{"id":1346,"date":"2019-02-25T22:18:36","date_gmt":"2019-02-25T22:18:36","guid":{"rendered":"http:\/\/matematika.besaba.com\/?p=1346"},"modified":"2023-11-02T18:52:14","modified_gmt":"2023-11-02T18:52:14","slug":"sutazne-ulohy-kategorii-z4-z9-2","status":"publish","type":"post","link":"https:\/\/matematika.besaba.com\/?p=1346","title":{"rendered":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z4-Z9"},"content":{"rendered":"<p class=\"hh2\"><a name=\"top\"><\/a>51. ro\u010dn\u00edk matematickej olympi\u00e1dy 2001-2002<br \/>\nS\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z4-Z9<\/p>\n<p><span class=\"povinne\">aktualizovan\u00e9 25.2.2019 23:25<\/span><\/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\">Z4<\/span><\/td>\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=\"#z41\">dom\u00e1ce<\/a><\/span><\/td>\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=\"#z42\">dom\u00e1ce<\/a><\/span><\/td>\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=\"5\" 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><!--more--><\/p>\n<p><a name=\"z41\"><\/a><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Z4-I-1<\/strong><br \/>\nAni\u010dka m\u00e1 na\u0161etren\u00fdch 129 Sk, 72 Sk v\u0161ak dlhuje Borisovi. D\u00e1vid si na\u0161etril 145 Sk, ale 76 Sk dlhuje Ani\u010dke. Boris m\u00e1 na\u0161etren\u00fdch 163 Sk a 89 Sk dlhuje D\u00e1vidovi. Ktor\u00e9 z det\u00ed bude ma\u0165 najviac kor\u00fan, ke\u010f sa vyrovnaj\u00fa v\u0161etky dl\u017eoby? Ko\u013eko to bude?<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z4-I-2<\/strong><br \/>\nZ papiera sme vystrihli v\u0161etky obd\u013a\u017eniky, ktor\u00e9 sa daj\u00fa posklada\u0165 zo \u0161tvor\u010dekov 1 cm x 1 cm a ktor\u00fdch obvody s\u00fa men\u0161ie ako 13 cm. Narysuj najv\u00e4\u010d\u0161\u00ed \u0161tvorec, ktor\u00fd m\u00f4\u017eeme zlo\u017ei\u0165 z vystrihnut\u00fdch obd\u013a\u017enikov. Obd\u013a\u017eniky sa nesm\u00fa prekr\u00fdva\u0165, nemus\u00edme ale pri skladan\u00ed pou\u017ei\u0165 v\u0161etky.<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z4-I-3<\/strong><br \/>\nV \u0161kole rie\u0161ili \u0161tvrt\u00e1ci tak\u00fato \u00falohu: <em>&#8222;Mysl\u00edm si \u010d\u00edslo. Ak ho vyn\u00e1sob\u00edm 5 a k tomuto n\u00e1sobku pripo\u010d\u00edtam \u010d\u00edslo 6, vyjde mi \u010d\u00edslo 11. Ak\u00e9 \u010d\u00edslo si mysl\u00edm?&#8220;<\/em> Za dom\u00e1cu \u00falohu potom dostali vyrie\u0161i\u0165 &#8222;t\u00fa ist\u00fa&#8220; \u00falohu <em>&#8222;Mysl\u00edm si \u010d\u00edslo. Ak ho vyn\u00e1sob\u00edm &#8230; a k tomuto n\u00e1sobku pripo\u010d\u00edtam &#8230;, vyjde mi \u010d\u00edslo &#8230; Ak\u00e9 \u010d\u00edslo si mysl\u00edm?&#8220;<\/em>, ale s \u010d\u00edslami 8, 23 a 5.Vyrie\u0161 &#8222;\u0161kolsk\u00fa \u00falohu&#8220;. <br \/>\nNap\u00ed\u0161, ako vyzeralo zadanie dom\u00e1cej \u00falohy, ak vie\u0161, \u017ee mala rie\u0161enie a vyrie\u0161 ju.<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z4-I-4<\/strong><br \/>\nVp\u00ed\u0161te do trojuholn\u00edkov na obr\u00e1zku \u010d\u00edsla tak, aby aspo\u0148 v jednom trojuholn\u00edku bolo \u010d\u00edslo 279, s\u00fa\u010det vp\u00edsan\u00fdch \u010d\u00edsel bol v ka\u017edom koso\u0161tvorci zlo\u017eenom z dvoch trojuholn\u00edkov ten ist\u00fd, s\u00fa\u010det v\u0161etk\u00fdch vp\u00edsan\u00fdch \u010d\u00edsel bol 4200.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"51\/51z4a1.gif\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z4-I-5<\/strong><br \/>\nV\u010dera som v na\u0161ej z\u00e1hrade nar\u00e1tala 136 lienok. Ka\u017ed\u00e1 z nich mala dve, \u0161es\u0165 alebo sedem bodiek. V\u0161etky dvojbodkov\u00e9 lienky mali spolu to\u013eko bodiek ako v\u0161etky \u0161es\u0165bodkov\u00e9. V\u0161etky \u0161es\u0165bodkov\u00e9 mali spolu rovnako ve\u013ea bodiek ako v\u0161etky sedembodkov\u00e9. Ko\u013eko ak\u00fdch lienok som videla v z\u00e1hrade?<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"51\/51z4a2.gif\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z4-I-6<\/strong><br \/>\nDo 4.A chod\u00ed 27 \u017eiakov. \u0160iesti z nich s\u00fa jedin\u00e1\u010dikovia, dvan\u00e1sti maj\u00fa mlad\u0161ieho s\u00farodenca, \u0161estn\u00e1sti star\u0161ieho s\u00farodenca, z dvoji\u010diek, troji\u010diek ani in\u00fdch &#8222;viac-i\u010diek&#8220; nie je nik. Ko\u013eko det\u00ed tejto triedy nem\u00e1 mlad\u0161ieho s\u00farodenca? Ko\u013ek\u00ed maj\u00fa len jedn\u00e9ho s\u00farodenca?<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\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=\"z42\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Z4-II-1<\/strong><br \/>\nVp\u00ed\u0161te do \u0161ed\u00fdch trojuholn\u00ed\u010dkov na obr\u00e1zku \u010d\u00edsla tak, aby s\u00fa\u010det \u010d\u00edsel vp\u00edsan\u00fdch do \u013eubovo\u013en\u00fdch dvoch \u0161ed\u00fdch trojuholn\u00ed\u010dkov so spolo\u010dn\u00fdm vrcholom bol 238.