{"id":1089,"date":"2015-09-10T20:10:09","date_gmt":"2015-09-10T20:10:09","guid":{"rendered":"http:\/\/matematika.besaba.com\/?p=1089"},"modified":"2016-09-05T21:41:28","modified_gmt":"2016-09-05T21:41:28","slug":"sutazne-ulohy-kategorii-z5-z9-4","status":"publish","type":"post","link":"https:\/\/matematika.besaba.com\/?p=1089","title":{"rendered":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9"},"content":{"rendered":"<p class=\"hh2\"><a name=\"top\"><\/a>65. ro\u010dn\u00edk matematickej olympi\u00e1dy 2015-2016<br \/> S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9<\/p>\n<p><span class=\"povinne\">aktualizovan\u00e9 15.8.2016 17:00<\/span><\/p>\n<p><!--more--><\/p>\n<p><center><script type=\"text\/javascript\" async=\"true\" charset=\"utf-8\" src=\"http:\/\/sk.search.etargetnet.com\/generic\/advert.php?g=ref:48528,area:600x70,tabl:4,divid:,design_name:green,border_color:ffffff,border_style:none,background_opacity:100,background_color:ffffff,title_color:426000,text_color:000000,url_color:426000,h_title_color:426000,h_text_color:000000,h_url_color:426000,freespace:0,logo_type:3,logo:1,title_underline:0,url_underline:0,h_title_underline:1,h_url_underline:1,nourl:,fsi:11,font:verdana,require_login_pass:,login_pass:,show_in_fb:,show_in_fb_text:,allow_youtube_ads:1\" ><\/script><\/center><\/p>\n<table style=\"width: 90%;\" border=\"1\" cellspacing=\"2\" cellpadding=\"5\" align=\"center\">\n<tbody>\n<tr align=\"center\" bgcolor=\"#c0c0c0\">\n<td><span class=\"a\">Z5<\/span><\/td>\n<td><span class=\"a\">Z6<\/span><\/td>\n<td><span class=\"a\">Z7<\/span><\/td>\n<td><span class=\"a\">Z8<\/span><\/td>\n<td><span class=\"a\">Z9<\/span><\/td>\n<\/tr>\n<tr align=\"center\">\n<td><span class=\"a\"><a href=\"#z51\">dom\u00e1ce<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z61\">dom\u00e1ce<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z71\">dom\u00e1ce<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z81\">dom\u00e1ce<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z91\">dom\u00e1ce<\/a><\/span><\/td>\n<\/tr>\n<tr align=\"center\">\n<td><span class=\"a\"><a href=\"#z52\">obvodn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z62\">obvodn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z72\">obvodn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z82\">obvodn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z92\">obvodn\u00e9<\/a><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"4\" bgcolor=\"#c0c0c0\">\u00a0<\/td>\n<td align=\"center\"><span class=\"a\"><a href=\"#z93\">krajsk\u00e9<\/a><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"background-color: #ffff99;\" colspan=\"5\" align=\"center\"><span class=\"rs\">Zadania \u00faloh dom\u00e1ceho kola na