{"id":514,"date":"2011-09-19T15:24:40","date_gmt":"2011-09-19T15:24:40","guid":{"rendered":"http:\/\/matematika.okamzite.eu\/?p=514"},"modified":"2011-09-19T15:24:40","modified_gmt":"2011-09-19T15:24:40","slug":"sutazne-ulohy-kategorii-z4-z9","status":"publish","type":"post","link":"https:\/\/matematika.besaba.com\/?p=514","title":{"rendered":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9"},"content":{"rendered":"<p class=\"hh2\"><a name=\"top\"><\/a>61. ro\u010dn\u00edk matematickej olympi\u00e1dy 2011-2012<\/p>\n<p><span class=\"povinne\">aktualizovan\u00e9 11.4.2012 22:50<\/span><\/p>\n<p style=\"text-align: center;\"><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<table style=\"width: 90%;\" border=\"1\" cellspacing=\"2\" cellpadding=\"2\" 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\"><\/td>\n<td align=\"center\"><span class=\"a\"><a href=\"#z92\">krajsk\u00e9<\/a><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"background-color: #ffff99;\" colspan=\"5\" align=\"center\"><span class=\"rs\">Zadania dom\u00e1ceho kola na stiahnutie vo form\u00e1te PDF:\u00a0\u00a0\u00a0<a href=\"\/61\/61_moz59.pdf\"><img loading=\"lazy\" decoding=\"async\" src=\"\/img\/down.png\" border=\"0\" alt=\"\" width=\"48\" height=\"48\" align=\"absmiddle\" \/><\/a> (ulo\u017ei\u0165 ako &#8230;)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><!--more--><\/p>\n<p><a name=\"z51\"><\/a><\/p>\n<form> <input onclick=\"window.open('61\/61z5ir.pdf','Rie\u0161enie','width =700,height=500')\" name=\"butt51\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z5-I-1<\/strong><\/p>\n<p>Traja kamar\u00e1ti Pankr\u00e1c, Serv\u00e1c a Bonif\u00e1c i\u0161li cez pr\u00e1zdniny na no\u010dn\u00fa prech\u00e1dzku pr\u00edrodn\u00fdm labyrintom. Pri vstupe dostal ka\u017ed\u00fd svie\u010dku a vydal sa in\u00fdm smerom ako zvy\u0161n\u00ed dvaja. V\u0161etci traja labyrintom \u00faspe\u0161ne pre\u0161li, ale ka\u017ed\u00fd i\u0161iel inou cestou. V nasleduj\u00facej \u0161tvorcovej sieti s\u00fa vyzna\u010den\u00e9 ich cesty. Vieme, \u017ee Pankr\u00e1c nikdy ne\u0161iel na juh a \u017ee Serv\u00e1c nikdy ne\u0161iel na z\u00e1pad. Ko\u013eko metrov pre\u0161iel v labyrinte Bonif\u00e1c, ke\u010f vieme, \u017ee Pankr\u00e1c pre\u0161iel presne 500 m?<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/61\/61z5a1.png\" alt=\"\" width=\"432\" height=\"270\" \/><\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z5-I-2<\/strong><\/p>\n<p>Do ka\u017ed\u00e9ho nevyplnen\u00e9ho \u0161tvor\u010deka dopl\u0148te \u010d\u00edslo 1, 2 alebo 3 tak, aby v ka\u017edom st\u013apci a riadku bolo ka\u017ed\u00e9 z t\u00fdchto \u010d\u00edsel pr\u00e1ve raz a aby boli splnen\u00e9 dodato\u010dn\u00e9 po\u017eiadavky v ka\u017edej vyzna\u010denej oblasti.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/61\/61z5a2.png\" alt=\"\" width=\"289\" height=\"272\" \/><\/p>\n<p><em>(Ak vo vyzna\u010denej oblasti po\u017eadujeme ur\u010dit\u00fd podiel, m\u00e1me na mysli podiel, ktor\u00fd z\u00edskame vydelen\u00edm v\u00e4\u010d\u0161ieho \u010d\u00edsla men\u0161\u00edm. Podobne pracujeme aj s rozdielom.)<\/em><\/p>\n<div class=\"pri\">(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z5-I-3<\/strong><\/p>\n<p>Julka pripravuje pre svoje kamar\u00e1tky ob\u010derstvenie &#8211; chleb\u00ed\u010dky. Natrie ich zemiakov\u00fdm \u0161al\u00e1tom a navrch chce da\u0165 e\u0161te pr\u00edsady: \u0161unku, tvrd\u00fd syr, pl\u00e1tok vaj\u00ed\u010dka a pr\u00fa\u017eok nakladanej papriky. Nechce v\u0161ak, aby niektor\u00e9 dva chleb\u00ed\u010dky obsahovali \u00faplne rovnak\u00fa kombin\u00e1ciu pr\u00edsad. Ak\u00fd najv\u00e4\u010d\u0161\u00ed po\u010det navz\u00e1jom r\u00f4znych chleb\u00ed\u010dkov m\u00f4\u017ee nachysta\u0165, ak \u017eiadny z nich nem\u00e1 ma\u0165 v\u0161etky \u0161tyri pr\u00edsady a \u017eiadny z nich nie je iba so \u0161al\u00e1tom (t.j. bez \u010fal\u0161\u00edch pr\u00edsad)?<\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z5-I-4<\/strong><\/p>\n<p>Na obr\u00e1zku je nakreslen\u00e1 stavba zlepen\u00e1 z rovnako ve\u013ek\u00fdch kociek. Stavba je vlastne ve\u013ek\u00e1 kocka s tromi rovn\u00fdmi tunelmi, ktor\u00fdmi sa d\u00e1 pozera\u0165 skrz a ktor\u00e9 maj\u00fa v\u0161ade rovnak\u00fd prierez. Z ko\u013ek\u00fdch kociek je stavba zlepen\u00e1?<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/61\/61z5a3.gif\" alt=\"\" width=\"198\" height=\"189\" \/><\/p>\n<div class=\"pri\">(M. Krej\u010dov\u00e1)<\/div>\n<p><strong>Z5-I-5<\/strong><\/p>\n<p>V rozpr\u00e1vke o siedmich zhavranen\u00fdch bratoch bolo sedem bratov, z ktor\u00fdch ka\u017ed\u00fd sa narodil presne rok a pol po predch\u00e1dzaj\u00facom bratovi. Ke\u010f bol najstar\u0161\u00ed z bratov pr\u00e1ve \u0161tyrikr\u00e1t star\u0161\u00ed ako najmlad\u0161\u00ed, matka v\u0161etk\u00fdch bratov zakliala. Ko\u013eko rokov mali jednotliv\u00ed bratia, ke\u010f ich matka zakliala?