{"id":962,"date":"2014-09-07T13:24:49","date_gmt":"2014-09-07T13:24:49","guid":{"rendered":"http:\/\/matematika.okamzite.eu\/?p=962"},"modified":"2015-04-08T20:18:41","modified_gmt":"2015-04-08T20:18:41","slug":"sutazne-ulohy-kategorii-z5-z9-3","status":"publish","type":"post","link":"https:\/\/matematika.besaba.com\/?p=962","title":{"rendered":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9"},"content":{"rendered":"<p class=\"hh2\"><a name=\"top\"><\/a>64. ro\u010dn\u00edk matematickej olympi\u00e1dy 2014-2015<br \/> S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9<\/p>\n<p><span class=\"povinne\">aktualizovan\u00e9 8.4.2015 22:15<\/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<tr>\n<td style=\"background-color: #ffff99;\" colspan=\"5\" align=\"center\"><span class=\"rs\">Zadania \u00faloh dom\u00e1ceho kola na stiahnutie vo form\u00e1te PDF: <a href=\"\/64\/64_z59.pdf\" target=\"_blank\"><img loading=\"lazy\" decoding=\"async\" src=\"\/img\/down.png\" alt=\"\" width=\"48\" height=\"48\" align=\"absmiddle\" border=\"0\" \/><\/a> (ulo\u017ei\u0165 ako &#8230;)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"z51\"><\/a><\/p>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=1253\" target=\"_blank\"><input name=\"butt51\" type=\"button\" value=\"Zadanie\" \/><\/a><\/form>\n<p><strong>Z5-I-1<\/strong><br \/> Chlapci si medzi sebou menili zn\u00e1mky, gu\u013e\u00f4\u010dky a lopti\u010dky. Za 8 gu\u013e\u00f4\u010dok je 10 zn\u00e1mok, za 4 lopti\u010dky je 15 zn\u00e1mok. Ko\u013eko gu\u013e\u00f4\u010dok je za jednu lopti\u010dku?<\/p>\n<div class=\"pri\">(M. Krej\u010dov\u00e1)<\/div>\n<p><strong>Z5-I-2<\/strong><br \/> \u017dab\u00ed princ sa z\u00fa\u010dastnil skokanskej s\u00fa\u0165a\u017ee, pri ktorej sa sk\u00e1kalo po kame\u0148och rozmiestnen\u00fdch ako na obr\u00e1zku. Bolo dovolen\u00e9 sk\u00e1ka\u0165 len na najbli\u017e\u0161ie kamene v\u00fdchodn\u00fdm alebo ju\u017en\u00fdm smerom. Ka\u017ed\u00fd skok na v\u00fdchod bol ocenen\u00fd dvoma bodmi, ka\u017ed\u00fd skok na juh bol ocenen\u00fd piatimi bodmi. \u017dab\u00ed princ z\u00edskal 14 bodov. Ur\u010dte v\u0161etky mo\u017en\u00e9 cesty, kadia\u013e mohol sk\u00e1ka\u0165.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"..\/..\/64\/64z5a1.png\" alt=\"\" width=\"428\" height=\"134\" border=\"0\" \/><\/p>\n<div class=\"pri\">(E. Pat\u00e1kov\u00e1)<\/div>\n<p><strong>Z5-I-3<\/strong><br \/> Z \u010d\u00edsla 215 m\u00f4\u017eeme vytvori\u0165 \u0161tvorcifern\u00e9 \u010d\u00edslo t\u00fdm, \u017ee medzi jeho cifry vp\u00ed\u0161eme \u010fal\u0161iu cifru. Takto sme vytvorili dve \u0161tvorcifern\u00e9 \u010d\u00edsla, ktor\u00fdch rozdiel bol 120. Ak\u00e9 dve \u0161tvorcifern\u00e9 \u010d\u00edsla to mohli by\u0165? Ur\u010dte aspo\u0148 jedno rie\u0161enie.<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z5-I-4<\/strong><br \/> N\u00e1jdite najv\u00e4\u010d\u0161ie \u010d\u00edslo tak\u00e9, \u017ee<\/p>\n<ul>\n<li>\u017eiadna cifra sa v \u0148om neopakuje,<\/li>\n<li>s\u00fa\u010din ka\u017ed\u00fdch dvoch cifier je nep\u00e1rny,<\/li>\n<li>s\u00fa\u010det v\u0161etk\u00fdch cifier je p\u00e1rny.<\/li>\n<\/ul>\n<div class=\"pri\">(M. Mach)<\/div>\n<p><strong>Z5-I-5<\/strong><br \/> Na obr\u00e1zku je \u0161tvorec rozdelen\u00fd na 25 \u0161tvor\u010dekov. Vyfarbite \u0161tvor\u010deky piatimi farbami tak, aby platilo:<\/p>\n<ul>\n<li>ka\u017ed\u00fd \u0161tvor\u010dek je vyfarben\u00fd jednou farbou,<\/li>\n<li>v \u017eiadnom riadku ani v \u017eiadnom st\u013apci nie s\u00fa dva \u0161tvor\u010deky rovnakej farby,<\/li>\n<li>na \u017eiadnej z oboch uhloprie\u010dok nie s\u00fa dva \u0161tvor\u010deky rovnakej farby,<\/li>\n<li>\u017eiadne dva rovnako zafarben\u00e9 \u0161tvor\u010deky sa nedot\u00fdkaj\u00fa stranou ani vrcholom.<\/li>\n<\/ul>\n<table border=\"2\" cellspacing=\"1\" cellpadding=\"1\" align=\"center\">\n<tbody>\n<tr>\n<td width=\"15\">\u00a0<\/td>\n<td width=\"15\">\u00a0<\/td>\n<td width=\"15\">\u00a0<\/td>\n<td width=\"15\">\u00a0<\/td>\n<td width=\"15\">\u00a0<\/td>\n<\/tr>\n<tr>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<\/tr>\n<tr>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<\/tr>\n<tr>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<\/tr>\n<tr>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z5-I-6<\/strong><br \/> Na medaile, ktor\u00e1 m\u00e1 tvar kruhu s priemerom 20 cm, je narysovan\u00fd snehuliak tak, \u017ee s\u00fa splnen\u00e9 nasleduj\u00face po\u017eiadavky:<\/p>\n<ul>\n<li>snehuliak je zlo\u017een\u00fd z troch kruhov ako na obr\u00e1zku,<\/li>\n<li>priemery v\u0161etk\u00fdch kruhov vyjadren\u00e9 v cm s\u00fa celo\u010d\u00edseln\u00e9,<\/li>\n<li>priemer ka\u017ed\u00e9ho v\u00e4\u010d\u0161ieho kruhu je o 2 cm v\u00e4\u010d\u0161\u00ed ako priemer kruhu predch\u00e1dzaj\u00faceho.