stiahnutie vo form\u00e1te PDF: <a href=\"\/65\/65_z59.pdf\" target=\"_blank\"><img loading=\"lazy\" decoding=\"async\" src=\"\/img\/down.png\" alt=\"\" width=\"48\" height=\"48\" align=\"absmiddle\" border=\"0\" \/><\/a> (ulo\u017ei\u0165 ako &#8230;)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"z51\"><\/a><\/p>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=1499\" target=\"_blank\"><input name=\"butt51z\" type=\"button\" value=\"Zadanie\" \/><\/a>\u00a0\u00a0<a href=\"http:\/\/skmo.sk\/dokument.php?id=1733\" target=\"_blank\"><input name=\"butt51r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z5-I-1<\/strong><br \/>\u010c\u00edsla 1, 2, 3, 4, 5, 6, 7, 8 a 9 cestovali vlakom. Vlak mal tri vag\u00f3ny a v ka\u017edom sa viezli pr\u00e1ve tri \u010d\u00edsla. \u010c\u00edslo 1 sa viezlo v prvom vag\u00f3ne a v poslednom vag\u00f3ne boli v\u0161etky \u010d\u00edsla nep\u00e1rne. Sprievodca cestou spo\u010d\u00edtal s\u00fa\u010det \u010d\u00edsel v prvom, druhom aj poslednom vag\u00f3ne a zaka\u017ed\u00fdm mu vy\u0161iel rovnak\u00fd s\u00fa\u010det. Ur\u010dte, ako mohli by\u0165 \u010d\u00edsla do vag\u00f3nov rozdelen\u00e9.<\/p>\n<div class=\"pri\">(Veronika Huc\u00edkov\u00e1)<\/div>\n<p><strong>Z5-I-2<\/strong><br \/>Marta niesla svojej chorej kamar\u00e1tke Majke 7 jab\u013ak, 6 hru\u0161iek a 3 pomaran\u010de. Cestou<br \/> ale dva kusy ovocia zjedla. Ur\u010dte, ktor\u00e9 z nasleduj\u00facich situ\u00e1ci\u00ed mohli nasta\u0165 a ak\u00e9 dva kusy ovocia by Marta v takom pr\u00edpade musela zjes\u0165:<\/p>\n<p><strong>a)<\/strong>\u00a0\u00a0Majka nedostala \u017eiadny pomaran\u010d.<br \/> <strong>b)<\/strong>\u00a0\u00a0Majka dostala menej hru\u0161iek ako pomaran\u010dov.<br \/> <strong>c)<\/strong>\u00a0\u00a0Majka dostala rovnak\u00fd po\u010det jab\u013ak, hru\u0161iek aj pomaran\u010dov.<br \/> <strong>d)<\/strong>\u00a0\u00a0Majka dostala rovnak\u00fd po\u010det kusov ovocia dvojak\u00e9ho druhu.<br \/> <strong>e)<\/strong>\u00a0\u00a0Majka dostala viac jab\u013ak ako ostatn\u00fdch kusov ovocia dokopy.<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z5-I-3<\/strong><br \/>Mami\u010dka vyprala \u0161tvorcov\u00e9 utierky a ve\u0161ia ich ved\u013ea seba na bieliz\u0148ov\u00fa \u0161n\u00faru natiahnut\u00fa medzi dvoma stromami. Pou\u017eila \u0161n\u00faru s d\u013a\u017ekou 7,5 metra, pri\u010dom na uviazanie okolo kme\u0148ov potrebovala na ka\u017edej strane 8 dm. V\u0161etky utierky maj\u00fa \u0161\u00edrku 45 cm. Medzi krajnou utierkou a kme\u0148om mami\u010dka nech\u00e1va medzeru aspo\u0148 10 cm, utierky sa jej neprekr\u00fdvaj\u00fa a nem\u00e1 ich zlo\u017een\u00e9 ani skr\u010den\u00e9. Ko\u013eko najviac utierok m\u00f4\u017ee takto zavesi\u0165 na natiahnut\u00fa \u0161n\u00faru?