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"51\/51z4b1.gif\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z4-II-2<\/strong><br \/>\nMajka, Vojto a Pe\u0165o sa hrali gu\u013eky. Na za\u010diatku hry mala Majka o 6 guliek menej ako Pe\u0165o a o 7 viac ako Vojto. Na konci hry mal Vojto o 8 guliek menej ako Majka a o 9 viac ako Pe\u0165o. Zistite, ko\u013eko guliek mala Majka na za\u010diatku, ak ich na konci hry mala 20..<\/p>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\u00e1)<\/div>\n<p><strong>Z4-II-3<\/strong><br \/>\nRi\u0161ko vystrihol z papiera obd\u013a\u017enik, ktor\u00fd sa d\u00e1 posklada\u0165 zo \u0161tvor\u010dekov 1 cm \u00d7 1 cm a ktor\u00e9ho obvod je 18 cm. Potom vystrihol z papiera e\u0161te nieko\u013eko \u0161tvor\u010dekov 1cm \u00d7 1cm. Z obd\u013a\u017enika a v\u0161etk\u00fdch \u0161tvor\u010dekov zlo\u017eil ve\u013ek\u00fd \u0161tvorec s obvodom 24 cm. Ko\u013eko najmenej \u0161tvor\u010dekov vystrihol Ri\u0161ko?<br \/>\n<em>(Pri skladan\u00ed sa obd\u013a\u017enik a \u0161tvorce nesm\u00fa prekr\u00fdva\u0165 a mus\u00edme v\u0161etky pou\u017ei\u0165.)<\/em><\/p>\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=\"z51\"><\/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<p><strong>Z5-I-1<\/strong><br \/>\nPavol m\u00e1 v stavebnici dreven\u00e9 kocky a kv\u00e1dre. Hrana ka\u017edej kocky meria 3 cm. Ka\u017ed\u00fd kv\u00e1der m\u00e1 rozmery 5 cm, 5 cm, 7 cm. Z celej stavebnice postavil Pavol ve\u017eu vysok\u00fa 50 cm. Ko\u013eko najmenej dielov m\u00f4\u017ee ma\u0165 v stavebnici? Ko\u013eko najviac ich m\u00f4\u017ee ma\u0165? (Ve\u017eu stavia tak, \u017ee v ka\u017edej vrstve je bu\u010f len 1 kocka alebo len 1 kv\u00e1der.)<\/p>\n<div class=\"pri\">(Dillingerov\u00e1)<\/div>\n<p><strong>Z5-I-2<\/strong><br \/>\nPlamienkovci vyr\u00e1baj\u00fa svie\u010dky pre cel\u00e9 Svetielkovo. Vosk roztopia vo ve\u013ekom hrnci a v\u0161etok vylej\u00fa do pripraven\u00fdch foriem. Z ka\u017edej formy vyber\u00fa 5 svie\u010dok a o\u010disten\u00edm formy od zvy\u0161kov vosku z\u00edskaj\u00fa materi\u00e1l na v\u00fdrobu e\u0161te 1 svie\u010dky. V\u0161etky zvy\u0161ky spolu roztopia a rovnako ako predt\u00fdm vyr\u00e1baj\u00fa \u010fal\u0161ie svie\u010dky. Tento postup opakuj\u00fa dovtedy, k\u00fdm sa d\u00e1 voskom naplni\u0165 cel\u00e1 forma. Pri minut\u00ed v\u0161etk\u00e9ho vosku z prv\u00e9ho rozt\u00e1pania vyrobili Plamienkovci 360 svie\u010dok. Ko\u013eko svie\u010dok vyrobili pri druhom rozt\u00e1pan\u00ed? Ko\u013eko svie\u010dok vyrobili celkom?<\/p>\n<div class=\"pri\">(Dillingerov\u00e1 &#8211; U\u0161iakov\u00e1)<\/div>\n<p><strong>Z5-I-3<\/strong><br \/>\n<em>Pestr\u00e9 \u010d\u00edslo<\/em> je tak\u00e9, ktor\u00e9 nem\u00e9 \u017eiadne dve cifry rovnak\u00e9. <em>Zrkadlov\u00fdm obrazom<\/em> \u010d\u00edsla 102 958 je \u010d\u00edslo 859 201. Ak\u00fd najmen\u0161\u00ed a ak\u00fd najv\u00e4\u010d\u0161\u00ed 5-cifern\u00fd v\u00fdsledok m\u00f4\u017eeme dosta\u0165 pri spr\u00e1vnom s\u010d\u00edtan\u00ed dvoch <em>pestr\u00fdch<\/em> \u0161tvorcifern\u00fdch \u010d\u00edsel, z ktor\u00fdch jedno je <em>zrkadlov\u00fdm obrazom<\/em> druh\u00e9ho?<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z5-I-4<\/strong><br \/>\nZa prekro\u010denie r\u00fdchlosti d\u00e1vaj\u00fa v Slowlande ve\u013ek\u00e9 pokuty. Za ka\u017ed\u00fd km\/h navy3e oproti maxim\u00e1lnej povolenej r\u00fdchlosti zaplat\u00edte 400 Sk. Pol\u00edcia zastavila p\u00e1na Quicka a povedala mu: &#8222;I\u0161li ste r\u00fdchlos\u0165ou 93 km\/h. Keby ste i\u0161li e\u0161te o 7 km\/h r\u00fdchlej\u0161ie, zaplatili by ste pokutu 18 000 Sk.&#8220;<\/p>\n<p>\u00a0\u00a0\u00a0\u00a0<strong>a)<\/strong>\u00a0Ak\u00e1 je maxim\u00e1lna povolen\u00e1 r\u00fdchlos\u0165 v Slowlande?<br \/>\n\u00a0\u00a0\u00a0\u00a0<strong>b)<\/strong>\u00a0Ko\u013eko zaplatil p\u00e1n Quick za prekro\u010denie r\u00fdchlosti?<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z5-I-5<\/strong><br \/>\nNa obr\u00e1zku s\u00fa zn\u00e1zornen\u00e9 v\u0161etky vrcholy dvoch \u0161tvorcov. Zisti obsah ich spolo\u010dnej \u010dasti. (Jeden \u0161tvor\u010dek siete m\u00e1 obsah 25 mm<sup>2<\/sup>.)<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"51\/51z5a1.gif\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z5-I-6<\/strong><br \/>\nTramt\u00e1rijsk\u00e9 vlaky maj\u00fa ru\u017eov\u00e9 a modr\u00e9 vag\u00f3niky, ktor\u00e9 musia by\u0165 zoraden\u00e9 tak, aby \u017eiadne dva vag\u00f3niky rovnakej farby neboli ved\u013ea seba. Nov\u00fd zora\u010fova\u010d to nevedel a za lokomot\u00edvu zaradil najprv 5 ru\u017eov\u00fdch a potom 5 modr\u00fdch vag\u00f3nov.<br \/>\nV akom najkrat\u0161om \u010dase to teraz m\u00f4\u017ee napravi\u0165, ak vie na pomocnej ko\u013eaji vymeni\u0165 poradie dvoch susedn\u00fdch vag\u00f3nov a jedna tak\u00e1to v\u00fdmena mu trv\u00e1 10 min\u00fat?