<\/li>\n<\/ul>\n<p>Ur\u010dte v\u00fd\u0161ku \u010do najv\u00e4\u010d\u0161ieho snehuliaka s uveden\u00fdmi vlastnos\u0165ami.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"64\/64z5a2.png\" alt=\"\" width=\"158\" height=\"158\" border=\"0\" \/><\/p>\n<div class=\"pri\">(L. Dedkov\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=1333\" target=\"_blank\"><input name=\"butt52\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z5-II-1<\/strong><br \/> V k\u00fazeln\u00edckom baz\u00e1re si k\u00fazeln\u00edci medzi sebou vymie\u0148ali k\u00fazeln\u00edcke klob\u00faky, pali\u010dky a pl\u00e1\u0161te. Za 4 pali\u010dky je 6 pl\u00e1\u0161\u0165ov a za 5 pali\u010diek je 5 klob\u00fakov. Ko\u013eko pl\u00e1\u0161\u0165ov je za 5 pali\u010diek a 1 klob\u00fak?<\/p>\n<div class=\"pri\">(Veronika Huc\u00edkov\u00e1)<\/div>\n<p><strong>Z5-II-2<\/strong><br \/> Juro mal tri zhodn\u00e9 obd\u013a\u017eniky. Najsk\u00f4r ich k sebe prilo\u017eil tak ako na obr\u00e1zku a dostal obd\u013a\u017enik, ktor\u00fd mal obvod 20 cm. Potom ich k sebe prilo\u017eil inak a dostal obd\u013a\u017enik s in\u00fdm obvodom. Ak\u00fd obvod mal tento obd\u013a\u017enik?<\/p>\n<table border=\"2\" width=\"70\" align=\"center\">\n<tbody>\n<tr>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"pri\">(Erika Novotn\u00e1)<\/div>\n<p><strong>Z5-II-3<\/strong><br \/> Z \u010d\u00edsla 215 m\u00f4\u017eeme vytvori\u0165 \u0161tvorcifern\u00e9 \u010d\u00edslo tak, \u017ee medzi jeho cifry vp\u00ed\u0161eme ak\u00fako\u013evek \u010fal\u0161iu cifru. Takto sme vytvorili dve \u0161tvorcifern\u00e9 \u010d\u00edsla, ktor\u00fdch s\u00fa\u010det bol 4 360. Ak\u00e9 dve \u0161tvorcifern\u00e9 \u010d\u00edsla to mohli by\u0165? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Libor \u0160im\u016fnek)<\/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=1373\" target=\"_blank\"><input name=\"butt61\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z6-I-1<\/strong><br \/> Erika a Peter dostali kocku, ktor\u00e1 mala ka\u017ed\u00fa stenu rozdelen\u00fa na \u0161tyri rovnak\u00e9 \u0161tvorce (pozri obr\u00e1zok). Peter tvrdil, \u017ee sa daj\u00fa do v\u0161etk\u00fdch \u0161tvorcov vp\u00edsa\u0165 \u010d\u00edsla 1 alebo 2 tak, aby na ka\u017edej zo \u0161iestich stien bol in\u00fd s\u00fa\u010det. Erika naopak tvrdila, \u017ee to mo\u017en\u00e9 nie je. Rozhodnite, kto z nich mal pravdu.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"64\/64z6a2.png\" alt=\"\" width=\"79\" height=\"79\" border=\"0\" \/><\/p>\n<div class=\"pri\">(E. Novotn\u00e1)<\/div>\n<p><strong>Z6-I-2<\/strong><br \/> Jan\u00ed\u010dko a Walter zbierali aut\u00ed\u010dka. Walter mal aut\u00ed\u010dka ulo\u017een\u00e9 v skrinke na troch poli\u010dk\u00e1ch. Najviac aut\u00ed\u010dok st\u00e1lo na hornej poli\u010dke, na prostrednej ich bolo o tri menej ako na hornej a na spodnej poli\u010dke ich bolo o tri menej ako na prostrednej. Pritom na jednej z t\u00fdchto poli\u010diek bolo 15 aut\u00ed\u010dok. Ke\u010f si Jan\u00ed\u010dko zbierku prezrel, povedal Walterovi: &#8222;Myslel som si, \u017ee ke\u010f m\u00e1m viac ako 20 aut\u00ed\u010dok, tak ich m\u00e1m ve\u013ea. Teraz ale vid\u00edm, \u017ee ty m\u00e1\u0161 dvakr\u00e1t viac aut\u00ed\u010dok ako ja!&#8220; Ko\u013eko aut\u00ed\u010dok mal vo svojej zbierke Jan\u00ed\u010dko?<\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z6-I-3<\/strong><br \/> P\u00e1n Karfi\u00f3l m\u00e1 obd\u013a\u017enikov\u00fa z\u00e1hradu rozdelen\u00fa na 9 pravouholn\u00edkov\u00fdch z\u00e1honov ako na obr\u00e1zku. Pri piatich z\u00e1honoch s\u00fa zap\u00edsan\u00e9 ve\u013ekosti ich obvodov v metroch. Ur\u010dte obvod celej z\u00e1hrady p\u00e1na Karfi\u00f3la.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"64\/64z6a3.png\" alt=\"\" width=\"147\" height=\"111\" border=\"0\" \/><\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z6-I-4<\/strong><br \/> Katka, Barbora a Adela sa dohadovali, ktor\u00e9 dvojcifern\u00e9 \u010d\u00edslo je najkraj\u0161ie. Katka vravela, \u017ee to je to jej, preto\u017ee je delite\u013en\u00e9 \u0161tyrmi, a ke\u010f ho nap\u00ed\u0161e pospiatky, dostane in\u00e9 dvojcifern\u00e9 \u010d\u00edslo, ktor\u00e9 je tie\u017e delite\u013en\u00e9 \u0161tyrmi. Barbora tvrdila, \u017ee je to ur\u010dite to jej, preto\u017ee jedna z jeho cifier je n\u00e1sobkom druhej. Adela o svojom ob\u013e\u00fabenom \u010d\u00edsle prezradila, \u017ee sa d\u00e1 rozlo\u017ei\u0165 na s\u00fa\u010din \u0161tyroch prvo\u010d\u00edsel. Nakoniec kamar\u00e1tky zistili, \u017ee hovoria v\u0161etky o tom istom \u010d\u00edsle. Ur\u010dte, ktor\u00e9 \u010d\u00edslo to bolo.