<\/p>\n<div class=\"pri\">(Lenka Dedkov\u00e1)<\/div>\n<p><strong>Z5-I-4<\/strong><br \/>Ke\u010f p\u00e1n Baran zakladal chov, mal bielych oviec o 8 viac ako \u010diernych. V s\u00fa\u010dasnosti m\u00e1 bielych oviec \u0161tyrikr\u00e1t viac ako na za\u010diatku a \u010diernych trikr\u00e1t viac ako na za\u010diatku. Bielych oviec je teraz o 42 viac ako \u010diernych. Ko\u013eko teraz p\u00e1n Baran chov\u00e1 bielych a \u010diernych oviec dokopy?<\/p>\n<div class=\"pri\">(\u013dubo\u0161 \u0160im\u016fnek)<\/div>\n<p><strong>Z5-I-5<\/strong><br \/>\u0160tvorcov\u00e1 sie\u0165 sa sklad\u00e1 zo \u0161tvorcov so stranou d\u013a\u017eky 1 cm. Narysujte do nej aspo\u0148 tri r\u00f4zne \u00fatvary tak\u00e9, aby ka\u017ed\u00fd mal obsah 6 cm<sup>2<\/sup> a obvod 12 cm a aby ich strany spl\u00fdvali s priamkami siete.<\/p>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z5-I-6<\/strong><br \/>V nepriestupnom roku bolo 53 nedie\u013e. Na ak\u00fd de\u0148 t\u00fd\u017ed\u0148a pripadol \u0160tedr\u00fd de\u0148?<\/p>\n<div class=\"pri\">(Marta Volfov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z52\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=1737\" target=\"_blank\"><input name=\"butt52\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z5-II-1<\/strong><br \/>Mami\u010dka zav\u00e1ra slivky do flia\u0161 tak, \u017ee slivky z jednej f\u013ea\u0161e jej vysta\u010dia bu\u010f na 16 \u0161at\u00f4\u010dok, alebo na 4 kol\u00e1\u010diky, alebo na polovicu plechu ovocn\u00fdch rezov. V \u0161pajze m\u00e1 4 tak\u00e9 f\u013ea\u0161e a chce upiec\u0165 jeden plech ovocn\u00fdch rezov a 6 kol\u00e1\u010dikov. Na ko\u013eko \u0161at\u00f4\u010dok jej vysta\u010dia zvy\u0161n\u00e9 slivky?<\/p>\n<div class=\"pri\">(Michaela Petrov\u00e1)<\/div>\n<p><strong>Z5-II-2<\/strong><br \/>Str\u00fdko Fero mal dve z\u00e1hrady: mrkvov\u00e1 mala tvar \u0161tvorca, jahodov\u00e1 mala tvar obd\u013a\u017enika. Pritom \u0161\u00edrka jahodovej z\u00e1hrady bola trikr\u00e1t men\u0161ia ako \u0161\u00edrka mrkvovej z\u00e1hrady a d\u013a\u017eka jahodovej z\u00e1hrady bola o 8 metrov v\u00e4\u010d\u0161ia ako d\u013a\u017eka mrkvovej z\u00e1hrady. Ke\u010f str\u00fdko z\u00e1hrady oplotil, zistil, \u017ee obe mali rovnak\u00fd obvod. Ur\u010dte rozmery mrkvovej a jahodovej z\u00e1hrady.<\/p>\n<div class=\"pri\">(Erika Novotn\u00e1)<\/div>\n<p><strong>Z5-II-3<\/strong><br \/>Jeden mesiac mal \u0161tyri pondelky, p\u00e4\u0165 nedie\u013e a jeden piatok trin\u00e1steho. Na ktor\u00fd de\u0148 v t\u00fd\u017edni pripadol toho roku Nov\u00fd rok? Ktor\u00fdm d\u0148om v t\u00fd\u017edni bude Nov\u00fd rok v roku nasleduj\u00facom?