<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\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=\"51\/51z5iir.pdf\" target=\"_blank\" rel=\"noopener\"><input name=\"butt52r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z5-II-1<\/strong><br \/>\n<em>Pestr\u00e9 \u010d\u00edslo<\/em> je tak\u00e9, ktor\u00e9 nem\u00e1 \u017eiadne dve cifry rovnak\u00e9. <em>Zrkadlov\u00fdm obrazom pestr\u00e9ho \u010d\u00edsla<\/em> 102 958 je \u010d\u00edslo 859 201. Mysl\u00edm si dve dvojcifern\u00e9 \u010d\u00edsla. Obe s\u00fa pestr\u00e9 a jedno je zrkadlov\u00fdm obrazom druh\u00e9ho. Ich s\u00fa\u010din je trojcifern\u00e9 \u010d\u00edslo a nie je pestr\u00e9. Ak\u00e9 \u010d\u00edsla si m\u00f4\u017eem myslie\u0165? Je viac mo\u017enost\u00ed?<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z5-II-2<\/strong><br \/>\n47-hlav\u00e1 dra\u010dica bola nakupova\u0165. Pre ka\u017ed\u00fa svoju hlavu k\u00fapila v akcii 1 \u0161\u00e1l po 50 Sk, 1 \u010diapku po 40 Sk, 1 kefku na zuby po 30 Sk, 1 hrebe\u0148 po 20 Sk, a 1 ma\u0161\u013eu po 10 Sk. Pri pokladni jej na\u00fa\u010dtovali 7 000 Sk a dra\u010dica hne\u010f vedela, \u017ee sa museli pom\u00fdli\u0165. A naozaj, pri kontrole zistila, \u017ee jej na\u00fa\u010dtovali 1 hrebe\u0148 a 1 \u0161\u00e1l navy\u0161e, zato 4 rovnak\u00e9 veci zar\u00e1ta\u0165 zabudli.<\/p>\n<p>\u00a0\u00a0\u00a0\u00a0<strong>a)<\/strong>\u00a0Ko\u013eko mala dra\u010dica spr\u00e1vne zaplati\u0165?<br \/>\n\u00a0\u00a0\u00a0\u00a0<strong>b)<\/strong>\u00a0\u010co jej zabudli zar\u00e1ta\u0165?<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z5-II-3<\/strong><br \/>\nNa obr\u00e1zku s\u00fa zn\u00e1zornen\u00e9 v\u0161etky vrcholy dvoch \u0161tvorcov. Vypo\u010d\u00edtajte obsah spolo\u010dnej \u010dasti t\u00fdchto dvoch \u0161tvorcov. (Strana \u0161tvor\u010deka siete meria 5 mm.)<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"51\/51z5b1.gif\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\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<p><strong>Z6-I-1<\/strong><br \/>\nMoja mama sa narodila 16.3. 1948. Je to pekn\u00fd d\u00e1tum, plat\u00ed toti\u017e <strong>16 x 3 = 48<\/strong>. V ktor\u00fdch rokoch 20. storo\u010dia bolo najmenej tak\u00fdch pekn\u00fdch d\u00e1tumov? N\u00e1jdite v\u0161etky rie\u0161enia.<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z6-I-2<\/strong><br \/>\nDominik sa hral s plastel\u00ednou a vymodeloval kv\u00e1der s rozmermi 6 cm x 3 cm x 19 cm. Potom ho znova roztla\u010dil a z tejto plastel\u00edny vymodeloval tri r\u00f4zne ve\u013ek\u00e9 kocky. Na svoje prekvapenie zistil, \u017ee ve\u013ekosti v\u0161etk\u00fdch hr\u00e1n v centimetroch boli cel\u00e9 \u010d\u00edsla. Ak\u00e9 rozmery mali Dominikove kocky?<\/p>\n<div class=\"pri\">(Hozov\u00e1)<\/div>\n<p><strong>Z6-I-3<\/strong><br \/>\nPrirodzen\u00e9 \u010d\u00edslo je <em>vesel\u00e9<\/em>, ak je delite\u013en\u00e9 9 alebo 13, <em>smutn\u00e9<\/em>, ak je delite\u013en\u00e9 12, <em>hladn\u00e9<\/em>, ak obsahuje aspo\u0148 jednu nulu, <em>mal\u00e9<\/em>, ak je dvojcifern\u00e9 a <em>ve\u013ek\u00e9<\/em>, ak je trojcifern\u00e9 men\u0161ie ne\u017e 200. Ako ve\u013ek\u00fd obsah m\u00f4\u017ee ma\u0165 obd\u013a\u017enik, ktor\u00e9ho \u0161\u00edrka je mal\u00e1, smutn\u00e1, hladn\u00e1 a d\u013a\u017eka je ve\u013ek\u00e1, vesel\u00e1, hladn\u00e1 a jeho obvod je tie\u017e hladn\u00fd?<\/p>\n<div class=\"pri\">(Pt\u00e1\u010dkov\u00e1)<\/div>\n<p><strong>Z6-I-4<\/strong><br \/>\nDopl\u0148 do &#8222;s\u00fa\u010dinovej pyram\u00eddy&#8220; na obr\u00e1zku prirodzen\u00e9 \u010d\u00edsla tak, aby najv\u00e4\u010d\u0161ie doplnen\u00e9 \u010d\u00edslo bolo 315 a \u017eiadne dve doplnen\u00e9 \u010d\u00edsla neboli rovnak\u00e9. Ko\u013ek\u00fdmi r\u00f4znymi sp\u00f4sobmi sa to d\u00e1 spravi\u0165?<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"51\/51z6a1.gif\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z6-I-5<\/strong><br \/>\nIvan dostal \u0161peci\u00e1lnu bielo-hned\u00fa \u010dokol\u00e1du ako na obr\u00e1zku. Zisti hmotnos\u0165 bielej \u010dasti, ak cel\u00e1 \u010dokol\u00e1da m\u00e1 tri rovnako \u0161irok\u00e9 riadky a tri rovnako \u0161irok\u00e9 st\u013apce a cel\u00e1 v\u00e1\u017ei 144 gramov.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"51\/51z6a2.gif\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Dillingerov\u00e1)<\/div>\n<p><strong>Z6-I-6<\/strong><br \/>\nTraja dalmat\u00ednci a 2 \u0161pici v\u00e1\u017eia to\u013eko ako 14 jazve\u010d\u00edkov. Jeden dalmat\u00ednec v\u00e1\u017ei to\u013eko ako 1 \u0161pic a 3 jazve\u010d\u00edci. Ko\u013eko jazve\u010d\u00edkov vyv\u00e1\u017ei 101 dalmat\u00edncov? (Psi jedn\u00e9ho plemena s\u00fa v\u017edy rovnako \u0165a\u017ek\u00ed.)