<\/p>\n<div class=\"pri\">(L. Dedkov\u00e1)<\/div>\n<p><strong>Z6-I-5<\/strong><br \/> Ur\u010dte, ko\u013eko r\u00f4znych rie\u0161en\u00ed m\u00e1 nasleduj\u00faci algebrogram. Ka\u017ed\u00e9 p\u00edsmeno zodpoved\u00e1 jednej cifre od 0 do 5, r\u00f4zne p\u00edsmena zodpovedaj\u00fa r\u00f4znym cifr\u00e1m, rovnak\u00e9 rovnak\u00fdm.<\/p>\n<p class=\"r\">K O S A<br \/> <span style=\"text-decoration: underline;\">S A K O<\/span><br \/> B A B A<\/p>\n<div class=\"pri\">(K. Pazourek)<\/div>\n<p><strong>Z6-I-6<\/strong><br \/> Skauti na v\u00fdlete hrali hru. V lese bolo rozmiestnen\u00fdch 8 stanov\u00edsk prepojen\u00fdch \u0161pag\u00e1tmi tak, ako na obr\u00e1zku. Na ka\u017edom stanovisku sa vyd\u00e1valo jedno p\u00edsmenko, pr\u00edpadne poml\u010dka. Stanovisk\u00e1 sa daj\u00fa pozd\u013a\u017e \u0161pag\u00e1tov prebehn\u00fa\u0165 tak, \u017ee z\u00edskan\u00e9 znaky tvoria re\u0165azec<\/p>\n<p class=\"r\">ANANAS-KOKOS-MANGO.<\/p>\n<p>Prira\u010fte jednotliv\u00fdm stanovisk\u00e1m zodpovedaj\u00face znaky.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"..\/..\/64\/64z6a1.png\" alt=\"\" width=\"124\" height=\"117\" border=\"0\" \/><\/p>\n<div class=\"pri\">(M. 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=\"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=1377\" target=\"_blank\"><input name=\"butt62z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=1380\" target=\"_blank\"><input name=\"butt62r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z6-II-1<\/strong><br \/>Fabi\u00e1n m\u00e1 \u0161tyri karti\u010dky, na ka\u017ed\u00fa z nich nap\u00edsal jedno cel\u00e9 kladn\u00e9 \u010d\u00edslo men\u0161ie ako 10. \u010c\u00edsla nap\u00edsal \u0161tyrmi r\u00f4znymi farbami, pri\u010dom plat\u00ed:<\/p>\n<ul>\n<li>S\u00fa\u010din zelen\u00e9ho a \u017elt\u00e9ho \u010d\u00edsla je zelen\u00e9 \u010d\u00edslo.<\/li>\n<li>Modr\u00e9 \u010d\u00edslo je rovnak\u00e9 ako \u010derven\u00e9 \u010d\u00edslo.<\/li>\n<li>S\u00fa\u010din \u010derven\u00e9ho a modr\u00e9ho \u010d\u00edsla je dvojcifern\u00e9 \u010d\u00edslo zap\u00edsan\u00e9 zelenou a \u017eltou cifrou (v tomto porad\u00ed).<\/li>\n<\/ul>\n<p>Ur\u010dte tieto \u0161tyri \u010d\u00edsla.<\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z6-II-2<\/strong><br \/>Na de\u0148 det\u00ed otvorili v ZOO bludisko so \u0161iestimi stanoviskami, na ktor\u00fdch sa rozd\u00e1vali cukr\u00edky. Na jednom stanovisku sa pri ka\u017edom vstupe rozd\u00e1valo 5 cukr\u00edkov, na dvoch stanovisk\u00e1ch sa rozd\u00e1vali 3 cukr\u00edky a na troch stanovisk\u00e1ch 1 cukr\u00edk. Juraj najsk\u00f4r vst\u00fapil na stanovisko ozna\u010den\u00e9 na obr\u00e1zku \u0161\u00edpkou a pokra\u010doval tak, \u017ee ka\u017edou cesti\u010dkou pre\u0161iel nanajv\u00fd\u0161 raz. Ur\u010dte, ko\u013eko najviac cukr\u00edkov mohol Juraj dosta\u0165.<\/p>\n<div class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"64\/64z6b1.png\" alt=\"\" width=\"81\" height=\"87\" border=\"0\" \/><\/div>\n<\/p>\n<div class=\"pri\">(E. Novotn\u00e1)<\/div>\n<p><strong>Z6-II-3<\/strong><br \/>Jano mal \u0161tyri zhodn\u00e9 trojuholn\u00edky. Skladal z nich r\u00f4zne \u00fatvary, a to tak, \u017ee trojuholn\u00edky k sebe prikladal stranami rovnakej d\u013a\u017eky. Najsk\u00f4r zlo\u017eil \u00fatvar z troch trojuholn\u00edkov ako na obr\u00e1zku 3a, ktor\u00fd mal obvod 43 cm. Potom \u00fatvar rozobral a zlo\u017eil in\u00fd \u00fatvar z troch trojuholn\u00edkov, ktor\u00fd mal obvod 35 cm. Nakoniec zo v\u0161etk\u00fdch \u0161tyroch trojuholn\u00edkov zlo\u017eil \u010fal\u0161\u00ed \u00fatvar ako na obr\u00e1zku 3b, a ten mal obvod 46 cm. Ur\u010dte d\u013a\u017eky str\u00e1n trojuholn\u00edkov.<\/p>\n<div class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"64\/64z6b2.png\" alt=\"\" width=\"340\" height=\"92\" border=\"0\" \/><\/div>\n<\/p>\n<div class=\"pri\">(E. Pat\u00e1kov\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=1251\"><input name=\"butt51\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=1374\" target=\"_blank\"><input name=\"butt61\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z7-I-1<\/strong><br \/> \u013dubo\u0161, Martin a ich kamar\u00e1tka Erika \u0161etria na hra\u010dku. \u013dubo\u0161 a Martin prispeli do spolo\u010dnej pokladni\u010dky rovnak\u00fdm mno\u017estvom eur, Erika prispela inou sumou. Keby Erika prispela len tretinou z toho, \u010do do pokladni\u010dky dodala, celkom by mali polovicu zo sumy, ktor\u00e1 je v pokladni\u010dke teraz. Ko\u013ekokr\u00e1t viac eur do pokladni\u010dky dodala Erika ako \u013dubo\u0161?