<\/p>\n<div class=\"pri\">(Marta Volfov\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 \/>Archeol\u00f3govia zistili, \u017ee vlajka b\u00e1jneho matematick\u00e9ho kr\u00e1\u013eovstva bola rozdelen\u00e1 na \u0161es\u0165 pol\u00ed\u010dok, tak ako na obr\u00e1zku. V skuto\u010dnosti bola vlajka trojfarebn\u00e1 a ka\u017ed\u00e9 pol\u00ed\u010dko bolo vyfarben\u00e9 jednou farbou. Vedci u\u017e vyb\u00e1dali, \u017ee na vlajke bola pou\u017eit\u00e1 \u010derven\u00e1, biela a modr\u00e1 farba, \u017ee vn\u00fatorn\u00e9 obd\u013a\u017enikov\u00e9 pol\u00ed\u010dko bolo biele a \u017ee spolu nesusedili dve pol\u00ed\u010dka rovnakej farby. Ur\u010dte, ko\u013eko mo\u017enost\u00ed vzh\u013eadu vlajky musia archeol\u00f3govia<br \/> v tejto f\u00e1ze v\u00fdskumu zva\u017eova\u0165.<\/p>\n<p class=\"pc\"><img decoding=\"async\" src=\"65\/65z6a1.gif\" alt=\"\" border=\"0\" \/><\/p>\n<div class=\"pri\">(Veronika Huc\u00edkov\u00e1)<\/div>\n<p><strong>Z6-I-2<\/strong><br \/>Juraj i\u0161iel do slu\u017eby k \u010darodejn\u00edkovi. Ten mal v prvej pivnici viac m\u00fach ako pav\u00fakov, v druhej naopak. V ka\u017edej pivnici mali muchy a pav\u00faky dokopy 100 n\u00f4h. Ur\u010dte, ko\u013eko mohlo by\u0165 m\u00fach a pav\u00fakov v prvej a ko\u013eko v druhej pivnici.<\/p>\n<div class=\"pri\">(Marie Krej\u010dov\u00e1)<\/div>\n<p><strong>Z6-I-3<\/strong><br \/>Na obr\u00e1zku je \u0161tvorec <em>ABCD<\/em>, \u0161tvorec <em>EFGD<\/em> a obd\u013a\u017enik <em>HIJD<\/em>. Body <em>J<\/em> a <em>G<\/em> le\u017eia na strane <em>CD<\/em>, pri\u010dom plat\u00ed |<em>DJ<\/em>| < |<em>DG<\/em>|, a body <em>H<\/em> a <em>E<\/em> le\u017eia na strane <em>DA<\/em>, pri\u010dom plat\u00ed |<em>DH<\/em>| < |<em>DE<\/em>|. \u010ealej vieme, \u017ee |<em>DJ<\/em>| = |<em>GC<\/em>|. \u0160es\u0165uholn\u00edk <em>ABCGFE<\/em> m\u00e1 obvod 96 cm, \u0161es\u0165uholn\u00edk <em>EFGJIH<\/em> m\u00e1 obvod 60 cm a obd\u013a\u017enik <em>HIJD<\/em> m\u00e1 obvod 28 cm. Ur\u010dte obsah \u0161es\u0165uholn\u00edka <em>EFGJIH<\/em>.<\/p>\n<p class=\"pc\"><img decoding=\"async\" src=\"65\/65z6a2.png\" alt=\"\" border=\"0\" \/><\/p>\n<div class=\"pri\">(\u013dubo\u0161 \u0160im\u016fnek)<\/div>\n<p><strong>Z6-I-4<\/strong><br \/>Na obr\u00e1zku je obd\u013a\u017enik rozdelen\u00fd na 7 pol\u00ed\u010dok. Na ka\u017ed\u00e9 pol\u00ed\u010dko sa m\u00e1 nap\u00edsa\u0165 pr\u00e1ve jedno z \u010d\u00edsel 1, 2 a 3. Miro tvrd\u00ed, \u017ee sa to d\u00e1 spravi\u0165 tak, aby s\u00fa\u010det dvoch ved\u013ea seba nap\u00edsan\u00fdch \u010d\u00edsel bol zaka\u017ed\u00fdm in\u00fd. Zuzka naopak tvrd\u00ed, \u017ee to nie je mo\u017en\u00e9. Rozhodnite, kto z nich m\u00e1 pravdu.