<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z5-I-6<\/strong><\/p>\n<p>Janka a Hanka sa rady hraj\u00fa s modelmi zvieratiek. Hanka pre svoje kravi\u010dky postavila z uz\u00e1verov z PET flia\u0161 obd\u013a\u017enikov\u00fa ohradu, pozri obr\u00e1zok. Janka zo v\u0161etk\u00fdch svojich uz\u00e1verov zlo\u017eila pre ove\u010dky ohradu tvaru rovnostrann\u00e9ho trojuholn\u00edka. Potom ju rozobrala a postavila pre ne \u0161tvorcov\u00fa ohradu, takisto zo v\u0161etk\u00fdch svojich uz\u00e1verov. Ko\u013eko mohla ma\u0165 Janka uz\u00e1verov? N\u00e1jdite aspo\u0148 dve rie\u0161enia.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/61\/61z5a4.png\" alt=\"\" width=\"153\" height=\"99\" \/><\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td width=\"85%\" align=\"left\">\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> <input onclick=\"window.location='61\/61z5iir.pdf'\" name=\"butt52\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z5-II-1<\/strong><br \/>\nVojto si k\u00fapil 24 rovnako ve\u013ek\u00fdch \u0161tvorcov\u00fdch dla\u017ed\u00edc. Strana ka\u017edej dla\u017edice merala 40 cm. Zo v\u0161etk\u00fdch dla\u017ed\u00edc postavil pred svoju chatku obd\u013a\u017enikov\u00fa plo\u0161inu s najmen\u0161\u00edm mo\u017en\u00fdm obvodom. Ko\u013eko meral obvod tejto obd\u013a\u017enikovej plo\u0161iny?<br \/>\n<em>(Vojto dla\u017edice nerezal ani inak nel\u00e1mal.)<\/em><\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z5-II-2<\/strong><br \/>\nL\u00fape\u017en\u00edk Rumcajs u\u010dil Cip\u00fa\u0161ika p\u00edsa\u0165 \u010d\u00edsla. Ako prv\u00e9 nap\u00edsal Cip\u00fa\u0161ik \u010d\u00edslo 1, potom nap\u00edsal \u010d\u00edslo 2 a potom pokra\u010doval p\u00edsan\u00edm \u010fal\u0161\u00edch bezprostredne po sebe id\u00facich prirodzen\u00fdch \u010d\u00edsel. O chv\u00ed\u013eu to Cip\u00fa\u0161ika prestalo bavi\u0165 a prosil, aby prestali. Rumcajs sa nechal prehovori\u0165 a p\u00edsanie skon\u010dili. Ktor\u00e9 \u010d\u00edslo nap\u00edsal Cip\u00fa\u0161ik ako posledn\u00e9, ak sa cifra nula nach\u00e1dza v nap\u00edsan\u00fdch \u010d\u00edslach 35-kr\u00e1t?<\/p>\n<div class=\"pri\">(M. Krej\u010dov\u00e1)<\/div>\n<p><strong>Z5-II-3<\/strong><br \/>\nNa dr\u00f4te sed\u00ed 9 lastovi\u010diek, pri\u010dom vzdialenos\u0165 ka\u017ed\u00fdch dvoch susedn\u00fdch lastovi\u010diek je rovnak\u00e1. Vzdialenos\u0165 prvej a poslednej lastovi\u010dky je 720 cm.<\/p>\n<ul>\n<li> Ak\u00e1 je vzdialenos\u0165 dvoch susedn\u00fdch lastovi\u010diek?<\/li>\n<li> Ko\u013eko by na dr\u00f4te sedelo lastovi\u010diek, keby si medzi ka\u017ed\u00e9 dve susedn\u00e9 lastovi\u010dky, ktor\u00e9 pr\u00e1ve teraz sedia na dr\u00f4te, sadli \u010fal\u0161ie tri?<\/li>\n<\/ul>\n<div class=\"pri\">(M. Krej\u010dov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td width=\"85%\" align=\"left\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z61\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form> <input onclick=\"window.location='61\/61z6ir.pdf'\" name=\"butt61\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z6-I-1<\/strong><\/p>\n<p>Mirka \u010distila fazu\u013eu do polievky na papieri, ktor\u00fd vytiahla zo stol\u00edka svojej sestry. Na papieri boli nakreslen\u00e9 tri rovnako ve\u013ek\u00e9 kruhy, v ktor\u00fdch spolo\u010dn\u00e9 \u010dasti boli vyfarben\u00e9 sivou pastelkou. Do bielych \u010dast\u00ed jednotliv\u00fdch kruhov umiestnila to\u013eko fazuliek, ko\u013eko je nap\u00edsan\u00e9 na obr\u00e1zku. Ko\u013eko fazuliek m\u00e1 umiestni\u0165 do siv\u00fdch \u010dast\u00ed, aby bol v ka\u017edom kruhu rovnak\u00fd po\u010det fazuliek?<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/61\/61z6a1.png\" alt=\"\" width=\"174\" height=\"79\" \/><\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z6-I-2<\/strong><\/p>\n<p>Do hra\u010dk\u00e1rstva priviezli nov\u00e9 ply\u0161ov\u00e9 zvieratk\u00e1: v\u00e1\u017eky, p\u0161trosy a kraby. Ka\u017ed\u00e1 v\u00e1\u017eka m\u00e1 6 n\u00f4h a 4 kr\u00eddla, ka\u017ed\u00fd p\u0161tros m\u00e1 2 nohy a 2 kr\u00eddla a ka\u017ed\u00fd krab m\u00e1 8 n\u00f4h a 2 klepet\u00e1. Dohromady maj\u00fa tieto privezen\u00e9 hra\u010dky 118 n\u00f4h, 22 kr\u00eddiel a 22 klepiet. Ko\u013eko maj\u00fa dohromady hl\u00e1v?<\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z6-I-3<\/strong><\/p>\n<p>Na obr\u00e1zku je stavba zlepen\u00e1 z rovnak\u00fdch kociek. Stavba je vlastne ve\u013ek\u00e1 kocka s tromi rovn\u00fdmi tunelmi, ktor\u00fdmi sa d\u00e1 pozera\u0165 skrz a ktor\u00e9 maj\u00fa v\u0161ade rovnak\u00fd prierez. T\u00fato stavbu sme cel\u00fa ponorili do farby. Ko\u013eko kociek, z ktor\u00fdch je kocka zlo\u017een\u00e1, m\u00e1 zafarben\u00fa aspo\u0148 jednu stenu?<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/61\/61z5a3.gif\" alt=\"\" width=\"198\" height=\"189\" \/><\/p>\n<div class=\"pri\">(M. Krej\u010dov\u00e1)<\/div>\n<p><strong>Z6-I-4<\/strong><\/p>\n<p>Do ka\u017ed\u00e9ho nevyplnen\u00e9ho \u0161tvor\u010deka dopl\u0148te \u010d\u00edslo 1, 2, 3 alebo 4 tak, aby v ka\u017edom st\u013apci a riadku bolo ka\u017ed\u00e9 z t\u00fdchto \u010d\u00edsel pr\u00e1ve raz a aby boli splnen\u00e9 dodato\u010dn\u00e9 po\u017eiadavky v ka\u017edej vyzna\u010denej oblasti.