<\/p>\n<div class=\"pri\">(E. Pat\u00e1kov\u00e1)<\/div>\n<p><strong>Z7-I-2<\/strong><br \/> Lenka sa bavila t\u00fdm, \u017ee vy\u0165uk\u00e1vala na kalkula\u010dke \u010d\u00edsla. Pou\u017e\u00edvala iba cifry od 2 do 9 (pozri obr\u00e1zok) a \u010doskoro si v\u0161imla, \u017ee niektor\u00e9 z\u00e1pisy boli osovo s\u00famern\u00e9. Ur\u010dte po\u010det v\u0161etk\u00fdch nanajv\u00fd\u0161 trojcifern\u00fdch \u010d\u00edsel s uveden\u00fdmi vlastnos\u0165ami.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"64\/64z7a1.png\" alt=\"\" width=\"127\" height=\"38\" border=\"0\" \/><\/p>\n<div class=\"pri\">(L. Dedkov\u00e1)<\/div>\n<p><strong>Z7-I-3<\/strong><br \/> Pod\u013ea projektu bude dno baz\u00e9nu pokryt\u00e9 kamienkami troch farieb tak, ako ukazuje obr\u00e1zok (dno je navy\u0161e rozdelen\u00e9 na 25 zhodn\u00fdch pomocn\u00fdch \u0161tvorcov). Cena kamienkov na jednotku plochy sa pri jednotliv\u00fdch farb\u00e1ch l\u00ed\u0161i. Projektant po\u010d\u00edtal cenu kamienkov pou\u017eit\u00fdch na takto pokryt\u00e9 dno a na jeho prekvapenie sa za ka\u017ed\u00fd druh kamienkov utrat\u00ed rovnak\u00e1 suma. \u010ealej spo\u010d\u00edtal, \u017ee keby cel\u00fa plochu pokryl t\u00fdmi najlacnej\u0161\u00edmi kamienkami, boli by n\u00e1klady 1700\u20ac. Zistite, ak\u00e9 by boli n\u00e1klady, keby cel\u00e9 dno nechal<br \/> pokry\u0165 t\u00fdmi najdrah\u0161\u00edmi kamienkami.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"64\/64z7a2.png\" alt=\"\" width=\"129\" height=\"129\" border=\"0\" \/><\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z7-I-4<\/strong><br \/> Body <em>N, O, P<\/em> a <em>Q<\/em> s\u00fa vzh\u013eadom na trojuholn\u00edk <em>KLM<\/em> zadan\u00e9 nasleduj\u00facim sp\u00f4sobom:<\/p>\n<ul>\n<li>Body <em>N<\/em> a <em>O<\/em> s\u00fa postupne stredy str\u00e1n <em>KM<\/em> a <em>KL<\/em>,<\/li>\n<li>Vrchol <em>M<\/em> je stredom \u00fase\u010dky <em>NP<\/em>,<\/li>\n<li>Bod <em>Q<\/em> je priese\u010dn\u00edkom priamok <em>LM<\/em> a <em>OP<\/em>.<\/li>\n<\/ul>\n<p>Ur\u010dte, ak\u00fd je pomer d\u013a\u017eok \u00fase\u010diek <em>MQ<\/em> a <em>ML<\/em>.<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z7-I-5<\/strong><br \/> Na starom hrade b\u00fdva drak a v\u00e4zn\u00ed tam princezn\u00fa. Jano i\u0161iel princezn\u00fa oslobodi\u0165, na hrade objavil troje dvier s nasleduj\u00facimi n\u00e1pismi.<\/p>\n<p><strong>\u00a0\u00a0I: &#8222;Jasky\u0148a za dverami III je pr\u00e1zdna.\u201c<br \/> II: &#8222;Princezn\u00e1 je v priestore za dverami I.\u201c<br \/> III: &#8222;Pozor! Drak je v jaskyni za dverami II.\u201c<\/strong><\/p>\n<p>Dobr\u00e1 v\u00edla Janovi prezradila, \u017ee na dver\u00e1ch, za ktor\u00fdmi je princezn\u00e1, je n\u00e1pis pravdiv\u00fd, pri drakovi je nepravdiv\u00fd a na dver\u00e1ch pr\u00e1zdnej jaskyne m\u00f4\u017ee by\u0165 nap\u00edsan\u00e1 pravda aj lo\u017e. Jano m\u00e1 na oslobodenie princeznej iba jeden pokus. Ktor\u00e9 dvere m\u00e1 otvori\u0165?<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z7-I-6<\/strong><br \/> Matej m\u00e1 dve karti\u010dky, na ka\u017ed\u00fa z nich nap\u00edsal jedno dvojcifern\u00e9 \u010d\u00edslo. Ak zarad\u00ed men\u0161ie \u010d\u00edslo za v\u00e4\u010d\u0161ie, dostane \u0161tvorcifern\u00e9 \u010d\u00edslo, ktor\u00e9 je delite\u013en\u00e9 \u0161tyrmi a deviatimi. Ak zarad\u00ed naopak v\u00e4\u010d\u0161ie \u010d\u00edslo za men\u0161ie, dostane \u0161tvorcifern\u00e9 \u010d\u00edslo, ktor\u00e9 je delite\u013en\u00e9 piatimi a \u0161iestimi. Ko\u013eko dvoj\u00edc karti\u010diek mohol Matej vyrobi\u0165 tak, aby platili vy\u0161\u0161ie uveden\u00e9 podmienky? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Michaela Petrov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z72\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=1378\" target=\"_blank\"><input name=\"butt72z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=1381\" target=\"_blank\"><input name=\"butt762r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z7-II-1<\/strong><br \/>N\u00e1jdite v\u0161etky prirodzen\u00e9 \u010d\u00edsla od 90 do 150 tak\u00e9, \u017ee cifern\u00fd s\u00fa\u010det ich cifern\u00e9ho s\u00fa\u010dtu je rovn\u00fd 1.