<\/p>\n<div class=\"pri\">(Pt\u00e1\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=\"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=\"51\/51z678iir.pdf\" target=\"_blank\" rel=\"noopener\"><input name=\"butt62r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z6-II-1<\/strong><br \/>\nFerko sa rozhodol min\u00fa\u0165 v\u0161etky svoje \u00faspory na ma\u0161krty. Zistil, \u017ee si za ne m\u00f4\u017ee k\u00fapi\u0165 3 vetern\u00edky a 3 dl \u010dapovanej kofoly, alebo 18 dekagramov hrozienok v jogurte, pr\u00edpadne 12 dekagramov hrozienok v jogurte a pol litra kofoly. Nakoniec si k\u00fapil 1 vetern\u00edk a 6 dl kofoly. Na ko\u013eko gramov hrozienok v jogurte mu e\u0161te zv\u00fd\u0161ili peniaze?<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z6-II-2<\/strong><br \/>\nKatka rozstrihla \u0161tvorcov\u00fa \u0161atku nakreslen\u00fa na obr\u00e1zku pozd\u013a\u017e jej uhloprie\u010dky a vyrobila tak dve trojuholn\u00edkov\u00e9. Zisti, ak\u00e1 \u010das\u0165 ka\u017edej z nov\u00fdch \u0161atiek je biela, ak\u00e1 \u0161ed\u00e1 a ak\u00e1 \u010dierna, ak v p\u00f4vodnej \u0161atke bola \u010dierna <img loading=\"lazy\" decoding=\"async\" src=\"51\/51z6b2.gif\" width=\"11\" height=\"28\" align=\"absmiddle\" hspace=\"0\" \/> a \u0161ed\u00e1 <img loading=\"lazy\" decoding=\"async\" src=\"51\/51z6b3.gif\" width=\"10\" height=\"28\" align=\"absmiddle\" hspace=\"0\" \/> jej plochy.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"51\/51z6b1.gif\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z6-II-3<\/strong><br \/>\nMoje ob\u013e\u00faben\u00e9 \u010d\u00edsla s\u00fa tie, ktor\u00e9 sa po vyn\u00e1soben\u00ed svojim vlastn\u00fdm cifern\u00fdm s\u00fa\u010dtom desa\u0165kr\u00e1t zv\u00e4\u010d\u0161ia. S\u00fa\u010din troch mojich ob\u013e\u00faben\u00fdch \u010d\u00edsel je 71 668. Ktor\u00e9 s\u00fa to \u010d\u00edsla?<\/p>\n<div class=\"pri\">(Pt\u00e1\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=\"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<p><strong>Z7-I-1<\/strong><br \/>\nJanka plietla bratovi Luk\u00e1\u0161ovi \u0161\u00e1l. Luk\u00e1\u0161ovi sa \u0161\u00e1l v\u00f4bec nep\u00e1\u010dil, a tak v\u017edy ve\u010der potajomky z Jankin\u00fdch v ten de\u0148 upleten\u00fdch riadkov \u0161tvrtinu vyp\u00e1ral. Janka za\u010dala plies\u0165 v pondelok. V utorok uplietla o 24 radov viac ako v pondelok a v stredu dokonca dvakr\u00e1t to\u013eko ako v pondelok. Vo \u0161tvrtok uplietla u\u017e len 36 radov a hotov\u00fd 180 \u2013 radov\u00fd \u0161\u00e1l dala Luk\u00e1\u0161ovi. Ko\u013eko radov uplietla Janka v utorok? Ko\u013eko radov vyp\u00e1ral Luk\u00e1\u0161?<\/p>\n<div class=\"pri\">(Dillingerov\u00e1)<\/div>\n<p><strong>Z7-I-2<\/strong><br \/>\nV \u0161tvorcovej sieti na obr\u00e1zku s\u00fa zn\u00e1zornen\u00e9 v\u0161etky vrcholy dvoch \u0161tvorcov. Zisti obsah ich spolo\u010dnej \u010dasti.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"51\/51z7a1.gif\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z7-I-3<\/strong><br \/>\nMajo mal vyn\u00e1sobi\u0165 dve desatinn\u00e9 \u010d\u00edsla. Desatinn\u00e9 \u010diarky si pom\u00fdlil s bodkami a vyn\u00e1sobil \u0161tyri cel\u00e9 \u010d\u00edsla. Po\u010d\u00edtal bez chyby a vy\u0161lo mu 15 228. Spr\u00e1vny v\u00fdsledok v\u0161ak mal by\u0165 589,17. Ak\u00e9 \u010d\u00edsla mal n\u00e1sobi\u0165?<\/p>\n<div class=\"pri\">(Dillingerov\u00e1)<\/div>\n<p><strong>Z7-I-4<\/strong><br \/>\nV t\u00e1bore rozde\u013eovali ved\u00faci deti do skup\u00edn po 5, posledn\u00e1 skupina v\u0161ak bola ne\u00fapln\u00e1. Pri novej hre ich teda ved\u00faci sk\u00fasili rozdeli\u0165 po 8 a op\u00e4\u0165 to &#8222;nevy\u0161lo&#8220;. Pri druhej hre takto s\u00fa\u0165a\u017eilo o 4 skupiny menej ako pri prvej. V ne\u00fapln\u00fdch skupin\u00e1ch bol v\u017edy p\u00e1rny po\u010det det\u00ed. Ko\u013eko det\u00ed mohlo by\u0165 v t\u00e1bore?<\/p>\n<div class=\"pri\">(Dillingerov\u00e1)<\/div>\n<p><strong>Z7-I-5<\/strong><br \/>\nOzna\u010d vrcholy kocky cel\u00fdmi \u010d\u00edslami od 1 po 8 tak, aby pre ka\u017ed\u00fa jej stenu bol s\u00fa\u010det pr\u00edslu\u0161n\u00fdch \u010d\u00edsel<\/p>\n<p>\u00a0\u00a0\u00a0\u00a0<strong>a)<\/strong>\u00a0prvo\u010d\u00edslo,<br \/>\n\u00a0\u00a0\u00a0\u00a0<strong>b)<\/strong>\u00a0in\u00e9 prvo\u010d\u00edslo.<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z7-I-6<\/strong><br \/>\nStar\u00fd farm\u00e1r sa rozhodol, \u017ee v\u0161etok svoj majetok \u2013 st\u00e1do oviec \u2013 rozdel\u00ed medzi svoje deti. Najprv rozdelil st\u00e1do na dve \u010dasti v pomere 1:3. Men\u0161iu z nich dal prvoroden\u00e9mu, v\u00e4\u010d\u0161iu op\u00e4\u0165 rozdelil v tom istom pomere. Z nov\u00fdch \u010dast\u00ed men\u0161iu pridelil druhoroden\u00e9mu, v\u00e4\u010d\u0161iu znovu rozdelil v pomere 1:3. Takto pokra\u010doval, a\u017e k\u00fdm ka\u017ed\u00fd z jeho synov nedostal svoj diel a zost\u00e1vaj\u00facu \u010das\u0165 potom daroval svojej jedinej dc\u00e9re. Zisti, ko\u013eko mal farm\u00e1r oviec, ak vie\u0161, \u017ee prostredn\u00fd syn ich dostal 156. Ktor\u00e9 z det\u00ed dostalo najviac oviec?