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/61\/61z6a3.png\" alt=\"\" width=\"395\" height=\"275\" \/><\/p>\n<p><em>(Ak vo vyzna\u010denej oblasti po\u017eadujeme ur\u010dit\u00fd podiel, m\u00e1me na mysli podiel, ktor\u00fd z\u00edskame vydelen\u00edm v\u00e4\u010d\u0161ieho \u010d\u00edsla men\u0161\u00edm. Podobne pracujeme aj s rozdielom.)<\/em><\/p>\n<div class=\"pri\">(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z6-I-5<\/strong><\/p>\n<p>Ondro, Ma\u0165o a Kubo dostali na Vianoce od prarodi\u010dov ka\u017ed\u00fd jednu z nasleduj\u00facich hra\u010diek: ve\u013ek\u00e9 hasi\u010dsk\u00e9 auto, vrtu\u013en\u00edk na dia\u013ekov\u00e9 ovl\u00e1danie a stavebnicu Merkur. Bratranec Pe\u0165o doma hovoril:<\/p>\n<p>&#8222;Ondro dostal to ve\u013ek\u00e9 hasi\u010dsk\u00e9 auto. \u017delal si ho s\u00edce Kubo, ale ten ho nedostal. Ma\u0165o nem\u00e1 v ob\u013eube stavebnice, tak\u017ee Merkur nebol pre neho.&#8220;<\/p>\n<p>Uk\u00e1zalo sa, \u017ee Pe\u0165o sa dvakr\u00e1t m\u00fdlil v inform\u00e1cii, kto dostal \u010di nedostal dan\u00fd dar\u010dek a len raz povedal pravdu. Ako to teda s dar\u010dekmi bolo a kto teda dostal ak\u00fd dar\u010dek?<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z6-I-6<\/strong><\/p>\n<p>Marta, Libu\u0161ka a M\u00e1ria si vymysleli hru, ktor\u00fa by chceli hra\u0165 na obd\u013a\u017enikovom ihrisku zlo\u017eenom z 18 rovnak\u00fdch \u0161tvorcov, pozri obr\u00e1zok. Na hru potrebuj\u00fa ihrisko rozdeli\u0165 dvoma rovn\u00fdmi \u010diarami na tri rovnako ve\u013ek\u00e9 \u010dasti. Navy\u0161e tieto \u010diary musia obe prech\u00e1dza\u0165 t\u00fdm rohom ihriska, ktor\u00fd je na obr\u00e1zku v\u013eavo dole. Pora\u010fte diev\u010dat\u00e1m, ako maj\u00fa dokresli\u0165 \u010diary, aby sa mohli za\u010da\u0165 hra\u0165.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/61\/61z6a4.png\" alt=\"\" width=\"208\" height=\"107\" \/><\/p>\n<div class=\"pri\">(E. Troj\u00e1kov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td width=\"85%\" align=\"left\">\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> <input onclick=\"window.location='61\/61z6iir.pdf'\" name=\"butt62\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z6-II-1<\/strong><br \/>\nDanka a Janka dostali na narodeniny dve rovnako ve\u013ek\u00e9 biele kocky. Ka\u017ed\u00e1 kocka bola zlepen\u00e1 zo 125 mal\u00fdch kociek, tak ako vid\u00edte na obr\u00e1zku. Aby kocky rozoznali, dohodli sa, \u017ee ich oma\u013euj\u00fa. Danka vzala \u0161tetec a tri cel\u00e9 steny svojej kocky oma\u013eovala \u010dervenou farbou. Janka vzala \u0161tetec a tri cel\u00e9 steny svojej kocky ofarbila zelenou farbou. Po \u010dase sa diev\u010dat\u00e1 rozhodli, \u017ee ka\u017ed\u00fa z kociek rozkr\u00e1jaj\u00fa na jednotliv\u00e9 mal\u00e9 kocky. Ke\u010f to urobili, zistili, \u017ee po\u010det kociek, ktor\u00e9 maj\u00fa aspo\u0148 jednu stenu \u010derven\u00fa, je in\u00fd, ako po\u010det kociek, ktor\u00e9 maj\u00fa aspo\u0148 jednu stenu zelen\u00fa. Zistite, ak\u00fd je rozdiel t\u00fdchto po\u010dtov.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/61\/61z6b1.png\" alt=\"\" width=\"155\" height=\"156\" \/><\/p>\n<div class=\"pri\">(E. Novotn\u00e1)<\/div>\n<p><strong>Z6-II-2<\/strong><br \/>\nJurko zbiera podpisy zn\u00e1mych \u0161portovcov a spev\u00e1kov. Na podpisy si k\u00fapil zvl\u00e1\u0161tny zo\u0161it. Rozhodol sa, \u017ee podpisy bud\u00fa v\u017edy na ka\u017edom liste zo\u0161ita iba na prednej strane. Tieto strany postupne o\u010d\u00edsloval \u010d\u00edslami1, 3, 5, 7, 9, &#8230;  aby zistil, keby sa mu nejak\u00fd list zo zo\u0161ita stratil. V celom zo\u0161ite tak nap\u00edsal dokopy 125 cifier. Zistite:<\/p>\n<ul>\n<li> Ko\u013eko mal Jurkov zo\u0161it listov.<\/li>\n<li> Ko\u013eko jednotiek nap\u00edsal do zo\u0161ita.<\/li>\n<\/ul>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z6-II-3<\/strong><br \/>\nNa no\u010dn\u00fd pochod dostali dru\u017einy Vlkov a L\u00ed\u0161ok po jednej rovnakej svie\u010dke. Svie\u010dky zap\u00e1lili spolo\u010dne na \u0161tarte a vyrazili. Po\u010das pochodu ka\u017ed\u00fd \u010dlen dru\u017einy Vlkov niesol svie\u010dku tak\u00fa dobu, za ktor\u00fa sa jej d\u013a\u017eka skr\u00e1tila na polovicu. Vlci dobehli do cie\u013ea v momente, ke\u010f mal \u0161iesty \u010dlen dru\u017einy poda\u0165 svie\u010dku siedmemu. Od toho okamihu ich svie\u010dka cel\u00e1 dohorela o tri min\u00faty. L\u00ed\u0161ky dobehli do cie\u013ea za 2 hodiny a 57 min\u00fat.<\/p>\n<ul>\n<li> Ktor\u00e1 dru\u017eina bola v cieli ako prv\u00e1?<\/li>\n<li> O ko\u013eko min\u00fat dobehla v\u00ed\u0165azn\u00e1 dru\u017eina do cie\u013ea sk\u00f4r ako druh\u00e1 dru\u017eina?<\/li>\n<\/ul>\n<p>(<em>Svie\u010dka hor\u00ed rovnomerne: za rovnak\u00fd \u010das v\u017edy rovnak\u00fd k\u00fasok.<\/em>)<\/p>\n<div class=\"pri\">(M.Dillingerov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td width=\"85%\" align=\"left\">\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<form> <input onclick=\"window.location='61\/61z7ir.pdf'\" name=\"butt71\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z7-I-1<\/strong><\/p>\n<p>Trpasl\u00edci chodia na vodu k potoku. D\u017eb\u00e1nik ka\u017ed\u00e9ho z trpasl\u00edkov je inak ve\u013ek\u00fd: maj\u00fa objemy 3, 4, 5, 6, 7, 8 a 9 litrov. Trpasl\u00edci si d\u017eb\u00e1niky medzi sebou nepo\u017ei\u010diavaj\u00fa a v\u017edy ich prines\u00fa \u00faplne pln\u00e9 vody.