<\/p>\n<div class=\"pri\">(E. Pat\u00e1kov\u00e1)<\/div>\n<p><strong>Z7-II-2<\/strong><br \/>Siedme triedy z na\u0161ej \u0161koly s\u00fa\u0165a\u017eili v zbieran\u00ed vrchn\u00e1kov z PET flia\u0161. Trieda A nazbierala \u0161tvrtinu toho, \u010do triedy B a C dokopy, trieda B nazbierala p\u00e4tinu toho, \u010do triedy A a C dokopy, a trieda C nazbierala 570 vrchn\u00e1kov. Ur\u010dte, ko\u013eko vrchn\u00e1kov nazbierali tieto tri triedy spolu.<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z7-II-3<\/strong><br \/>Prebieha rekon\u0161trukcia n\u00e1mestia v tvare \u0161tvorca so stranou 20 metrov. Keby bolo cel\u00e9 vydl\u00e1\u017eden\u00e9 lacnej\u0161ou svetlou dla\u017ebou, boli by n\u00e1klady na materi\u00e1l 10 000 \u20ac. Keby bolo cel\u00e9 n\u00e1mestie pokryt\u00e9 drah\u0161ou tmavou dla\u017ebou, st\u00e1l by materi\u00e1l 30 000 \u20ac. Architekt v\u0161ak v centr\u00e1lnej \u010dasti n\u00e1mestia navrhol svetl\u00fa \u0161tvorcov\u00fa \u010das\u0165, ktor\u00e1 bude olemovan\u00e1 pruhom tmavej dla\u017eby o \u0161\u00edrke 2 metre, a vo vonkaj\u0161ej \u010dasti n\u00e1mestia bude rovnak\u00e1 svetl\u00e1 dla\u017eba ako uprostred (pozri obr\u00e1zok). Pod\u013ea tohto n\u00e1vrhu bud\u00fa n\u00e1klady na materi\u00e1l tmavej \u010dasti rovnak\u00e9 ako na celkov\u00fd materi\u00e1l svetl\u00fdch \u010dast\u00ed. Ur\u010dte:<\/p>\n<ul>\n<li>ko\u013eko stoj\u00ed materi\u00e1l na vydl\u00e1\u017edenie n\u00e1mestia pod\u013ea tohto projektu,<\/li>\n<li>ak\u00e1 dlh\u00e1 je strana svetl\u00e9ho \u0161tvorca v centr\u00e1lnej \u010dasti n\u00e1mestia.<\/li>\n<\/ul>\n<div class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"64\/64z7b3.png\" alt=\"\" width=\"132\" height=\"146\" border=\"0\" \/><\/div>\n<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/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=1250\"><input name=\"butt51\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=1375\" target=\"_blank\"><input name=\"butt61\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z8-I-1<\/strong><br \/> P\u00edsmenkov\u00fd logik je hra pre dvoch hr\u00e1\u010dov, ktor\u00e1 m\u00e1 nasleduj\u00face pravidl\u00e1:<\/p>\n<ol>\n<li>Prv\u00fd hr\u00e1\u010d si mysl\u00ed slovo zlo\u017een\u00e9 z piatich p\u00edsmen, v ktorom sa \u017eiadne p\u00edsmeno neopakuje.<\/li>\n<li>Druh\u00fd hr\u00e1\u010d nap\u00ed\u0161e nejak\u00e9 slovo z piatich p\u00edsmen.<\/li>\n<li>Prv\u00fd hr\u00e1\u010d odpovie dvoma \u010d\u00edslami &#8211; prv\u00e9 \u010d\u00edslo ud\u00e1va, ko\u013eko p\u00edsmen nap\u00edsan\u00e9ho slova sa zhoduje s myslen\u00fdm slovom, t.j. stoja z\u00e1rove\u0148 na spr\u00e1vnom mieste, druh\u00e9 \u010d\u00edslo ud\u00e1va, ko\u013eko p\u00edsmen nap\u00edsan\u00e9ho slova je obsiahnut\u00fdch v myslenom slove, ale nestoja na spr\u00e1vnom mieste.<\/li>\n<li>Kroky 2 a 3 sa opakuj\u00fa, k\u00fdm druh\u00fd hr\u00e1\u010d myslen\u00e9 slovo neuh\u00e1dne.<\/li>\n<\/ol>\n<p>Z\u00e1znam jednej hry dvoch kamar\u00e1tov vyzeral nasledovne:<\/p>\n<table border=\"0\" width=\"120\" align=\"center\">\n<tbody class=\"r\">\n<tr>\n<td>SONET<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>MUDRC<\/td>\n<td>0<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>PLAST<\/td>\n<td>0<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>KMOTR<\/td>\n<td>0<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>ATOLY<\/td>\n<td>1<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>DOGMA<\/td>\n<td>0<\/td>\n<td>2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>V nasleduj\u00facom \u0165ahu bolo myslen\u00e9 slovo uh\u00e1dnut\u00e9. Ur\u010dte, ktor\u00e9 slovo to bolo.<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z8-I-2<\/strong><br \/> S\u00fa\u010det v\u0161etk\u00fdch delite\u013eov ist\u00e9ho nep\u00e1rneho \u010d\u00edsla je 78. Ur\u010dte, ak\u00fd je s\u00fa\u010det v\u0161etk\u00fdch delite\u013eov dvojn\u00e1sobku tohto nezn\u00e1meho \u010d\u00edsla.<\/p>\n<div class=\"pri\">(K. Pazourek)<\/div>\n<p><strong>Z8-I-3<\/strong><br \/> V lichobe\u017en\u00edku <em>KLMN<\/em> plat\u00ed, \u017ee<\/p>\n<ul>\n<li>strany <em>KL<\/em> a <em>MN<\/em> s\u00fa rovnobe\u017en\u00e9,<\/li>\n<li>\u00fase\u010dky <em>KL<\/em> a <em>KM<\/em> s\u00fa zhodn\u00e9,<\/li>\n<li>\u00fase\u010dky <em>KN<\/em>, <em>NM<\/em> a <em>ML<\/em> s\u00fa navz\u00e1jom zhodn\u00e9.<\/li>\n<\/ul>\n<p>Ur\u010dte ve\u013ekos\u0165 uhla <em>KNM<\/em>.