<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\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=\"51\/51z678iir.pdf\" target=\"_blank\" rel=\"noopener\"><input name=\"butt72r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z7-II-1<\/strong><br \/>\nBonboni\u00e9ra s krabicou tvaru kv\u00e1dra bola pln\u00e1 cukr\u00edkov, pekne ulo\u017een\u00fdch v riadkoch a st\u013apcoch. Mi\u0161ko z nich nieko\u013eko zjedol a tie, \u010do mu ostali, preusporiadal tak, \u017ee bez jedn\u00e9ho miesta nimi zaplnil 3 cel\u00e9 riadky. Cukr\u00edky z ne\u00fapln\u00e9ho riadku zjedol a zvy\u0161n\u00e9 znovu preusporiadal. Tentoraz nimi zaplnil bez jedn\u00e9ho miesta 5 cel\u00fdch st\u013apcov. Nevyzeralo to pekne, preto cukr\u00edky z ne\u00fapln\u00e9ho st\u013apca zjedol a v bonboni\u00e9re mu ostala u\u017e len tretina z p\u00f4vodn\u00e9ho po\u010dtu cukr\u00edkov. Zisti<\/p>\n<p>\u00a0\u00a0\u00a0\u00a0<strong>a)<\/strong>\u00a0ko\u013eko cukr\u00edkov bolo v plnej bonboni\u00e9re,<br \/>\n\u00a0\u00a0\u00a0\u00a0<strong>b)<\/strong>\u00a0ko\u013eko cukr\u00edkov zjedol Mi\u0161ko predt\u00fdm, ako ich prv\u00fdkr\u00e1t preusporiadal.<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z7-II-2<\/strong><br \/>\nKatka si z papiera vystrihla 20 obd\u013a\u017enikov. Prv\u00fd z nich mal rozmery 1 cm x 2 cm, druh\u00fd 2 cm x 3 cm, tret\u00ed 3 cm x 4 cm at\u010f. Potom ich za\u010dala priklada\u0165 rovnako dlh\u00fdmi stranami k sebe tak, aby sa neprekr\u00fdvali.<\/p>\n<p>\u00a0\u00a0\u00a0\u00a0<strong>a)<\/strong>\u00a0Narysuj mnohouholn\u00edk, ktor\u00fd Katka zlo\u017eila z prv\u00fdch \u0161iestich svojich obd\u013a\u017enikov.<br \/>\n\u00a0\u00a0\u00a0\u00a0<strong>b)<\/strong>\u00a0Zisti obvod Katkinho mnohouholn\u00edka, zlo\u017een\u00e9ho zo v\u0161etk\u00fdch vystrihnut\u00fdch obd\u013a\u017enikov.<\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z7-II-3<\/strong><br \/>\nPani u\u010dite\u013eka dala de\u0165om na dom\u00e1cu \u00falohu vypo\u010d\u00edta\u0165 dva pr\u00edklady na n\u00e1sobenie cel\u00fdch z\u00e1porn\u00fdch \u010d\u00edsel. Fero si ich oba nap\u00edsal bez z\u00e1tvoriek a doma si potom myslel, \u017ee s\u00fa to pr\u00edklady na od\u010d\u00edtanie (bodky si nev\u0161imol). Takto zmenen\u00e9 pr\u00edklady bezchybne vypo\u010d\u00edtal. Jeden z jeho v\u00fdsledkov sa od toho &#8222;spr\u00e1vneho&#8220; l\u00ed\u0161il znamienkom, druh\u00fd zas bol -182 n\u00e1sobkom jeho prevr\u00e1tenej hodnoty. Ak\u00e9 pr\u00edklady Fero dostal na dom\u00e1cu \u00falohu?<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\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<p><strong>Z8-I-1<\/strong><br \/>\nMi\u0161o si mysl\u00ed trojcifern\u00e9 \u010d\u00edslo. Jurovi prezradil, \u017ee cifern\u00fd s\u00fa\u010det myslen\u00e9ho \u010d\u00edsla je 8. Petrovi prezradil len cifern\u00fd s\u00fa\u010din myslen\u00e9ho \u010d\u00edsla. Peter spr\u00e1vne ur\u010dil, \u017ee tak\u00fdch \u010d\u00edsel je 6 a povedal to Jurovi. Ten vyhl\u00e1sil: &#8222;U\u017e viem, ak\u00e9 s\u00fa cifry tvojho \u010d\u00edsla, ale e\u0161te mi to nesta\u010d\u00ed. Mi\u0161o obom chlapcom povedal:&#8220; Druh\u00e1 mocnina poslednej cifry nie je delite\u013eom myslen\u00e9ho \u010d\u00edsla.&#8220; To u\u017e chlapcom sta\u010dilo. Ktor\u00e9 \u010d\u00edslo si Mi\u0161o myslel?<\/p>\n<div class=\"pri\">(Dillingerov\u00e1)<\/div>\n<p><strong>Z8-I-2<\/strong><br \/>\nPresne o polnoci z 31.12.2000 na 1.1.2001 mali v Plav\u00e1kove sl\u00e1vnostne otvori\u0165 nov\u00fd, 160 cm hlbok\u00fd baz\u00e9n tvaru kv\u00e1dra. Vodu do neho v\u0161ak za\u010dali nap\u00fa\u0161\u0165a\u0165 u\u017e 30.12. Graf na obr\u00e1zku zn\u00e1zor\u0148uje, ako sa menila v\u00fd\u0161ka vodnej hladiny v z\u00e1vislosti od \u010dasu.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"51\/51z8a1.gif\" alt=\"\" border=\"0\" \/><\/div>\n<p>\u00a0\u00a0\u00a0\u00a0<strong>a)<\/strong>\u00a0Stihli v\u010das napusti\u0165 baz\u00e9n a\u017e po horn\u00fd okraj?<br \/>\n\u00a0\u00a0\u00a0\u00a0<strong>b)<\/strong>\u00a0Kedy presne za\u010dali baz\u00e9n nap\u00fa\u0161\u0165a\u0165?<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z8-I-3<\/strong><br \/>\nStar\u00fd farm\u00e1r sa rozhodol, \u017ee v\u0161etok svoj majetok \u2013 st\u00e1do oviec \u2013 rozdel\u00ed medzi svoje deti. Najprv rozdelil st\u00e1do na dve \u010dasti v pomere 1:3. Men\u0161iu z nich dal prvoroden\u00e9mu, v\u00e4\u010d\u0161iu op\u00e4\u0165 rozdelil v tom istom pomere. Z nov\u00fdch \u010dast\u00ed men\u0161iu pridelil druhoroden\u00e9mu, v\u00e4\u010d\u0161iu znovu rozdelil v pomere 1:3. Takto pokra\u010doval, a\u017e k\u00fdm ka\u017ed\u00fd z jeho synov nedostal svoj diel a zost\u00e1vaj\u00facu \u010das\u0165 potom daroval svojej jedinej dc\u00e9re.<br \/>\nZistite, ko\u013eko mal farm\u00e1r synov, ak len jeden z nich dostal viac oviec, ako jeho dc\u00e9ra.<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z8-I-4<\/strong><br \/>\nUr\u010dite najv\u00e4\u010d\u0161\u00ed mo\u017en\u00fd po\u010det lopti\u010diek, o priemere 100 mm, ktor\u00e9 je mo\u017en\u00e9 ulo\u017ei\u0165 do \u0161katule tvaru kv\u00e1dra o rozmeroch 100 cm x 100 cm x 10 cm.