<\/p>\n<ul>\n<li>K\u00fdchal prinesie vo svojom d\u017eb\u00e1niku viac vody ako Smejko.<\/li>\n<li>Spachto\u0161 by musel \u00eds\u0165 po vodu trikr\u00e1t, aby priniesol pr\u00e1ve to\u013eko vody, ko\u013eko Pla\u0161ko v jednom svojom d\u017eb\u00e1niku.<\/li>\n<li>Vedkov d\u017eb\u00e1nik je len o dva litre v\u00e4\u010d\u0161\u00ed ako Smejkov.<\/li>\n<li>S\u00e1m K\u00fdblik prinesie to\u013eko vody, ko\u013eko Spachto\u0161 a Smejko dokopy.<\/li>\n<li>Ke\u010f id\u00fa po vodu Vedko a K\u00fdblik, prines\u00fa rovnako vody ako Dudro\u0161, K\u00fdchal a Smejko dokopy.<\/li>\n<\/ul>\n<p>Ko\u013eko vody prines\u00fa K\u00fdchal a K\u00fdblik dohromady?<\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z7-I-2<\/strong><\/p>\n<p>Na obr\u00e1zku je \u0161tvorec <em>ABCD<\/em>, v ktorom s\u00fa umiestnen\u00e9 \u0161tyri zhodn\u00e9 rovnoramenn\u00e9 trojuholn\u00edky <em>ABE<\/em>, <em>BCF<\/em>, <em>CDG<\/em> a <em>DAH<\/em>, v\u0161etky vyfarben\u00e9 sivou. Strany \u0161tvorca <em>ABCD<\/em> s\u00fa z\u00e1klad\u0148ami t\u00fdchto rovnoramenn\u00fdch trojuholn\u00edkov. Vieme, \u017ee siv\u00e9 plochy \u0161tvorca <em>ABCD<\/em> maj\u00fa dokopy rovnak\u00fd obsah ako biela plocha \u0161tvorca. \u010ealej vieme, \u017ee |<em>HF<\/em>| = 12 cm. Ur\u010dte d\u013a\u017eku strany \u0161tvorca <em>ABCD<\/em>.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/61\/61z7a1.png\" alt=\"\" width=\"183\" height=\"163\" \/><\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z7-I-3<\/strong><\/p>\n<p>Sedem bezprostredne po sebe id\u00facich cel\u00fdch \u010d\u00edsel st\u00e1lo v rade, zoraden\u00e9 od najmen\u0161ieho po najv\u00e4\u010d\u0161ie. Po chv\u00edli sa \u010d\u00edsla za\u010dali nudi\u0165, a tak sa najsk\u00f4r vymenilo prv\u00e9 s posledn\u00fdm. Potom sa druh\u00e9 najv\u00e4\u010d\u0161ie posunulo \u00faplne na za\u010diatok radu a nakoniec sa najv\u00e4\u010d\u0161ie z \u010d\u00edsel postavilo do stredu. Na svoju ve\u013ek\u00fa spokojnos\u0165 sa tak ocitlo ved\u013ea \u010d\u00edsla, ktor\u00e9 bolo presne jeho polovicou. Ktor\u00fdch sedem \u010d\u00edsel mohlo st\u00e1\u0165 p\u00f4vodne v rade?<\/p>\n<div class=\"pri\">(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z7-I-4<\/strong><\/p>\n<p>U\u010dite\u013eka Smoliarska pripravovala previerku pre svoju triedu v troch verzi\u00e1ch, aby \u017eiaci nemohli odpisova\u0165. V ka\u017edej verzii zadala tri hrany kv\u00e1dra v centimetroch a \u00falohou bolo vypo\u010d\u00edta\u0165 jeho objem. \u00dalohy si ale dopredu neprerie\u0161ila, a tak netu\u0161ila, \u017ee v\u00fdsledok je vo v\u0161etk\u00fdch troch verzi\u00e1ch rovnak\u00fd. Do zadan\u00ed \u017eiakom nap\u00edsala tieto d\u013a\u017eky hr\u00e1n: 12, 18, 20, 24, 30, 33 a 70. Z deviatich celo\u010d\u00edseln\u00fdch d\u013a\u017eok hr\u00e1n, ktor\u00e9 u\u010dite\u013eka Smoliarska zadala, sme v\u00e1m teda prezradili iba sedem a ani sme v\u00e1m neprezradili, ktor\u00e9 d\u013a\u017eky patria do toho ist\u00e9ho zadania. Podar\u00ed sa v\u00e1m napriek tomu ur\u010di\u0165 zost\u00e1vaj\u00face dve d\u013a\u017eky hr\u00e1n?<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z7-I-5<\/strong><\/p>\n<p>Jeden vn\u00fatorn\u00fd uhol trojuholn\u00edka m\u00e1 50\u00b0. Ak\u00fd ve\u013ek\u00fd uhol zvieraj\u00fa osi dvoch zost\u00e1vaj\u00facich vn\u00fatorn\u00fdch uhlov trojuholn\u00edka?<\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z7-I-6<\/strong><\/p>\n<p>H\u013ead\u00e1me \u0161es\u0165cifern\u00fd \u010d\u00edseln\u00fd k\u00f3d, o ktorom vieme, \u017ee:<\/p>\n<ul>\n<li>\u017eiadna cifra v \u0148om nie je viackr\u00e1t,<\/li>\n<li>obsahuje aj 0, t\u00e1 v\u0161ak nie je na predposlednom mieste,<\/li>\n<li>vo svojom z\u00e1pise nem\u00e1 nikdy ved\u013ea seba dve p\u00e1rne ani dve nep\u00e1rne cifry,<\/li>\n<li>susedn\u00e9 cifry sa od seba l\u00ed\u0161ia aspo\u0148 o 3,<\/li>\n<li>ke\u010f \u010d\u00edslo rozdel\u00edme na tri dvoj\u010d\u00edslia, tak prv\u00e9 aj druh\u00e9 dvoj\u010d\u00edslie s\u00fa obe n\u00e1sobkom tretieho, teda posledn\u00e9ho dvoj\u010d\u00edslia.<\/li>\n<\/ul>\n<p>Ur\u010dte h\u013eadan\u00fd k\u00f3d.<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td width=\"85%\" align=\"left\">\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> <input onclick=\"window.location='61\/61z7iir.pdf'\" name=\"butt72\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z7-II-1<\/strong><br \/>\nPeter a Karol spolu hrali ve\u013ea parti\u00ed d\u00e1my. Dohodli sa, \u017ee za v\u00fdhru si hr\u00e1\u010d pripo\u010d\u00edta 3 body, za prehru si 2 body od\u010d\u00edta a za rem\u00edzu sa \u017eiadne body nepripo\u010d\u00edtavaj\u00fa ani neod\u010dituj\u00fa. Petrova sestra chcela vedie\u0165, ko\u013eko u\u017e Peter a Karol odohrali parti\u00ed a kto vedie, ale dozvedela sa iba, \u017ee Peter \u0161es\u0165kr\u00e1t vyhral, dvakr\u00e1t remizoval a nieko\u013ekokr\u00e1t prehral a Karol m\u00e1 pr\u00e1ve 9 bodov. Zistite, ko\u013eko parti\u00ed chlapci hrali a kto pr\u00e1ve teraz vedie.