dkov\u00e1)<\/div>\n<p><strong>Z7-II-3<\/strong><br \/>\n\u00datvar na obr\u00e1zku je zlo\u017een\u00fd z \u00f4smich zhodn\u00fdch \u0161tvorcov a je rozdelen\u00fd \u00fase\u010dkami na p\u00e4\u0165 farebne odl\u00ed\u0161en\u00fdch \u010dast\u00ed. Pritom bod <em>X<\/em> je stredom \u00fase\u010dky <em>KJ<\/em>, bod <em>Y<\/em> je stredom \u00fase\u010dky <em>EX<\/em> a \u00fase\u010dka <em>BZ<\/em> je zhodn\u00e1 s <em>BC<\/em>. Obsah \u010diernej \u010dasti \u00fatvaru je 7,5 cm<sup>2<\/sup>. Ur\u010dte obsahy zvy\u0161n\u00fdch \u0161tyroch \u010dast\u00ed.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"68\/68z7b1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1, Monika Dillingerov\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=3024\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt81z\" type=\"button\" value=\"Zadanie\" \/><\/a><input name=\"butt81r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z8-I-1<\/strong><br \/>\nFero a D\u00e1vid sa denne stret\u00e1vaj\u00fa vo v\u00fd\u0165ahu. Raz r\u00e1no zistili, \u017ee ke\u010f vyn\u00e1sobia svoje s\u00fa\u010dasn\u00e9 veky, dostan\u00fa 238. Keby to ist\u00e9 urobili za \u0161tyri roky, bol by tento s\u00fa\u010din 378. Ur\u010dte s\u00fa\u010det s\u00fa\u010dasn\u00fdch vekov Fera a D\u00e1vida.<\/p>\n<div class=\"pri\">(Michaela Petrov\u00e1)<\/div>\n<p><strong>Z8-I-2<\/strong><br \/>\nDo triedy pribudol nov\u00fd \u017eiak, o ktorom sa vedelo, \u017ee okrem angli\u010dtiny vie v\u00fdborne e\u0161te jeden cudz\u00ed jazyk. Traja spolu\u017eiaci sa dohadovali, ktor\u00fd jazyk to je.<br \/>\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Prv\u00fd s\u00fadil: <em>&#8222;Franc\u00faz\u0161tina to nie je.&#8220;<\/em><br \/>\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Druh\u00fd h\u00e1dal: <em>&#8222;Je to \u0161paniel\u010dina alebo nem\u010dina.&#8220;<\/em><br \/>\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Tret\u00ed usudzoval: <em>&#8222;Je to \u0161paniel\u010dina.&#8220;<\/em><br \/>\nVz\u00e1p\u00e4t\u00ed sa dozvedeli, \u017ee aspo\u0148 jeden z nich h\u00e1dal spr\u00e1vne a aspo\u0148 jeden nespr\u00e1vne. Ur\u010dte, ktor\u00fd z menovan\u00fdch jazykov nov\u00fd \u017eiak ovl\u00e1dal.<\/p>\n<div class=\"pri\">(Marta Volfov\u00e1)<\/div>\n<p><strong>Z8-I-3<\/strong><br \/>\nPeter narysoval pravideln\u00fd \u0161es\u0165uholn\u00edk, ktor\u00e9ho vrcholy le\u017eali na kru\u017enici d\u013a\u017eky 16 cm. Potom pre ka\u017ed\u00fd vrchol tohto \u0161es\u0165uholn\u00edka narysoval kru\u017enicu so stredom v tomto vrchole, ktor\u00e1 prech\u00e1dzala jeho dvoma susedn\u00fdmi vrcholmi. Vznikol tak \u00fatvar ako na obr\u00e1zku. Ur\u010dte obvod vyzna\u010den\u00e9ho kvietka.