<\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z8-I-4<\/strong><br \/> Adam m\u00e1 pln\u00fa krabicu gu\u013e\u00f4\u010dok, ktor\u00e9 s\u00fa ve\u013ek\u00e9 alebo mal\u00e9, \u010dierne alebo biele. Pomer po\u010dtu ve\u013ek\u00fdch a mal\u00fdch gu\u013e\u00f4\u010dok je 5 : 3. Medzi ve\u013ek\u00fdmi gu\u013e\u00f4\u010dkami je pomer po\u010dtu \u010diernych a bielych gu\u013e\u00f4\u010dok 1 : 2, medzi mal\u00fdmi gu\u013e\u00f4\u010dkami je pomer po\u010dtu \u010diernych a bielych 1 : 8. Ak\u00fd je pomer po\u010dtu v\u0161etk\u00fdch \u010diernych a v\u0161etk\u00fdch bielych gu\u013e\u00f4\u010dok?<\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z8-I-5<\/strong><br \/> Priemer zn\u00e1mok, ktor\u00e9 mali na vysved\u010den\u00ed \u017eiaci 8.A z matematiky, je presne 2,45. Ak by sme nepo\u010d\u00edtali jednotku a trojku s\u00farodencov Michala a Aleny, ktor\u00ed do triedy pri\u0161li pred mesiacom, bol by priemer presne 2,5. Ur\u010dte, ko\u013eko \u017eiakov m\u00e1 8.A.<\/p>\n<div class=\"pri\">(M. Dillingerov\u00e1)<\/div>\n<p><strong>Z8-I-6<\/strong><br \/> Pejko dostal od svojho p\u00e1na kv\u00e1der zlo\u017een\u00fd z navz\u00e1jom rovnak\u00fdch kociek cukru, ktor\u00fdch bolo najmenej 1000 a nanajv\u00fd\u0161 2000. Pejko kocky cukru odjedal po jednotliv\u00fdch vrstv\u00e1ch &#8211; prv\u00fd de\u0148 odjedol jednu vrstvu spredu, druh\u00fd de\u0148 jednu vrstvu sprava a tret\u00ed de\u0148 jednu vrstvu zhora. Pritom v t\u00fdchto troch vrstv\u00e1ch bol zaka\u017ed\u00fdm rovnak\u00fd po\u010det kociek. Zistite, ko\u013eko kociek mohol ma\u0165 darovan\u00fd kv\u00e1der. Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(E. Novotn\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"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=1379\" target=\"_blank\"><input name=\"butt82z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=1382\" target=\"_blank\"><input name=\"butt82r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z8-II-1<\/strong><br \/>V rovnostrannom trojuholn\u00edku <em>ABC<\/em> le\u017e\u00ed na strane <em>BC<\/em> bod <em>F<\/em>. Obsah trojuholn\u00edka <em>ABF<\/em> je trikr\u00e1t v\u00e4\u010d\u0161\u00ed ako obsah trojuholn\u00edka <em>ACF<\/em> a rozdiel obvodov t\u00fdchto dvoch trojuholn\u00edkov je 5 cm. Ur\u010dte d\u013a\u017eku strany trojuholn\u00edka <em>ABC<\/em>.<\/p>\n<div class=\"pri\">(E. Pat\u00e1kov\u00e1)<\/div>\n<p><strong>Z8-II-2<\/strong><br \/>Traja hudobn\u00edci Janek, Mike\u0161 a V\u00e1vra si zvy\u010dajne rozdelia spolo\u010dn\u00fd honor\u00e1r v pomere 4 : 5 : 6, najmenej dostane Janek a najviac V\u00e1vra. Tentoraz V\u00e1vra nehral dobre, a tak sa svojho podielu vzdal. Janek navrhol, \u017ee si V\u00e1vrovu \u010das\u0165 rozdelia s Mike\u0161om na polovice. Mike\u0161 v\u0161ak trval na tom, aby si aj t\u00fato \u010das\u0165 rozdelili nerovnomerne ako zvy\u010dajne, teda v pomere 4 : 5. Mike\u0161 by toti\u017e pod\u013ea Jankovho n\u00e1vrhu dostal o 4 \u20ac menej ako pod\u013ea svojho. Ur\u010dte v\u00fd\u0161ku spolo\u010dn\u00e9ho honor\u00e1ra.<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z8-II-3<\/strong><br \/>Ke\u010f jeden rozmer kv\u00e1dra zdvojn\u00e1sob\u00edme, druh\u00fd rozmer kv\u00e1dra predel\u00edme dvoma a tret\u00ed rozmer zv\u00e4\u010d\u0161\u00edme o 6 cm, dostaneme kocku, ktor\u00e1 m\u00e1 rovnak\u00fd povrch ako p\u00f4vodn\u00fd kv\u00e1der. Ur\u010dte rozmery tohto kv\u00e1dra.<\/p>\n<div class=\"pri\">(M. 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=\"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=1249\" target=\"_blank\"><input name=\"butt91\" type=\"button\" value=\"Zadanie\" \/><\/a><\/form>\n<p><strong>Z9-I-1<\/strong><br \/> Milena nazbierala do ko\u0161\u00edka posledn\u00e9 spadnut\u00e9 orechy a zavolala na partiu chlapcov, nech sa o ne podelia. Dala im ale podmienku: prv\u00fd si vezme 1 orech a desatinu zvy\u0161ku, druh\u00fd si vezme 2 orechy a desatinu nov\u00e9ho zvy\u0161ku, tret\u00ed si vezme 3 orechy a desatinu \u010fal\u0161ieho zvy\u0161ku a tak \u010falej. Takto sa podarilo rozobra\u0165 v\u0161etky orechy a pritom ka\u017ed\u00fd dostal rovnako ve\u013ea. Ur\u010dte, ko\u013eko Milena nazbierala orechov a ko\u013eko sa o ne delilo chlapcov.<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z9-I-2<\/strong><br \/> Lenka sa bavila t\u00fdm, \u017ee vy\u0165uk\u00e1vala na kalkula\u010dke \u010d\u00edsla, pri\u010dom pou\u017e\u00edvala iba cifry od 2 do 9 (pozri obr\u00e1zok). Z\u00e1pisy niektor\u00fdch \u010d\u00edsel mali t\u00fa vlastnos\u0165, \u017ee ich obraz v osovej alebo stredovej s\u00famernosti bol op\u00e4\u0165 z\u00e1pisom nejak\u00e9ho \u010d\u00edsla. Ur\u010dte po\u010det v\u0161etk\u00fdch nanajv\u00fd\u0161 trojcifern\u00fdch \u010d\u00edsel s uveden\u00fdmi vlastnos\u0165ami.