<\/p>\n<div class=\"pri\">(Krej\u010dov\u00e1)<\/div>\n<p><strong>Z8-I-5<\/strong><br \/>\nNa n\u00e1kladnom automobile sa pneumatiky na predn\u00fdch koles\u00e1ch opotrebuj\u00fa po 15 000 km, na zadn\u00fdch, zdvojen\u00fdch koles\u00e1ch po 20 000 km. \u0160of\u00e9r pr\u00e1ve k\u00fapil s\u00fapravu \u0161iestich nov\u00fdch pneumat\u00edk. Ko\u013eko kilometrov maxim\u00e1lne m\u00f4\u017ee na nich najazdi\u0165?<\/p>\n<div class=\"pri\">(Krej\u010dov\u00e1)<\/div>\n<p><strong>Z8-I-6<\/strong><br \/>\nUhloprie\u010dky delia koso\u0161tvorec s obvodom 40 cm na 4 trojuholn\u00edky s celo\u010d\u00edseln\u00fdmi d\u013a\u017ekami str\u00e1n. Ak\u00fd najv\u00e4\u010d\u0161\u00ed obvod m\u00f4\u017ee ma\u0165<\/p>\n<p>\u00a0\u00a0\u00a0\u00a0<strong>a)<\/strong>\u00a0trojuholn\u00edk<br \/>\n\u00a0\u00a0\u00a0\u00a0<strong>b)<\/strong>\u00a0\u0161tvoruholn\u00edk<br \/>\n\u00a0\u00a0\u00a0\u00a0<strong>c)<\/strong>\u00a0p\u00e4\u0165uholn\u00edk zlo\u017een\u00fd z t\u00fdchto trojuholn\u00edkov?<\/p>\n<p>(Trojuholn\u00edky sa nesm\u00fa prekr\u00fdva\u0165 a v\u0161etky 4 je treba pou\u017ei\u0165.)<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\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=\"51\/51z678iir.pdf\" target=\"_blank\" rel=\"noopener\"><input name=\"butt82r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z8-II-1<\/strong><br \/>\nDo spodn\u00e9ho riadku tejto \u010dudnej s\u010d\u00edtacej pyram\u00eddy dopl\u0148te \u010d\u00edsla 1, 1, 2, 2, 3, 3 a 4 tak, aby hore vy\u0161lo \u010do najv\u00e4\u010d\u0161ie \u010d\u00edslo.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"51\/51z8b1.gif\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z8-II-2<\/strong><br \/>\nV pravidelnom \u0161es\u0165uholn\u00edku s obsahom 144 cm<sup>2<\/sup> sme vyzna\u010dili v\u0161etky uhloprie\u010dky. \u0160es\u0165uholn\u00edk sa n\u00e1m takto &#8222;rozpadol&#8220; na nieko\u013eko trojuholn\u00edkov a \u0161tvoruholn\u00edkov.<\/p>\n<p>\u00a0\u00a0\u00a0\u00a0<strong>a)<\/strong>\u00a0Zistite, na ko\u013eko \u010dast\u00ed ho uhloprie\u010dky rozdelili.<br \/>\n\u00a0\u00a0\u00a0\u00a0<strong>b)<\/strong>\u00a0Ur\u010dte obsah pravideln\u00e9ho \u0161es\u0165uholn\u00edka, ktor\u00fd vznikne zlo\u017een\u00edm v\u0161etk\u00fdch \u0161tvoruholn\u00edkov\u00fdch \u010dast\u00ed dan\u00e9ho \u0161es\u0165uholn\u00edka.<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z8-II-3<\/strong><br \/>\nPri ohni sedeli n\u00e1\u010deln\u00edci troch spriatelen\u00fdch indi\u00e1nskych kme\u0148ov s tromi rovnak\u00fdmi fajkami. Mali bojov\u00fa poradu a faj\u010dili. Prv\u00fd z nich vyfaj\u010d\u00ed cel\u00fa fajku za desa\u0165 min\u00fat, druh\u00fd za pol hodiny a tret\u00ed za hodinu. V okamihu, ke\u010f niektor\u00fd z nich bude ma\u0165 pr\u00e1zdnu fajku, musia bojov\u00fa poradu ukon\u010di\u0165. Ako si maj\u00fa n\u00e1\u010deln\u00edci medzi sebou vymie\u0148a\u0165 fajky, aby sa mohli radi\u0165 \u010do najdlh\u0161ie? Ako dlho by v takomto pr\u00edpade trvala ich bojov\u00e1 porada?<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\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<p><strong>Z9-I-1<\/strong><br \/>\n\u0160tvoruholn\u00edk, ktor\u00fd nem\u00e1 \u017eiadne dve strany rovnako dlh\u00e9, budeme vola\u0165 nerovnakostrann\u00fd. Pravideln\u00fd 12-uholn\u00edk m\u00e1 obsah 81 cm<sup>2<\/sup>. Narysujte v\u0161etky tak\u00e9 tvarovo r\u00f4zne nerovnakostrann\u00e9 \u0161tvoruholn\u00edky s vrcholmi vo vrcholoch tohto 12-uholn\u00edka a zistite obsah ka\u017ed\u00e9ho z nich.<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z9-I-2<\/strong><br \/>\nV hokejovom turnaji Zelen\u00ed 3 kr\u00e1t vyhrali a umiestnili sa na 1. mieste s celkov\u00fdm sk\u00f3re 7:1. \u010cerven\u00ed dosiahli sk\u00f3re 2:3 a Modr\u00ed 3:3. Posledn\u00ed Hned\u00ed prehrali v\u0161etky tri z\u00e1pasy a ich celkov\u00e9 sk\u00f3re bolo 1:6. Vypl\u0148te tabu\u013eku z\u00e1pasov, ak e\u0161te viete, \u017ee Zelen\u00ed porazili \u010cerven\u00fdch 3:0 a \u017ee \u010cerven\u00ed i Modr\u00ed pr\u00e1ve raz vyhrali, raz prehrali a raz remizovali.<\/p>\n<p><center><\/p>\n<table border=\"1\" width=\"357\" cellspacing=\"1\" cellpadding=\"4\">\n<tbody>\n<tr align=\"center\" valign=\"top\">\n<td>&nbsp;<\/td>\n<td><b>Zelen\u00ed<\/b><\/td>\n<td><b>\u010cerven\u00ed<\/b><\/td>\n<td><b>Modr\u00ed<\/b><\/td>\n<td><b>Hned\u00ed<\/b><\/td>\n<td><b>Sk\u00f3re<\/b><\/td>\n<\/tr>\n<tr align=\"center\" valign=\"top\">\n<td><b>Zelen\u00ed<\/b><\/td>\n<td><b>X<\/b><\/td>\n<td>&nbsp;<\/td>\n<td>&nbsp;<\/td>\n<td>&nbsp;<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr align=\"center\" valign=\"top\">\n<td><b>\u010cerven\u00ed<\/b><\/td>\n<td>&nbsp;<\/td>\n<td><b>X<\/b><\/td>\n<td>&nbsp;<\/td>\n<td>&nbsp;<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr align=\"center\" valign=\"top\">\n<td><b>Modr\u00ed<\/b><\/td>\n<td>&nbsp;<\/td>\n<td>&nbsp;<\/td>\n<td><b>X<\/b><\/td>\n<td>&nbsp;<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr align=\"center\" valign=\"top\">\n<td><b>Hned\u00ed<\/b><\/td>\n<td>&nbsp;<\/td>\n<td>&nbsp;<\/td>\n<td>&nbsp;<\/td>\n<td><b>X<\/b><\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><\/center><\/p>\n<div class=\"pri\">(Volfov\u00e1)<\/div>\n<p><strong>Z9-I-3<\/strong><br \/>\nOsovo s\u00famern\u00fd p\u00e4\u0165uholn\u00edk s obsahom 27 cm<sup>2<\/sup> m\u00e1 pr\u00e1ve tri vn\u00fatorn\u00e9 uhly prav\u00e9 a pr\u00e1ve 3 strany zhodn\u00e9. Zistite ve\u013ekosti \u010fal\u0161\u00edch dvoch vn\u00fatorn\u00fdch uhlov p\u00e4\u0165uholn\u00edka a d\u013a\u017eku niektorej z trojice zhodn\u00fdch str\u00e1n.