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"68\/68z8a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Erika Novotn\u00e1)<\/div>\n<p><strong>Z8-I-4<\/strong><br \/>\nNa \u0161tyroch karti\u010dk\u00e1ch boli \u0161tyri r\u00f4zne cifry, z ktor\u00fdch jedna bola nula. Vojto z karti\u010diek zlo\u017eil \u010do najv\u00e4\u010d\u0161ie \u0161tvorcifern\u00e9 \u010d\u00edslo, Martin potom \u010do najmen\u0161ie \u0161tvorcifern\u00e9 \u010d\u00edslo. Adam zap\u00edsal na tabu\u013eu rozdiel Vojtovho a Martinovho \u010d\u00edsla. Potom Vojto z karti\u010diek zlo\u017eil \u010do najv\u00e4\u010d\u0161ie trojcifern\u00e9 \u010d\u00edslo a Martin \u010do najmen\u0161ie trojcifern\u00e9 \u010d\u00edslo. Adam op\u00e4\u0165 zap\u00edsal na tabu\u013eu rozdiel Vojtovho a Martinovho \u010d\u00edsla. Potom Vojto s Martinom podobne zlo\u017eili dvojcifern\u00e9 \u010d\u00edsla a Adam zap\u00edsal na tabu\u013eu ich rozdiel. Nakoniec Vojto vybral \u010do najv\u00e4\u010d\u0161ie jednocifern\u00e9 \u010d\u00edslo a Martin \u010do najmen\u0161ie nenulov\u00e9 jednocifern\u00e9 \u010d\u00edslo a Adam zap\u00edsal ich rozdiel. Ke\u010f Adam s\u010d\u00edtal v\u0161etky \u0161tyri rozdiely na tabuli, vy\u0161lo mu 9090. Ur\u010dte \u0161tyri cifry na karti\u010dk\u00e1ch.<\/p>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\u00e1)<\/div>\n<p><strong>Z8-I-5<\/strong><br \/>\nKr\u00e1\u013e dal mur\u00e1rovi V\u00e1clavovi za \u00falohu postavi\u0165 m\u00far siln\u00fd 25 cm, dlh\u00fd 50 m a vysok\u00fd 2 m. Ak by V\u00e1clav pracoval bez prest\u00e1vky a rovnak\u00fdm tempom, postavil by m\u00far za 26 hod\u00edn. Pod\u013ea platn\u00fdch kr\u00e1\u013eovsk\u00fdch nariaden\u00ed v\u0161ak mus\u00ed V\u00e1clav dodr\u017eiava\u0165 nasleduj\u00face podmienky:<\/p>\n<ul>\n<li>Po\u010das pr\u00e1ce mus\u00ed spravi\u0165 pr\u00e1ve \u0161es\u0165 polhodinov\u00fdch prest\u00e1vok.<\/li>\n<li>Na za\u010diatku pr\u00e1ce a po ka\u017edej polhodinovej prest\u00e1vke, ke\u010f je dostato\u010dne odd\u00fdchnut\u00fd, m\u00f4\u017ee pracova\u0165 o \u0161tvrtinu r\u00fdchlej\u0161ie ako norm\u00e1lnym tempom, ale nie dlh\u0161ie ako jednu hodinu.<\/li>\n<li>Medzi prest\u00e1vkami mus\u00ed pracova\u0165 najmenej 3\/4 hodiny.<\/li>\n<\/ul>\n<p>Za ak\u00fd najkrat\u0161\u00ed \u010das m\u00f4\u017ee V\u00e1clav splni\u0165 zadan\u00fa \u00falohu?<\/p>\n<div class=\"pri\">(Jakub Norek)<\/div>\n<p><strong>Z8-I-6<\/strong><br \/>\nV lichobe\u017en\u00edku <em>KLMN<\/em> m\u00e1 z\u00e1klad\u0148a <em>KL<\/em> ve\u013ekos\u0165 40 cm a z\u00e1klad\u0148a <em>MN<\/em> m\u00e1 ve\u013ekos\u0165 16 cm. Bod <em>P<\/em> le\u017e\u00ed na \u00fase\u010dke <em>KL<\/em> tak, \u017ee \u00fase\u010dka <em>NP<\/em> rozde\u013euje lichobe\u017en\u00edk na dve \u010dasti s rovnak\u00fdmi obsahmi. Ur\u010dte ve\u013ekos\u0165 \u00fase\u010dky <em>KP<\/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=\"http:\/\/skmo.sk\/dokument.php?id=3132\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt82z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=3129\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt82r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z8-II-1<\/strong><br \/>\nV dvojposchodovom dome, ktor\u00fd m\u00e1 obytn\u00fa \u010das\u0165 okrem 1. a 2. poschodia aj na pr\u00edzem\u00ed, b\u00fdva 35 \u013eud\u00ed nad niek\u00fdm a 45 \u013eud\u00ed b\u00fdva pod niek\u00fdm. Pritom na 1. poschod\u00ed b\u00fdva jedna tretina v\u0161etk\u00fdch os\u00f4b \u017eij\u00facich v dome. Ko\u013eko os\u00f4b b\u00fdva v dome celkom?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z8-II-2<\/strong><br \/>\nPre ko\u013eko kladn\u00fdch cel\u00fdch \u010d\u00edsel men\u0161\u00edch ako 1000 plat\u00ed, \u017ee medzi \u010d\u00edslami 2, 3, 4, 5, 6, 7, 8 a 9 je pr\u00e1ve jedno, ktor\u00e9 nie je jeho delite\u013eom?<\/p>\n<div class=\"pri\"> (Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z8-II-3<\/strong><br \/>\nV lichobe\u017en\u00edku <em>ABCD<\/em> so z\u00e1klad\u0148ami <em>AB<\/em> a <em>CD<\/em> plat\u00ed, \u017ee |<em>AD<\/em>| = |<em>CD<\/em>|, |<em>AB<\/em>| = 2|<em>CD<\/em>|, |<em>BC<\/em>| = 24 cm a |<em>AC<\/em>| = 10 cm. Vypo\u010d\u00edtajte obsah lichobe\u017en\u00edka <em>ABCD<\/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=3023\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt91z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=3048\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt91r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-I-1<\/strong><br \/>\nN\u00e1jdite v\u0161etky kladn\u00e9 cel\u00e9 \u010d\u00edsla x a y, pre ktor\u00e9 plat\u00ed<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"68\/68z9a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Al\u017ebeta Bohinikov\u00e1)<\/div>\n<p><strong>Z9-I-2<\/strong><br \/>\nV rovnostrannom trojuholn\u00edku <em>ABC<\/em> je <em>K<\/em> stredom strany <em>AB<\/em>, bod <em>L<\/em> le\u017e\u00ed v tretine strany <em>BC<\/em> bli\u017e\u0161ie bodu <em>C<\/em> a bod <em>M<\/em> le\u017e\u00ed v tretine strany <em>AC<\/em> bli\u017e\u0161ie bodu <em>A<\/em>. Ur\u010dte, ak\u00fa \u010das\u0165 obsahu trojuholn\u00edka <em>ABC<\/em> zaber\u00e1 trojuholn\u00edk <em>KLM<\/em>.<\/p>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\u00e1)<\/div>\n<p><strong>Z9-I-3<\/strong><br \/>\nV na\u0161om meste s\u00fa \u0161tyri kin\u00e1, ktor\u00fdm sa hovor\u00ed pod\u013ea svetov\u00fdch str\u00e1n. O ich otv\u00e1rac\u00edch hodin\u00e1ch je zn\u00e1me, \u017ee:<\/p>\n<ul>\n<li>ka\u017ed\u00fd de\u0148 je otvoren\u00e9 aspo\u0148 jedno kino,<\/li>\n<li>ak je otvoren\u00e9 ju\u017en\u00e9 kino, tak nie je otvoren\u00e9 severn\u00e9 kino,<\/li>\n<li>nikdy nie je otvoren\u00e9 s\u00fa\u010dasne severn\u00e9 a v\u00fdchodn\u00e9 kino,<\/li>\n<li>ak je otvoren\u00e9 v\u00fdchodn\u00e9 kino, tak je otvoren\u00e9 aj ju\u017en\u00e9 alebo severn\u00e9 kino (alebo obe).