<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z7-II-2<\/strong><br \/>\nV trojuholn\u00edku ABC sa osi jeho vn\u00fatorn\u00fdch uhlov pret\u00ednaj\u00fa v bode S. Uhol BSC m\u00e1 ve\u013ekos\u0165 130\u00b0 a uhol ASC m\u00e1 120\u00b0. Zistite ve\u013ekosti v\u0161etk\u00fdch vn\u00fatorn\u00fdch uhlov trojuholn\u00edka ABC.<\/p>\n<div class=\"pri\">(E. Pat\u00e1kov\u00e1)<\/div>\n<p><strong>Z7-II-3<\/strong><br \/>\nTrojcifern\u00e9 prirodzen\u00e9 \u010d\u00edslo budeme naz\u00fdva\u0165 p\u00e1rnomiln\u00e9, ak obsahuje dve p\u00e1rne cifry a cifru 1. Trojcifern\u00e9 prirodzen\u00e9 \u010d\u00edslo budeme naz\u00fdva\u0165 nep\u00e1rnomiln\u00e9, ak obsahuje dve nep\u00e1rne cifry a cifru 2. Zistite, ko\u013eko je v\u0161etk\u00fdch trojcifern\u00fdch p\u00e1rnomiln\u00fdch a ko\u013eko nep\u00e1rnomiln\u00fdch \u010d\u00edsel.<\/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 width=\"85%\" align=\"left\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z81\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form> <input onclick=\"window.location='61\/61z8ir.pdf'\" name=\"butt81\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z8-I-1<\/strong><\/p>\n<p>Kore\u0161ponden\u010dn\u00e1 matematick\u00e1 s\u00fa\u0165a\u017e prebieha v troch kol\u00e1ch, ktor\u00fdch n\u00e1ro\u010dnos\u0165 sa stup\u0148uje. Do druh\u00e9ho kola postupuj\u00fa len t\u00ed rie\u0161itelia, ktor\u00ed boli \u00faspe\u0161n\u00ed v prvom kole, do tretieho kola postupuj\u00fa len \u00faspe\u0161n\u00ed rie\u0161itelia druh\u00e9ho kola. V\u00ed\u0165azom je ka\u017ed\u00fd, kto je \u00faspe\u0161n\u00fdm rie\u0161ite\u013eom posledn\u00e9ho, teda tretieho kola. V poslednom ro\u010dn\u00edku tejto s\u00fa\u0165a\u017ee bolo presne 14% rie\u0161ite\u013eov \u00faspe\u0161n\u00fdch v prvom kole, presne 25% rie\u0161ite\u013eov druh\u00e9ho kola post\u00fapilo do tretieho kola a presne 8% rie\u0161ite\u013eov tretieho kola zv\u00ed\u0165azilo. Ak\u00fd je najmen\u0161\u00ed po\u010det s\u00fa\u0165a\u017eiacich, ktor\u00ed sa mohli z\u00fa\u010dastni\u0165 prv\u00e9ho kola? Ko\u013eko by bolo v takomto pr\u00edpade v\u00ed\u0165azov?<\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z8-I-2<\/strong><\/p>\n<p>Je dan\u00fd rovnoramenn\u00fd trojuholn\u00edk <em>ABC<\/em> so z\u00e1klad\u0148ou <em>AB<\/em> dlhou 10 cm a ramenami dlh\u00fdmi 20 cm. Bod <em>S<\/em> je stred z\u00e1kladne <em>AB<\/em>. Rozde\u013ete trojuholn\u00edk <em>ABC<\/em> \u0161tyrmi priamkami prech\u00e1dzaj\u00facimi bodom <em>S<\/em> na p\u00e4\u0165 \u010dast\u00ed s rovnak\u00fdm obsahom. Zistite, ak\u00e9 dlh\u00e9 \u00faseky vytn\u00fa tieto priamky na ramen\u00e1ch trojuholn\u00edka <em>ABC<\/em>.<\/p>\n<div class=\"pri\">(E. Troj\u00e1kov\u00e1)<\/div>\n<p><strong>Z8-I-3<\/strong><\/p>\n<p>H\u013ead\u00e1me p\u00e4\u0165cifern\u00e9 \u010d\u00edslo s nasleduj\u00facimi vlastnos\u0165ami: je to palindr\u00f3m (t.j. \u010d\u00edta sa odzadu rovnako ako odpredu), je delite\u013en\u00e9 dvan\u00e1stimi a vo svojom z\u00e1pise obsahuje cifru 2 bezprostredne za cifrou 4. Ur\u010dte v\u0161etky mo\u017en\u00e9 \u010d\u00edsla, ktor\u00e9 vyhovuj\u00fa zadan\u00fdm podmienkam.<\/p>\n<div class=\"pri\">(M. Mach)<\/div>\n<p><strong>Z8-I-4<\/strong><\/p>\n<p>Na stred hrn\u010diarskeho kruhu sme polo\u017eili kocku, ktor\u00e1 mala na ka\u017edej svojej stene nap\u00edsan\u00e9 jedno prirodzen\u00e9 \u010d\u00edslo. Tesne predt\u00fdm, ako sme kruh rozto\u010dili, sme z miesta, kde stoj\u00edme, videli tri steny kocky a teda len tri \u010d\u00edsla. Ich s\u00fa\u010det bol 42. Po oto\u010den\u00ed hrn\u010diarskeho kruhu o 90\u00b0 sme z rovnak\u00e9ho miesta videli tri steny s \u010d\u00edslami, ktor\u00e9 mali s\u00fa\u010det 34 a po oto\u010den\u00ed o \u010fal\u0161\u00edch 90\u00b0 sme videli tri \u010d\u00edsla so s\u00fa\u010dtom 53.<\/p>\n<p><strong>1.<\/strong> Ur\u010dte s\u00fa\u010det troch \u010d\u00edsel, ktor\u00e9 z n\u00e1\u0161ho miesta uvid\u00edme, ke\u010f sa kruh oto\u010d\u00ed e\u0161te o \u010fal\u0161\u00edch 90\u00b0.<\/p>\n<p><strong>2.<\/strong> Kocka cel\u00fd \u010das le\u017eala na stene s \u010d\u00edslom 6. Ur\u010dte maxim\u00e1lny mo\u017en\u00fd s\u00fa\u010det v\u0161etk\u00fdch \u0161iestich \u010d\u00edsel na kocke.<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z8-I-5<\/strong><\/p>\n<p>Pankr\u00e1c, Serv\u00e1c a Bonif\u00e1c s\u00fa traja bratia, ktor\u00ed maj\u00fa P, S a B rokov. Vieme, \u017ee P, S a B s\u00fa prirodzen\u00e9 \u010d\u00edsla men\u0161ie ako 16, pre ktor\u00e9 plat\u00ed:<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/61\/61z8a1.gif\" alt=\"\" width=\"95\" height=\"90\" \/><\/p>\n<p>Ur\u010dte vek v\u0161etk\u00fdch troch bratov.<\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z8-I-6<\/strong><\/p>\n<p>Janka narysovala obd\u013a\u017enik s obvodom 22 cm a d\u013a\u017ekami str\u00e1n vyjadren\u00fdmi v centimetroch cel\u00fdmi \u010d\u00edslami. Potom obd\u013a\u017enik rozdelila bezo zvy\u0161ku na tri obd\u013a\u017eniky, z ktor\u00fdch jeden mal rozmery 2 cm \u00d7 6 cm. S\u00fa\u010det obvodov v\u0161etk\u00fdch troch obd\u013a\u017enikov bol o 18 cm v\u00e4\u010d\u0161\u00ed ako obvod p\u00f4vodn\u00e9ho obd\u013a\u017enika. Ak\u00e9 rozmery mohol ma\u0165 p\u00f4vodn\u00fd obd\u013a\u017enik? N\u00e1jdite v\u0161etky rie\u0161enia.