<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z9-I-4<\/strong><br \/>\nMoja mama sa narodila 16.3. 1948. Je to <em>pekn\u00fd d\u00e1tum<\/em>, plat\u00ed toti\u017e <strong>16 \u00d7 3 = 48<\/strong>. V ktor\u00fdch rokoch 20. storo\u010dia bolo najviac tak\u00fdchto <em>pekn\u00fdch d\u00e1tumov<\/em>?<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z9-I-5<\/strong><br \/>\nKu ka\u017edej stene kocky s hranou d\u013a\u017eky <em>a<\/em> sme prilepili tak\u00fd pravideln\u00fd 4-bok\u00fd ihlan, \u017ee vzniklo teleso s 12 stenami. Ko\u013eko m\u00e1 toto teleso vrcholov? O ko\u013eko % m\u00e1 zlepen\u00e9 teleso v\u00e4\u010d\u0161\u00ed objem ako p\u00f4vodn\u00e1 kocka?<\/p>\n<div class=\"pri\">(Dillingerov\u00e1)<\/div>\n<p><strong>Z9-I-6<\/strong><br \/>\nOstrov obrov m\u00e1 rovnako ve\u013ea obyvate\u013eov ako Ostrov trpasl\u00edkov. Ani na jednom z t\u00fdchto ostrovov ne\u017eij\u00fa dve rovnako \u0165a\u017ek\u00e9 bytosti a okrem dvoch obrov a dvoch trpasl\u00edkov m\u00e1 ka\u017ed\u00fd na svojom ostrove dvoch kamar\u00e1tov, z ktor\u00fdch je jeden o 2 kg od neho \u0165a\u017e\u0161\u00ed a druh\u00fd o 2 kg od neho \u013eah\u0161\u00ed. Dvaja naj\u0165a\u017e\u0161\u00ed trpasl\u00edci v\u00e1\u017eia spolu to\u013eko, \u010do naj\u013eah\u0161\u00ed obor, traja &#8222;prostredn\u00ed&#8220; trpasl\u00edci v\u00e1\u017eia spolu to\u013eko, \u010do prostredn\u00fd obor a \u0161tyria naj\u013eah\u0161\u00ed trpasl\u00edci to\u013eko, \u010do \u00f4smy naj\u0165a\u017e\u0161\u00ed obor. Zistite, ko\u013eko obyvate\u013eov m\u00e1 Ostrov trpasl\u00edkov a o ko\u013eko kg je naj\u0165a\u017e\u0161\u00ed obor \u0165a\u017e\u0161\u00ed, ako naj\u013eah\u0161\u00ed trpasl\u00edk.<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\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=\"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=\"51\/51z9iir.pdf\" target=\"_blank\" rel=\"noopener\"><input name=\"butt92r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-II-1<\/strong><br \/>\nMilka si myslela dvojcifern\u00e9 \u010d\u00edslo. Zmenila poradie cifier a dostala \u010d\u00edslo, ktor\u00e9 sa od p\u00f4vodn\u00e9ho l\u00ed\u0161ilo o 75%. Ak\u00e9 \u010d\u00edslo si mohla myslie\u0165?<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z9-II-2<\/strong><br \/>\nAndrea triedi trojuholn\u00edky, ktor\u00fdch strany vyjadren\u00e9 v cm s\u00fa cel\u00e9 \u010d\u00edsla, na <em>\u00fazke<\/em>, <em>\u0161irok\u00e9<\/em> a <em>ostatn\u00e9<\/em>. <em><strong>\u00dazky<\/strong> trojuholn\u00edk<\/em> je tak\u00fd, v ktorom je s\u00fa\u010det d\u013a\u017eok niektor\u00fdch dvoch str\u00e1n len o 1 cm v\u00e4\u010d\u0161\u00ed, ako d\u013a\u017eka tretej strany. <em><strong>\u0160irok\u00fd<\/strong> trojuholn\u00edk<\/em> je tak\u00fd, v ktorom je s\u00fa\u010det d\u013a\u017eok niektor\u00fdch dvoch str\u00e1n dvojn\u00e1sobkom d\u013a\u017eky tretej strany. N\u00e1jdite<\/p>\n<p>\u00a0\u00a0\u00a0\u00a0<strong>a)<\/strong>\u00a0v\u0161etky <strong><em>\u00fazke<\/em><\/strong> trojuholn\u00edky, ktor\u00fdch dve strany meraj\u00fa 6 cm a 8 cm,<br \/>\n\u00a0\u00a0\u00a0\u00a0<strong>b)<\/strong>\u00a0v\u0161etky <strong><em>\u0161irok\u00e9<\/em><\/strong> trojuholn\u00edky, ktor\u00fdch dve strany meraj\u00fa 4 cm a 3 cm,<br \/>\n\u00a0\u00a0\u00a0\u00a0<strong>c)<\/strong>\u00a0v\u0161etky trojuholn\u00edky, ktor\u00fdch jedna strana meria 5 cm a s\u00fa <strong><em>\u0161irok\u00e9<\/em><\/strong> aj <strong><em>\u00fazke<\/em><\/strong> s\u00fa\u010dasne.<\/p>\n<div class=\"pri\">(Dillingerov\u00e1)<\/div>\n<p><strong>Z9-II-3<\/strong><br \/>\nDvaja cyklisti vy\u0161li s\u00fa\u010dasne z toho ist\u00e9ho miesta na v\u00fdlet. Pre\u0161li t\u00fa ist\u00fa trasu a vr\u00e1tili sa naraz domov. Cestou obaja odpo\u010d\u00edvali. Prv\u00fd i\u0161iel dvakr\u00e1t tak dlho, ako druh\u00fd odpo\u010d\u00edval. Druh\u00fd i\u0161iel \u0161tyrikr\u00e1t tak dlho, ako odpo\u010d\u00edval prv\u00fd. Kto z nich jazd\u00ed na bicykli r\u00fdchlej\u0161ie a ko\u013ekokr\u00e1t?