<\/li>\n<\/ul>\n<p>Vydali sme sa do ju\u017en\u00e9ho kina a zistili sme, \u017ee je zatvoren\u00e9. Ktor\u00e9 zo zvy\u0161n\u00fdch k\u00edn je ur\u010dite otvoren\u00e9?<\/p>\n<div class=\"pri\">(Monika Dillingerov\u00e1)<\/div>\n<p><strong>Z9-I-4<\/strong><br \/>\nHotelier chcel vybavi\u0165 jed\u00e1le\u0148 nov\u00fdmi stoli\u010dkami. V katal\u00f3gu si vybral typ stoli\u010dky. A\u017e pri zad\u00e1van\u00ed objedn\u00e1vky sa od v\u00fdrobcu dozvedel, \u017ee v r\u00e1mci z\u013eavovej akcie pon\u00fakaj\u00fa ka\u017ed\u00fa \u0161tvrt\u00fa stoli\u010dku za polovi\u010dn\u00fa cenu a \u017ee teda oproti pl\u00e1nu m\u00f4\u017ee u\u0161etri\u0165 za sedem a pol stoli\u010dky. Hotelier si spo\u010d\u00edtal, \u017ee za p\u00f4vodne pl\u00e1novan\u00fa \u010diastku m\u00f4\u017ee zaobstara\u0165 o dev\u00e4\u0165 stoli\u010diek viac, ako zam\u00fd\u0161\u013eal. Ko\u013eko stoli\u010diek chcel hotelier p\u00f4vodne k\u00fapi\u0165?<\/p>\n<div class=\"pri\">(Libor \u0160im\u016fnek)<\/div>\n<p><strong>Z9-I-5<\/strong><br \/>\nAdam a Eva vytv\u00e1rali dekor\u00e1cie z navz\u00e1jom zhodn\u00fdch bielych kruhov. Adam pou\u017eil \u0161tyri kruhy, ktor\u00e9 polo\u017eil tak, \u017ee sa ka\u017ed\u00fd dot\u00fdkal dvoch in\u00fdch kruhov. Medzi ne potom vlo\u017eil in\u00fd kruh, ktor\u00fd sa dot\u00fdkal v\u0161etk\u00fdch \u0161tyroch bielych kruhov, a ten vyfarbil \u010dervenou. Eva pou\u017eila tri kruhy, ktor\u00e9 polo\u017eila tak, \u017ee sa dot\u00fdkali navz\u00e1jom. Medzi ne potom vlo\u017eila in\u00fd kruh, ktor\u00fd sa dot\u00fdkal v\u0161etk\u00fdch troch bielych kruhov, a ten vyfarbila zelenou. Eva si v\u0161imla, \u017ee jej zelen\u00fd kruh a Adamov \u010derven\u00fd kruh s\u00fa r\u00f4zne ve\u013ek\u00e9, a za\u010dali spolu zis\u0165ova\u0165, ako sa l\u00ed\u0161ia. Vyjadrite polomery \u010derven\u00e9ho a zelen\u00e9ho kruhu v\u0161eobecne pomocou polomeru bielych kruhov.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"68\/68z9a2.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Marie Krej\u010dov\u00e1)<\/div>\n<p><strong>Z9-I-6<\/strong><br \/>\nPrirodzen\u00e9 \u010d\u00edslo <em>N<\/em> nazveme <em>bombastick\u00e9<\/em>, ak neobsahuje vo svojom z\u00e1pise \u017eiadnu nulu a ak \u017eiadne men\u0161ie prirodzen\u00e9 \u010d\u00edslo nem\u00e1 rovnak\u00fd s\u00fa\u010din cifier ako \u010d\u00edslo <em>N<\/em>. Karol sa najsk\u00f4r zauj\u00edmal o <em>bombastick\u00e9<\/em> prvo\u010d\u00edsla a tvrdil, \u017ee ich nie je ve\u013ea. Vyp\u00ed\u0161te v\u0161etky dvojcifern\u00e9 <em>bombastick\u00e9<\/em> prvo\u010d\u00edsla. Potom Karol zvolil jedno <em>bombastick\u00e9<\/em> \u010d\u00edslo a prezradil n\u00e1m, \u017ee obsahuje cifru 3 a \u017ee iba jedna z jeho \u010fal\u0161\u00edch cifier je p\u00e1rna. O ktor\u00fa p\u00e1rnu cifru sa mohlo jedna\u0165?