<\/p>\n<div class=\"pri\">(M. Dillingerov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td width=\"85%\" align=\"left\">\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> <input onclick=\"window.location='61\/61z8iir.pdf'\" name=\"butt82\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z8-II-1<\/strong><br \/>\nDivadeln\u00fd s\u00fabor uviedol po\u010das sez\u00f3ny tridsa\u0165kr\u00e1t \u201eVe\u010der pln\u00fd improviz\u00e1cie\u201c. Filom\u00e9na, obdivovate\u013eka hlavn\u00e9ho hrdinu, si na za\u010diatku sez\u00f3ny vypo\u010d\u00edtala, ko\u013eko by dokopy minula pe\u0148az\u00ed, keby chodila na ka\u017ed\u00e9 predstavenie. Po nieko\u013ek\u00fdch predstaveniach v\u0161ak vstupn\u00e9 ne\u010dakane vzr\u00e1stlo o 6\u20ac. Nesk\u00f4r z\u00edskal s\u00fabor sponzora a t\u00fato nov\u00fa cenu zn\u00ed\u017eil o 8,50\u20ac. Na konci sez\u00f3ny Filom\u00e9na mohla poveda\u0165, \u017ee nevynechala ani jedno predstavenie a na vstupn\u00e9 minula presne to\u013eko, ko\u013eko si vypo\u010d\u00edtala na za\u010diatku sez\u00f3ny. Ko\u013ekokr\u00e1t i\u0161la Filom\u00e9na na predstavenie za vstupn\u00e9 v p\u00f4vodnej v\u00fd\u0161ke?<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z8-II-2<\/strong><br \/>\nN\u00e1jdite najmen\u0161ie tak\u00e9 prirodzen\u00e9 \u010d\u00edslo, \u017ee jeho polovica je delite\u013en\u00e1 tromi, jeho tretina je delite\u013en\u00e1 \u0161tyrmi, jeho \u0161tvrtina je delite\u013en\u00e1 jeden\u00e1stimi a jeho polovica d\u00e1va po delen\u00ed siedmimi zvy\u0161ok 5.<\/p>\n<div class=\"pri\">(E. Pat\u00e1kov\u00e1)<\/div>\n<p><strong>Z8-II-3<\/strong><br \/>\nJe dan\u00fd trojuholn\u00edk <em>ABC<\/em>, ktor\u00e9ho n\u00e1\u010drt vid\u00edte na obr\u00e1zku. Na strane <em>AB<\/em> le\u017e\u00ed bod <em>X<\/em> a na strane <em>BC<\/em> le\u017e\u00ed bod <em>Y<\/em> tak, \u017ee <em>CX<\/em> je \u0165a\u017enica, <em>AY<\/em> je v\u00fd\u0161ka a <em>XY<\/em> je stredn\u00e1 prie\u010dka trojuholn\u00edka <em>ABC<\/em>. Vypo\u010d\u00edtajte obsah trojuholn\u00edka vyzna\u010den\u00e9ho na obr\u00e1zku sivou farbou, ak obsah trojuholn\u00edka <em>ABC<\/em> je 24 cm<sup>2<\/sup>.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/61\/61z8b1.png\" alt=\"\" width=\"213\" height=\"158\" \/><\/p>\n<div class=\"pri\">(M.Dillingerov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td width=\"85%\" align=\"left\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z91\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form> <input onclick=\"window.location='61\/61z9ir.pdf'\" name=\"butt91\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z9-I-1<\/strong><\/p>\n<p>Pokladn\u00ed\u010dka v gal\u00e9rii pred\u00e1va n\u00e1v\u0161tevn\u00edkom vstupenky s \u010d\u00edslom pod\u013ea toho, ko\u013ek\u00ed v porad\u00ed v ten de\u0148 pri\u0161li. Prv\u00fd n\u00e1v\u0161tevn\u00edk dostane vstupenku s \u010d\u00edslom 1, druh\u00fd s \u010d\u00edslom 2, at\u010f. Po\u010das d\u0148a sa v\u0161ak minul \u017elt\u00fd papier, na ktor\u00fd sa vstupenky tla\u010dili, preto musela pokladn\u00ed\u010dka pokra\u010dova\u0165 tla\u010den\u00edm na \u010derven\u00fd papier. Za cel\u00fd de\u0148 predala rovnako ve\u013ea \u017elt\u00fdch vstupeniek ako \u010derven\u00fdch. Ve\u010der zistila, \u017ee s\u00fa\u010det \u010d\u00edsel na \u017elt\u00fdch vstupenk\u00e1ch bol o 1 681 men\u0161\u00ed ako s\u00fa\u010det \u010d\u00edsel na \u010derven\u00fdch vstupenk\u00e1ch. Ko\u013eko vstupeniek v ten de\u0148 predala?<\/p>\n<div class=\"pri\">(M. Mach)<\/div>\n<p><strong>Z9-I-2<\/strong><\/p>\n<p>Filom\u00e9na m\u00e1 mobil s nasleduj\u00facim rozmiestnen\u00edm tla\u010didiel:<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/61\/61z9a1.png\" alt=\"\" width=\"75\" height=\"100\" \/><\/p>\n<p>Dev\u00e4\u0165cifern\u00e9 telef\u00f3nne \u010d\u00edslo jej najlep\u0161ej kamar\u00e1tky Kl\u00e1ry m\u00e1 tieto vlastnosti:<\/p>\n<ul>\n<li>v\u0161etky cifry Kl\u00e1rinho telef\u00f3nneho \u010d\u00edsla s\u00fa r\u00f4zne,<\/li>\n<li>prv\u00e9 \u0161tyri cifry s\u00fa zoraden\u00e9 pod\u013ea ve\u013ekosti od najmen\u0161ej po najv\u00e4\u010d\u0161iu a stredy ich tla\u010didiel tvoria \u0161tvorec,<\/li>\n<li>stredy tla\u010didiel posledn\u00fdch \u0161tyroch cifier takisto tvoria \u0161tvorec,<\/li>\n<li>telef\u00f3nne \u010d\u00edslo je delite\u013en\u00e9 tromi aj piatimi.<\/li>\n<\/ul>\n<p>Ko\u013eko r\u00f4znych dev\u00e4\u0165cifern\u00fdch \u010d\u00edsel by mohlo by\u0165 Kl\u00e1rin\u00fdm telef\u00f3nnym \u010d\u00edslom?