<\/p>\n<div class=\"pri\">(\u010cernek)<\/div>\n<p><strong>Z9-II-4<\/strong><br \/>\nV Squarlande \u017eij\u00fa iba \u0161tvorce. A\u017e na dve v\u00fdnimky tam ka\u017ed\u00fd z nich m\u00e1 priate\u013ea, ktor\u00fd m\u00e1 od neho o 8 cm v\u00e4\u010d\u0161\u00ed obvod a priate\u013ea s obvodom o 8 cm men\u0161\u00edm. Priemern\u00e1 d\u013a\u017eka strany squarlandsk\u00e9ho \u0161tvorca je 15 cm. \u017diadne dva z nich nie s\u00fa zhodn\u00e9 a obvod najmen\u0161ieho je rovn\u00fd d\u013a\u017eke strany najv\u00e4\u010d\u0161ieho. Zistite<\/p>\n<p>\u00a0\u00a0\u00a0\u00a0<strong>a)<\/strong>\u00a0po\u010det obyvate\u013eov Squarlandu,<br \/>\n\u00a0\u00a0\u00a0\u00a0<strong>b)<\/strong>\u00a0rozmery najmen\u0161ieho a rozmery najv\u00e4\u010d\u0161ieho squarlandsk\u00e9ho \u0161tvorca,<br \/>\n\u00a0\u00a0\u00a0\u00a0<strong>c)<\/strong>\u00a0priemern\u00fd obsah &#8222;Squarlan\u010fana&#8220;.<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\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=\"51\/51z9iiir.pdf\" target=\"_blank\" rel=\"noopener\"><input name=\"butt92r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-III-1<\/strong><br \/>\nSkupinka det\u00ed si zorganizovala pi\u0161kvorkov\u00fd turnaj. Ka\u017ed\u00fd hral s ka\u017ed\u00fdm a celkom odohrali 136 z\u00e1pasov. Z toho pr\u00e1ve 66 z\u00e1pasov bolo typu diev\u010da &#8211; diev\u010da alebo chlapec &#8211; chlapec. Ko\u013eko bolo v skupinke chlapcov a ko\u013eko diev\u010dat?<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z9-III-2<\/strong><br \/>\nV Squarlande \u017eij\u00fa iba \u0161tvorce. A\u017e na dve v\u00fdnimky tam ka\u017ed\u00fd z nich m\u00e1 priate\u013ea, ktor\u00fd m\u00e1 od neho o 8 cm v\u00e4\u010d\u0161\u00ed obvod a priate\u013ea s obvodom o 8 cm men\u0161\u00edm. Priemern\u00fd obsah squarlandsk\u00e9ho \u0161tvorca je 116 cm<sup>2<\/sup>. \u017diadne dva z nich nie s\u00fa zhodn\u00e9 a obvod najmen\u0161ieho je rovn\u00fd d\u013a\u017eke strany najv\u00e4\u010d\u0161ieho. Zistite<\/p>\n<p>\u00a0\u00a0\u00a0\u00a0<strong>a)<\/strong>\u00a0po\u010det obyvate\u013eov Squarlandu,<br \/>\n\u00a0\u00a0\u00a0\u00a0<strong>b)<\/strong>\u00a0rozmery najmen\u0161ieho a rozmery najv\u00e4\u010d\u0161ieho squarlandsk\u00e9ho \u0161tvorca,<br \/>\n\u00a0\u00a0\u00a0\u00a0<strong>c)<\/strong>\u00a0priemern\u00fa d\u013a\u017eku strany &#8222;Squarlan\u010fana&#8220;.<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z9-III-3<\/strong><br \/>\nNa svojich potulk\u00e1ch objavil profesor Hmyzuli nov\u00fd druh chrob\u00e1ka. Pod\u013ea po\u010dtu jeho no\u017ei\u010diek a tvaru jednotliv\u00fdch \u010dl\u00e1nkov ho nazval Gu\u013e\u00f4\u010dkovcom Dvan\u00e1s\u0165n\u00f4\u017ekov\u00fdm: ka\u017ed\u00fd \u010dl\u00e1nok tohto chrob\u00e1ka m\u00e1 tvar gule a z ka\u017ed\u00e9ho \u010dl\u00e1nku okrem prv\u00e9ho (hlavi\u010dky) vyrastaj\u00fa dva p\u00e1ry no\u017ei\u010diek. Navy\u0161e, ka\u017ed\u00fd z \u010dl\u00e1nkov m\u00e1 o 21% v\u00e4\u010d\u0161\u00ed prierez, ako \u010dl\u00e1nok za n\u00edm. Zistite presn\u00fa d\u013a\u017eku chrob\u00e1ka, ak viete, \u017ee hlavi\u010dka m\u00e1 polomer 0,2662 cm.<\/p>\n<div class=\"pri\">(Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z9-III-4<\/strong><br \/>\nMarta triedi trojuholn\u00edky, ktor\u00fdch strany vyjadren\u00e9 v cm s\u00fa cel\u00e9 \u010d\u00edsla, na <em>vysok\u00e9<\/em>, <em>n\u00edzke<\/em> a <em>ostatn\u00e9<\/em>. <strong><em>Vysok\u00fd<\/em><\/strong> trojuholn\u00edk je tak\u00fd, v ktorom sa s\u00fa\u010det d\u013a\u017eok niektor\u00fdch dvoch str\u00e1n rovn\u00e1 \u0161tvorn\u00e1sobku d\u013a\u017eky tretej strany. <strong><em>N\u00edzky<\/em><\/strong> trojuholn\u00edk je tak\u00fd, v ktorom je s\u00fa\u010din d\u013a\u017eok niektor\u00fdch dvoch str\u00e1n dvojn\u00e1sobkom d\u013a\u017eky tretej strany. N\u00e1jdite<\/p>\n<p>\u00a0\u00a0\u00a0\u00a0<strong>a)<\/strong>\u00a0v\u0161etky <strong><em>vysok\u00e9<\/em><\/strong> trojuholn\u00edky, ktor\u00fdch jedna strana meria 6 cm,<br \/>\n\u00a0\u00a0\u00a0\u00a0<strong>b)<\/strong>\u00a0v\u0161etky <strong><em>n\u00edzke<\/em><\/strong> trojuholn\u00edky, ktor\u00fdch jedna strana meria 6 cm,<br \/>\n\u00a0\u00a0\u00a0\u00a0<strong>c)<\/strong>\u00a0v\u0161etky trojuholn\u00edky, ktor\u00fdch jedna strana meria 4 cm a s\u00fa <strong><em>vysok\u00e9<\/em><\/strong> a <strong><em>n\u00edzke<\/em><\/strong> s\u00fa\u010dasne.<\/p>\n<div class=\"pri\">(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 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>51. ro\u010dn\u00edk matematickej olympi\u00e1dy 2001-2002 S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z4-Z9 aktualizovan\u00e9 25.2.2019 23:25 Z4 Z5 Z6 Z7 Z8 Z9 dom\u00e1ce dom\u00e1ce dom\u00e1ce dom\u00e1ce dom\u00e1ce dom\u00e1ce dom\u00e1ce obvodn\u00e9 obvodn\u00e9 obvodn\u00e9 obvodn\u00e9 obvodn\u00e9 \u00a0 krajsk\u00e9<\/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":[27],"tags":[],"class_list":["post-1346","post","type-post","status-publish","format-standard","hentry","category-51-rocnik"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.6 - 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