<\/p>\n<div class=\"pri\">(Michal Rol\u00ednek)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"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=3050\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt92z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=3052\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt92r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-II-1<\/strong><br \/>\nMarienka nap\u00edsala na tabu\u013eu dve r\u00f4zne prirodzen\u00e9 \u010d\u00edsla. Marta si vzala karti\u010dku, na ktorej jednu stranu nap\u00edsala s\u00fa\u010det Marienkin\u00fdch \u010d\u00edsel a na druh\u00fa stranu ich s\u00fa\u010din. Jedno z \u010d\u00edsel na karti\u010dke bolo prvo\u010d\u00edslo a s\u00fa\u010det \u010d\u00edsel z oboch str\u00e1n karti\u010dky bol 97. Ktor\u00e9 \u010d\u00edsla nap\u00edsala Marienka na tabu\u013eu?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z9-II-2<\/strong><br \/>\nM\u00e1me kv\u00e1der, ktor\u00e9ho jedna hrana je p\u00e4\u0165kr\u00e1t dlh\u0161ia ako in\u00e1 jeho hrana. Keby sme v\u00fd\u0161ku kv\u00e1dra zv\u00e4\u010d\u0161ili o 2 cm, zv\u00e4\u010d\u0161il by sa jeho objem o 90 cm<sup>3<\/sup>. Keby sme v\u00fd\u0161ku takto zv\u00e4\u010d\u0161en\u00e9ho kv\u00e1dra zmenili na polovicu, bol by objem nov\u00e9ho kv\u00e1dra rovn\u00fd trom p\u00e4tin\u00e1m objemu p\u00f4vodn\u00e9ho kv\u00e1dra. Ak\u00e9 m\u00f4\u017eu by\u0165 rozmery p\u00f4vodn\u00e9ho kv\u00e1dra? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z9-II-3<\/strong><br \/>\nZ cifier 3, 4, 5, 7 a 9 boli vytvoren\u00e9 v\u0161etky mo\u017en\u00e9 trojcifern\u00e9 \u010d\u00edsla tak, aby sa ka\u017ed\u00e1 cifra vyskytovala v ka\u017edom \u010d\u00edsle nanajv\u00fd\u0161 raz. Ur\u010dte po\u010det takto vzniknut\u00fdch \u010d\u00edsel a ich celkov\u00fd s\u00fa\u010det.<\/p>\n<div class=\"pri\">(Marta Volfov\u00e1)<\/div>\n<p><strong>Z9-II-4<\/strong><br \/>\nV pravouhlom trojuholn\u00edku je polomer jemu op\u00edsanej kru\u017enice 14,5 cm a polomer jemu vp\u00edsanej kru\u017enice 6 cm. Ur\u010dte obvod tohto trojuholn\u00edka.<\/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=\"http:\/\/skmo.sk\/dokument.php?id=3116\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt93z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=3117\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt93r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-III-1<\/strong><br \/>\nAdela nap\u00edsala na tabu\u013eu dve kladn\u00e9 cel\u00e9 \u010d\u00edsla a dala Luk\u00e1\u0161ovi a Petrovi za \u00falohu ur\u010di\u0165 kladn\u00fd rozdiel druh\u00fdch mocn\u00edn t\u00fdchto dvoch \u010d\u00edsel. Luk\u00e1\u0161 namiesto toho ur\u010dil druh\u00fa mocninu rozdielu dan\u00fdch dvoch \u010d\u00edsel. Vy\u0161lo mu tak \u010d\u00edslo o 4 038 men\u0161ie ako Petrovi, ktor\u00fd v\u00fdpo\u010det vykonal spr\u00e1vne. Ktor\u00e9 dve \u010d\u00edsla mohla Adela nap\u00edsa\u0165 na tabu\u013eu? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\u00e1)<\/div>\n<p><strong>Z9-III-2<\/strong><br \/>\nNa ostrove \u017eij\u00fa dva druhy domorodcov: Poctivci, ktor\u00ed v\u017edy hovoria pravdu, a Klam\u00e1ri, ktor\u00ed v\u017edy klam\u00fa. Ke\u010f cudzinec stretol troch domorodcov, Alana, Bruna a Ctibora, sp\u00fdtal sa ich, do ktorej skupiny patria.<\/p>\n<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Alan ozn\u00e1mil: \u201dBruno je Klam\u00e1r.\u201c<br \/>\n&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Bruno povedal: \u201dAlan a Ctibor s\u00fa bu\u010f obaja Klam\u00e1ri, alebo obaja Poctivci.\u201c<br \/>\n&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Ctibor sa nevyjadril.<\/p>\n<p>Mohol cudzinec pri niektorom z t\u00fdchto domorodcov s istotou ur\u010di\u0165, \u010di je Poctivec, alebo Klam\u00e1r?<\/p>\n<div class=\"pri\">(Marta Volfov\u00e1)<\/div>\n<p><strong>Z9-III-3<\/strong><br \/>\nKe\u010f \u010d\u00edslo <em>X<\/em> vydel\u00edm \u010d\u00edslom <em>Y<\/em>, dostanem \u010d\u00edslo <em>Z<\/em> a zvy\u0161ok 27. Ke\u010f \u010d\u00edslo <em>X<\/em> vydel\u00edm \u010d\u00edslom <em>Z<\/em>, dostanem \u010d\u00edslo <em>1,1 \u00b7 Y<\/em> a zvy\u0161ok 3. Ktor\u00e9 \u010d\u00edsla <em>X<\/em>, <em>Y<\/em>, <em>Z<\/em> vyhovuj\u00fa uveden\u00fdm podmienkam? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z9-III-4<\/strong><br \/>\nDan\u00e1 je kru\u017enica so stredom <em>S<\/em> a polomerom 39 mm. Do kru\u017enice m\u00e1me vp\u00edsa\u0165 trojuholn\u00edk <em>ABC<\/em> tak, aby ve\u013ekos\u0165 strany <em>AC<\/em> bola 72 mm a bod <em>B<\/em> le\u017eal v polrovine ur\u010denej priamkou <em>AC<\/em> a bodom <em>S<\/em>. Zo zadan\u00fdch \u00fadajov vypo\u010d\u00edtajte, ak\u00fa ve\u013ekos\u0165 m\u00e1 ma\u0165 v\u00fd\u0161ka trojuholn\u00edka <em>ABC<\/em> z vrcholu <em>B<\/em>, aby \u00faloha mala dve rie\u0161enia.<\/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 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>68. ro\u010dn\u00edk matematickej olympi\u00e1dy 2018-2019 S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9 aktualizovan\u00e9 30.1.2020 10:25<\/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":[26],"tags":[],"class_list":["post-1306","post","type-post","status-publish","format-standard","hentry","category-68-rocnik"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.6 - 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