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"64\/64z7a1.png\" alt=\"\" width=\"127\" height=\"38\" border=\"0\" \/><\/p>\n<div class=\"pri\">(L. Dedkov\u00e1)<\/div>\n<p><strong>Z9-I-3<\/strong><br \/> Dar\u010dek je zabalen\u00fd do krabice, ktorej rozmery v cm s\u00fa 40\u00d730\u00d76. Krabica je previazan\u00e1 \u0161pag\u00e1tom ako na obr\u00e1zku. Ur\u010dte, ko\u013eko najmenej cm \u0161pag\u00e1tu je treba na previazanie krabice, ak na uzol a ma\u0161\u013eu sta\u010d\u00ed 20 cm.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"64\/64z9a1.png\" alt=\"\" width=\"186\" height=\"62\" border=\"0\" \/><\/p>\n<div class=\"pri\">(M. Krej\u010dov\u00e1)<\/div>\n<p><strong>Z9-I-4<\/strong><br \/> Peter, Martin a Juro triafali do zvl\u00e1\u0161tneho ter\u010da, ktor\u00fd mal iba tri pol\u00ed\u010dka s hodnotami 12, 18 a 30 bodov. V\u0161etci chlapci h\u00e1dzali rovnak\u00fdm po\u010dtom \u0161\u00edpok, v\u0161etky \u0161\u00edpky trafili do ter\u010da a v\u00fdsledky ka\u017ed\u00fdch dvoch chlapcov sa l\u00ed\u0161ili v jedinom hode. Petrov priemern\u00fd bodov\u00fd v\u00fdsledok bol o dva body lep\u0161\u00ed ako Martinov a ten bol o jeden bod lep\u0161\u00ed ako priemer Jurov. Ur\u010dte, ko\u013ek\u00fdmi \u0161\u00edpkami h\u00e1dzal ka\u017ed\u00fd z chlapcov.<\/p>\n<div class=\"pri\">(E. Novotn\u00e1)<\/div>\n<p><strong>Z9-I-5<\/strong><br \/> Jaro si k\u00fapil nov\u00e9 nohavice, ale boli pr\u00edli\u0161 dlh\u00e9. Ich d\u013a\u017eka bola vzh\u013eadom k Jarovej v\u00fd\u0161ke v pomere 5 : 8. Mami\u010dka mu nohavice skr\u00e1tila o 4 cm, \u010d\u00edm sa p\u00f4vodn\u00fd pomer zmen\u0161il o 4 %. Ur\u010dte, ak\u00fd vysok\u00fd je Jaro.<\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z9-I-6<\/strong><br \/> Nezn\u00e1me \u010d\u00edslo je delite\u013en\u00e9 pr\u00e1ve tromi r\u00f4znymi prvo\u010d\u00edslami. Ke\u010f tieto prvo\u010d\u00edsla zorad\u00edme vzostupne, plat\u00ed nasleduj\u00face:<\/p>\n<ul>\n<li>Rozdiel druh\u00e9ho a prv\u00e9ho prvo\u010d\u00edsla je polovicou rozdielu tretieho a druh\u00e9ho prvo\u010d\u00edsla.<\/li>\n<li>S\u00fa\u010din rozdielu druh\u00e9ho a prv\u00e9ho prvo\u010d\u00edsla s rozdielom tretieho a druh\u00e9ho prvo\u010d\u00edsla je n\u00e1sobkom 17.<\/li>\n<\/ul>\n<p>Ur\u010dte najmen\u0161ie \u010d\u00edslo, ktor\u00e9 m\u00e1 v\u0161etky vy\u0161\u0161ie uveden\u00e9 vlastnosti.<\/p>\n<div class=\"pri\">(K. Pazourek)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"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=1334\" target=\"_blank\"><input name=\"butt92\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-II-1<\/strong><br \/> Mysl\u00edm na nieko\u013eko bezprostredne po sebe id\u00facich prirodzen\u00fdch \u010d\u00edsel. Keby sme z nich vy\u0161krtli \u010d\u00edsla 70, 82 a 103, aritmetick\u00fd priemer \u010d\u00edsel by sa nezmenil. Keby sme namiesto toho vy\u0161krtli \u010d\u00edsla 122 a 123, aritmetick\u00fd priemer by sa zmen\u0161il presne o 1. Na ktor\u00e9 prirodzen\u00e9 \u010d\u00edsla mysl\u00edm?<\/p>\n<div class=\"pri\">(Libor \u0160im\u016fnek)<\/div>\n<p><strong>Z9-II-2<\/strong><br \/> Po sebe id\u00face prirodzen\u00e9 \u010d\u00edsla postupne pri\u010d\u00edtame a od\u010d\u00edtame pod\u013ea nasleduj\u00faceho n\u00e1vodu:<\/p>\n<div class=\"r\">1 + 2 \u2212 3 \u2212 4 + 5 + 6 \u2212 7 \u2212 8 + 9 + 10 \u2212 11 \u2212 12 + &#8230;<\/div>\n<p>Ur\u010dte, ak\u00e1 bude hodnota tak\u00e9hoto v\u00fdrazu, ak jeho posledn\u00fd \u010dlen je 2 015.<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z9-II-3<\/strong><br \/> Ani\u010dka dostala na narodeniny tortu v tvare obd\u013a\u017enika. Pomocou dvoch priamych rezov si odkrojila k\u00fasok, ktor\u00fd je na obr\u00e1zku vyzna\u010den\u00fd sivou. Ur\u010dte, ak\u00fa \u010das\u0165 torty si Ani\u010dka odkrojila.<\/p>\n<div class=\"pri\">(Al\u017ebeta Bohin\u00edkov\u00e1)<\/div>\n<p><strong>Z9-II-4<\/strong><br \/> Ist\u00fd obd\u013a\u017enik mal svoje rozmery vyjadren\u00e9 v decimetroch cel\u00fdmi \u010d\u00edslami. Potom rozmery trikr\u00e1t zmenil. Najsk\u00f4r jeden svoj rozmer zdvojn\u00e1sobil a druh\u00fd zmenil tak, aby mal rovnak\u00fd obsah ako na za\u010diatku. Potom jeden rozmer zv\u00e4\u010d\u0161il o 1 dm a druh\u00fd zmen\u0161il o 4 dm, pri\u010dom mal st\u00e1le tak\u00fd ist\u00fd obsah ako na za\u010diatku. Nakoniec svoj krat\u0161\u00ed rozmer zmen\u0161il o 1 dm, dlh\u0161\u00ed ponechal bezo zmeny. Ur\u010dte pomer str\u00e1n posledn\u00e9ho obd\u013a\u017enika.