<\/p>\n<div class=\"pri\">(K. Pazourek)<\/div>\n<p><strong>Z9-I-3<\/strong><\/p>\n<p>Alenka pozorovala veveri\u010dky na z\u00e1hradke, kde r\u00e1stli tieto tri stromy: smrek, buk a jed\u013ea. Veveri\u010dky sedeli pokojne na stromoch, tak\u017ee ich mohla spo\u010d\u00edta\u0165 &#8211; bolo ich 34. Ke\u010f presk\u00e1kalo 7 veveri\u010diek zo smreka na buk, bolo ich na buku rovnako ve\u013ea ako na oboch ihli\u010dnanoch dokopy. Potom e\u0161te presk\u00e1kalo 5 veveri\u010diek z jedle na buk, a vtedy bolo na jedli rovnako ve\u013ea veveri\u010diek ako na smreku. Na buku ich bolo vtedy dvakr\u00e1t viac, ako na jedli na \u00faplnom za\u010diatku. Ko\u013eko veveri\u010diek p\u00f4vodne sedelo na ka\u017edom strome?<\/p>\n<div class=\"pri\">(M. Mach)<\/div>\n<p><strong>Z9-I-4<\/strong><\/p>\n<p>V pravidelnom dvan\u00e1s\u0165uholn\u00edku ABCDEFGHIJKL vp\u00edsanom do kru\u017enice s polomerom 6 cm ur\u010dte obvod p\u00e4\u0165uholn\u00edka ACFHK.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/61\/61z9a2.png\" alt=\"\" width=\"194\" height=\"191\" \/><\/p>\n<div class=\"pri\">(K. Pazourek)<\/div>\n<p><strong>Z9-I-5<\/strong><\/p>\n<p>Pred viano\u010dn\u00fdm koncertom pon\u00fakali \u017eiaci na predaj 60 v\u00fdrobkov z hod\u00edn v\u00fdtvarnej v\u00fdchovy. Cenu si mohol ka\u017ed\u00fd z\u00e1kazn\u00edk ur\u010di\u0165 s\u00e1m a cel\u00fd v\u00fd\u0165a\u017eok i\u0161iel na dobro\u010dinn\u00e9 \u00fa\u010dely. Na za\u010diatku koncertu \u017eiaci spo\u010d\u00edtali, ko\u013eko centov v priemere ut\u0155\u017eili za jeden predan\u00fd v\u00fdrobok, a vy\u0161lo im cel\u00e9 \u010d\u00edslo. Ke\u010f\u017ee ale nepredali v\u0161etk\u00fdch 60 v\u00fdrobkov, pon\u00fakali ich aj po koncerte. Po koncerte si \u013eudia k\u00fapili e\u0161te sedem v\u00fdrobkov, za ktor\u00e9 dali dokopy 2 505 centov. T\u00fdm sa priemern\u00e1 tr\u017eba za jeden predan\u00fd v\u00fdrobok zv\u00fd\u0161ila na rovn\u00fdch 130 centov. Ko\u013eko v\u00fdrobkov potom ostalo nepredan\u00fdch?<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z9-I-6<\/strong><\/p>\n<p>V obd\u013a\u017enikovej z\u00e1hrade rastie brosky\u0148a. Tento strom je od dvoch susedn\u00fdch rohov z\u00e1hrady vzdialen\u00fd 5 metrov a 12 metrov a vzdialenos\u0165 medzi spom\u00ednan\u00fdmi dvoma rohmi je 13 metrov. \u010ealej vieme, \u017ee brosky\u0148a stoj\u00ed na uhloprie\u010dke z\u00e1hrady. Ak\u00e1 ve\u013ek\u00e1 m\u00f4\u017ee by\u0165 plocha z\u00e1hrady?<\/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 width=\"85%\" align=\"left\">\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> <input onclick=\"window.location='61\/61z9iir.pdf'\" name=\"butt92\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z9-II-1<\/strong><br \/>\nDan\u00fd je kosod\u013a\u017enik <em>ABCD<\/em> ako na obr\u00e1zku. Po strane <em>AB<\/em> sa pohybuje bod <em>E<\/em> a po strane <em>DC<\/em> sa pohybuje bod <em>G<\/em> tak, \u017ee \u00fase\u010dka <em>EG<\/em> je st\u00e1le rovnobe\u017en\u00e1 s <em>AD<\/em>. Ke\u010f bol priese\u010dn\u00edk <em>F<\/em> \u00fase\u010diek <em>EG<\/em> a <em>DB<\/em> v p\u00e4tine uhloprie\u010dky <em>DB<\/em> (bli\u017e\u0161ie k bodu <em>D<\/em>), bol obsah vyfarbenej \u010dasti kosod\u013a\u017enika o 1 cm<sup>2<\/sup> v\u00e4\u010d\u0161\u00ed, ako ke\u010f bol <em>F<\/em> v dvoch p\u00e4tin\u00e1ch <em>DB<\/em> (op\u00e4\u0165 bli\u017e\u0161ie k <em>D<\/em>). Ur\u010dte obsah kosod\u013a\u017enika <em>ABCD<\/em>.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/61\/61z5b1.png\" alt=\"\" width=\"261\" height=\"118\" \/><\/p>\n<div class=\"pri\">(E. Pat\u00e1kov\u00e1)<\/div>\n<p><strong>Z9-II-2<\/strong><br \/>\nSnehulienka m\u00e1 na z\u00e1hrade 101 sadrov\u00fdch trpasl\u00edkov zoraden\u00fdch pod\u013ea hmotnosti od naj\u0165a\u017e\u0161ieho po naj\u013eah\u0161ieho, pri\u010dom rozdiel hmotnost\u00ed ka\u017ed\u00fdch dvoch susedn\u00fdch trpasl\u00edkov je rovnak\u00fd. Raz Snehulienka trpasl\u00edkov v\u00e1\u017eila a zistila, \u017ee prv\u00fd (teda naj\u0165a\u017e\u0161\u00ed) trpasl\u00edk v\u00e1\u017ei presne 5 kg. Snehulienku v\u0161ak najviac prekvapilo, \u017ee ke\u010f na v\u00e1hu postavila v\u0161etk\u00fdch trpasl\u00edkov od 76. po 80., v\u00e1\u017eili dokopy rovnako ako v\u0161etci trpasl\u00edci od 96. po 101. Ko\u013eko v\u00e1\u017ei naj\u013eah\u0161\u00ed trpasl\u00edk?<\/p>\n<div class=\"pri\">(M. Mach)<\/div>\n<p><strong>Z9-II-3<\/strong><br \/>\nTuristick\u00fd oddiel usporiadal trojd\u0148ov\u00fd cyklistick\u00fd v\u00fdlet. Prv\u00fd de\u0148 chceli prejs\u0165 1\/3 celej pl\u00e1novanej trasy, ale pre\u0161li o 4 km menej ako chceli. Druh\u00fd de\u0148 chceli prejs\u0165 viac: polovicu zvy\u0161ku, ale pre\u0161li o 2 km menej ako chceli. Tret\u00ed de\u0148 v\u0161ak v\u0161etko dobehli a pre\u0161li 10\/11 zvy\u0161ku trasy a e\u0161te 4 km, tak\u017ee dorazili do pl\u00e1novan\u00e9ho cie\u013ea. Ak\u00e1 dlh\u00e1 bola trasa a ko\u013eko pre\u0161li prv\u00fd, druh\u00fd a tret\u00ed de\u0148?