<\/p>\n<div class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"64\/64z9b1.png\" alt=\"\" width=\"207\" height=\"195\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Erika Novotn\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z93\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=1364\" target=\"_blank\"><input name=\"butt93\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-III-1<\/strong><br \/> Peter, Martin a Juraj triafali do zvl\u00e1\u0161tneho ter\u010da, ktor\u00fd mal iba tri pol\u00ed\u010dka s navz\u00e1jom r\u00f4znymi hodnotami. Ka\u017ed\u00fd z chlapcov h\u00e1dzal celkom desa\u0165kr\u00e1t a v\u017edy trafil do ter\u010da. Bodov\u00fd zisk z prv\u00fdch \u00f4smich hodov mali v\u0161etci traja chlapci rovnak\u00fd. Pri posledn\u00fdch dvoch hodoch trafil Juraj dvakr\u00e1t pol\u00ed\u010dko s najni\u017e\u0161ou mo\u017enou hodnotou, Martin dvakr\u00e1t pol\u00ed\u010dko so strednou hodnotou a Peter dvakr\u00e1t pol\u00ed\u010dko s najvy\u0161\u0161ou hodnotou. Aritmetick\u00fd priemer v\u0161etk\u00fdch Martinov\u00fdch hodov bol o 1 v\u00e4\u010d\u0161\u00ed ako Jurajov priemer a Petrov priemer bol o 1 v\u00e4\u010d\u0161\u00ed ako Martinov priemer. Ur\u010dte v\u0161etky mo\u017en\u00e9 hodnoty pol\u00ed\u010dok na ter\u010di, ak viete, \u017ee jedna z nich bola 12.<\/p>\n<div class=\"pri\">(Erika Novotn\u00e1)<\/div>\n<p><strong>Z9-III-2<\/strong><br \/> V trojuholn\u00edku <em>ABC<\/em> le\u017eia na strane <em>AB<\/em> body <em>E<\/em> a <em>F<\/em>. Obsah trojuholn\u00edka <em>AEC<\/em> je 1 cm<sup>2<\/sup>, obsah trojuholn\u00edka <em>EFC<\/em> je 3 cm<sup>2<\/sup> a obsah trojuholn\u00edka <em>FBC<\/em> je 2 cm<sup>2<\/sup>. Bod <em>T<\/em> je \u0165a\u017eiskom trojuholn\u00edka <em>AFC<\/em> a bod <em>G<\/em> je priese\u010dn\u00edkom priamok <em>CT<\/em> a <em>AB<\/em>. Bod <em>R<\/em> je \u0165a\u017eiskom trojuholn\u00edka <em>EBC<\/em> a bod <em>H<\/em> je priese\u010dn\u00edkom priamok <em>CR<\/em> a <em>AB<\/em>. Ur\u010dte obsah trojuholn\u00edka <em>GHC<\/em>.<\/p>\n<div class=\"pri\">(Eva Pat\u00e1kov\u00e1)<\/div>\n<p><strong>Z9-III-3<\/strong><br \/>\nUr\u010dte, ak\u00e1 je posledn\u00e1 cifra s\u00fa\u010dinu v\u0161etk\u00fdch p\u00e1rnych prirodzen\u00fdch \u010d\u00edsel, ktor\u00e9 s\u00fa men\u0161ie ako 100 a ktor\u00e9 nie s\u00fa n\u00e1sobkami 10.<\/p>\n<div class=\"pri\">(Marta Volfov\u00e1)<\/div>\n<p><strong>Z9-III-4<\/strong><br \/>\nDan\u00fd je obd\u013a\u017enik <em>ABCD<\/em>, ktor\u00e9ho krat\u0161ia strana je <em>AB<\/em>. Ur\u010dte, pre ktor\u00e9 body <em>P<\/em> na priamke <em>AD<\/em> plat\u00ed, \u017ee os uhla <em>BPD<\/em> prech\u00e1dza bodom <em>C<\/em>. Svoje tvrdenie zd\u00f4vodnite a pop\u00ed\u0161te, ako by ste v\u0161etky tak\u00e9 body zostrojili.<\/p>\n<div class=\"pri\">(L. R\u016f\u017ei\u010dkov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><center><script type=\"text\/javascript\" async=\"true\" charset=\"utf-8\" src=\"http:\/\/sk.search.etargetnet.com\/generic\/advert.php?g=ref:48528,area:600x70,tabl:4,divid:,design_name:green,border_color:ffffff,border_style:none,background_opacity:100,background_color:ffffff,title_color:426000,text_color:000000,url_color:426000,h_title_color:426000,h_text_color:000000,h_url_color:426000,freespace:0,logo_type:3,logo:1,title_underline:0,url_underline:0,h_title_underline:1,h_url_underline:1,nourl:,fsi:11,font:verdana,require_login_pass:,login_pass:,show_in_fb:,show_in_fb_text:,allow_youtube_ads:1\" ><\/script><\/center><\/p>\n<p><script type=\"text\/javascript\"><!--\r\ngoogle_ad_client = \"pub-0508472359151885\";\r\n\/* 468x60, bola vytvoren\u00e1 12.11.2010 *\/\r\ngoogle_ad_slot = \"3127737040\";\r\ngoogle_ad_width = 468;\r\ngoogle_ad_height = 60;\r\n\/\/-->\r\n<\/script>\r\n<script type=\"text\/javascript\"\r\nsrc=\"http:\/\/pagead2.googlesyndication.com\/pagead\/show_ads.js\">\r\n<\/script><\/p>\n<!--CusAds0-->\n<div style=\"font-size: 0px; height: 0px; line-height: 0px; margin: 0; padding: 0; clear: both;\"><\/div>","protected":false},"excerpt":{"rendered":"<p>64. ro\u010dn\u00edk matematickej olympi\u00e1dy 2014-2015 S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9 aktualizovan\u00e9 8.4.2015 22:15<\/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":[13],"tags":[],"class_list":["post-962","post","type-post","status-publish","format-standard","hentry","category-64-rocnik"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.6 - 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