<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z9-II-4<\/strong><br \/>\nOrganiz\u00e1tor v\u00fdstavy &#8222;Staviam, stavia\u0161, staviame&#8220; rozdelil v\u00fdstavu na dve \u010dasti: &#8222;V\u00fdvoj stavebn\u00fdch techn\u00edk&#8220; a &#8222;Budovy z in\u00e9ho uhla&#8220;. Ke\u010f\u017ee ho zauj\u00edmala reakcia n\u00e1v\u0161tevn\u00edkov, vyplnil ka\u017ed\u00fd n\u00e1v\u0161tevn\u00edk pri odchode jednoduch\u00fd dotazn\u00edk. Vyplynuli z neho tieto zauj\u00edmav\u00e9 skuto\u010dnosti:<\/p>\n<ul>\n<li> medzi t\u00fdmi, ktor\u00fdm sa p\u00e1\u010dila prv\u00e1 \u010das\u0165, bolo 96% tak\u00fdch, ktor\u00fdm sa p\u00e1\u010dila aj druh\u00e1 \u010das\u0165;<\/li>\n<li> medzi t\u00fdmi, ktor\u00fdm sa p\u00e1\u010dila druh\u00e1 \u010das\u0165, bolo 60% tak\u00fdch, ktor\u00fdm sa p\u00e1\u010dila aj prv\u00e1 \u010das\u0165;<\/li>\n<li> medzi v\u0161etk\u00fdmi n\u00e1v\u0161tevn\u00edkmi bolo 59% tak\u00fdch, ktor\u00fdm sa nep\u00e1\u010dila ani prv\u00e1 ani druh\u00e1 \u010das\u0165.<\/li>\n<\/ul>\n<p>Ko\u013eko percent v\u0161etk\u00fdch n\u00e1v\u0161tevn\u00edkov uviedlo, \u017ee sa im p\u00e1\u010dila prv\u00e1 \u010das\u0165 v\u00fdstavy?<\/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 width=\"85%\" align=\"left\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z93\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form> <input onclick=\"window.location='61\/61z9iiir.pdf'\" name=\"butt93\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z9-III-1<\/strong><br \/>\nNa obr\u00e1zku je obd\u013a\u017enik <em>ABCD<\/em>, ktor\u00e9ho d\u013a\u017eky str\u00e1n <em>AB<\/em> a <em>BC<\/em> s\u00fa v pomere 7:5. Vn\u00fatri obd\u013a\u017enika <em>ABCD<\/em> le\u017eia body <em>X<\/em> a <em>Y<\/em> tak, \u017ee trojuholn\u00edky <em>ABX<\/em> a <em>CDY<\/em> s\u00fa pravouhl\u00e9 rovnoramenn\u00e9 s prav\u00fdmi uhlami pri vrcholoch <em>X<\/em> a <em>Y<\/em>. Spolo\u010dn\u00e1 siv\u00e1 plocha oboch trojuholn\u00edkov tvor\u00ed \u0161tvorec s obsahom 72 cm<sup>2<\/sup>.\u00a0 Ur\u010dte d\u013a\u017eky str\u00e1n <em>AB <\/em>a <em>BC <\/em>obd\u013a\u017enika <em>ABCD<\/em>.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/61\/61z9c1.gif\" alt=\"\" width=\"169\" height=\"126\" \/><\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z9-III-2<\/strong><br \/>\nMarienka mala desa\u0165 karti\u010diek, na ktor\u00e9 nap\u00edsala desa\u0165 po sebe id\u00facich prirodzen\u00fdch \u010d\u00edsel, na ka\u017ed\u00fa karti\u010dku pr\u00e1ve jedno. Ne\u0161\u0165astnou n\u00e1hodou v\u0161ak jednu karti\u010dku stratila. S\u00fa\u010det \u010d\u00edsel na zost\u00e1vaj\u00facich deviatich karti\u010dk\u00e1ch bol 2\u00a0012. Zistite, ak\u00e9 \u010d\u00edslo bolo nap\u00edsan\u00e9 na stratenej karti\u010dke.<\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z9-III-3<\/strong><br \/>\nZistite, ko\u013ek\u00fdmi r\u00f4znymi sp\u00f4sobmi sa daj\u00fa do jednotliv\u00fdch pol\u00ed\u010dok trojuholn\u00edka na obr\u00e1zku vp\u00edsa\u0165 \u010d\u00edsla 1, 2, 3, 4, 5, 6, 7, 8 a 9 tak, aby s\u00fa\u010det v ka\u017edom \u0161tvorpol\u00ed\u010dkovom trojuholn\u00edku bol 23 a aby na niektorom pol\u00ed\u010dku v smere ka\u017edej \u0161\u00edpky bolo vp\u00edsan\u00e9 \u010d\u00edslo zadan\u00e9 pri \u0161\u00edpke.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/61\/61z9c2.gif\" alt=\"\" width=\"159\" height=\"134\" \/><\/p>\n<div class=\"pri\">(E. Novotn\u00e1)<\/div>\n<p><strong>Z9-III-4<\/strong><br \/>\nVojto chcel na kalkula\u010dke s\u010d\u00edta\u0165 nieko\u013eko trojcifern\u00fdch prirodzen\u00fdch \u010d\u00edsel. Na prv\u00fd pokus dostal v\u00fdsledok 2\u00a0224. Pre kontrolu s\u010d\u00edtal tieto \u010d\u00edsla e\u0161te raz a vy\u0161lo mu 2\u00a0198. Preto s\u010d\u00edtal \u010d\u00edsla e\u0161te raz a teraz dostal s\u00fa\u010det 2\u00a0204. Piate pripo\u010d\u00edtavan\u00e9 \u010d\u00edslo bolo toti\u017e prekliate &#8211; Vojto pri ka\u017edom pokuse nestla\u010dil niektor\u00fa z jeho cifi\fer dostato\u010dne silno a do kalkula\u010dky zadal v\u017edy namiesto trojcifern\u00e9ho \u010d\u00edsla len dvojcifern\u00e9. \u017diadne \u010fal\u0161ie chyby pri s\u010ditovan\u00ed neurobil. Ak\u00fd je spr\u00e1vny s\u00fa\u010det Vojtov\u00fdch \u010d\u00edsel?<\/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 width=\"85%\" align=\"left\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><script type=\"text\/javascript\"><!--\r\ngoogle_ad_client = \"pub-0508472359151885\";\r\n\/* 468x60, bola vytvoren\u00e1 12.11.2010 *\/\r\ngoogle_ad_slot = \"3127737040\";\r\ngoogle_ad_width = 468;\r\ngoogle_ad_height = 60;\r\n\/\/-->\r\n<\/script>\r\n<script type=\"text\/javascript\"\r\nsrc=\"http:\/\/pagead2.googlesyndication.com\/pagead\/show_ads.js\">\r\n<\/script><\/p>\n<!--CusAds0-->\n<div style=\"font-size: 0px; height: 0px; line-height: 0px; margin: 0; padding: 0; clear: both;\"><\/div>","protected":false},"excerpt":{"rendered":"<p>61. ro\u010dn\u00edk matematickej olympi\u00e1dy 2011-2012 aktualizovan\u00e9 11.4.2012 22:50 Z5 Z6 Z7 Z8 Z9 dom\u00e1ce dom\u00e1ce dom\u00e1ce dom\u00e1ce dom\u00e1ce obvodn\u00e9 obvodn\u00e9 obvodn\u00e9 obvodn\u00e9 obvodn\u00e9 krajsk\u00e9 Zadania dom\u00e1ceho kola na stiahnutie vo form\u00e1te PDF:\u00a0\u00a0\u00a0 (ulo\u017ei\u0165 ako &#8230;)<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[10],"tags":[],"class_list":["post-514","post","type-post","status-publish","format-standard","hentry","category-61-rocnik"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.6 - 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