<\/p>\n<table border=\"1\" width=\"175\" cellspacing=\"0\" cellpadding=\"10\" align=\"center\">\n<tbody>\n<tr>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"pri\">(Veronika Huc\u00edkov\u00e1)<\/div>\n<p><strong>Z6-I-5<\/strong><br \/>P\u00e1n Cuketa mal obd\u013a\u017enikov\u00fa z\u00e1hradu, ktorej obvod bol 28 metrov. Obsah celej z\u00e1hrady vyplnili pr\u00e1ve \u0161tyri \u0161tvorcov\u00e9 z\u00e1hony, ktor\u00fdch rozmery v metroch boli vyjadren\u00e9 cel\u00fdmi \u010d\u00edslami. Ur\u010dte, ak\u00e9 rozmery mohla ma\u0165 z\u00e1hrada. N\u00e1jdite v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z6-I-6<\/strong><br \/>V z\u00e1mockej kuchyni pripravuj\u00fa rezancov\u00fa polievku v hrncoch a kotloch. V pondelok uvarili 25 hrncov a 10 kotlov polievky. V utorok uvarili 15 hrncov a 13 kotlov. V stredu uvarili 20 hrncov a vo \u0161tvrtok 30 kotlov. Pritom v pondelok a v utorok uvarili rovnak\u00e9 mno\u017estvo polievky. Ko\u013ekokr\u00e1t viac polievky uvarili vo \u0161tvrtok ako v stredu?<\/p>\n<div class=\"pri\">(Karel Pazourek)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z62\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=1865\" target=\"_blank\"><input name=\"butt62z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=1868\" target=\"_blank\"><input name=\"butt62r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z6-II-1<\/strong><br \/>Pani u\u010dite\u013eka nap\u00edsala na tabu\u013eu dve \u010d\u00edsla pod seba a vyvolala Adama, aby ich s\u010d\u00edtal. Adam ich spr\u00e1vne s\u010d\u00edtal a v\u00fdsledok 39 nap\u00edsal pod zadan\u00e9 \u010d\u00edsla. Pani u\u010dite\u013eka zotrela najvrchnej\u0161ie \u010d\u00edslo, a tak zvy\u0161n\u00e9 dve \u010d\u00edsla vytvorili nov\u00fd pr\u00edklad na s\u010d\u00edtanie. Tentoraz spr\u00e1vny v\u00fdsledok zap\u00edsala pod \u010d\u00edsla Barbora. Pani u\u010dite\u013eka op\u00e4\u0165 zotrela najvrchnej\u0161ie \u010d\u00edslo, novo vzniknut\u00fd pr\u00edklad na s\u010d\u00edtanie spr\u00e1vne vypo\u010d\u00edtal Cyril a vy\u0161lo mu 96. Ur\u010dte dve \u010d\u00edsla, ktor\u00e9 boli p\u00f4vodne nap\u00edsan\u00e9 na tabuli.<\/p>\n<div class=\"pri\">(Libor \u0160im\u016fnek)<\/div>\n<p><strong>Z6-II-2<\/strong><br \/>Vn\u00fatri obd\u013a\u017enika <em>ABGH<\/em> s\u00fa dva zhodn\u00e9 \u0161tvorce <em>CDEF<\/em> a <em>IJKL<\/em>. Strana <em>CF<\/em> prv\u00e9ho \u0161tvorca le\u017e\u00ed na strane <em>BG<\/em> obd\u013a\u017enika a strana <em>IL<\/em> druh\u00e9ho z nich le\u017e\u00ed na strane <em>H<\/em>A obd\u013a\u017enika. Obvod osemuholn\u00edka <em>ABCDEFGH<\/em> je 48 cm, obvod dvan\u00e1s\u0165uholn\u00edka <em>ABCDEFGHIJKL<\/em> je 58 cm. Tento dvan\u00e1s\u0165uholn\u00edk je s\u00famern\u00fd pod\u013ea vodorovnej osi a d\u013a\u017eky v\u0161etk\u00fdch jeho str\u00e1n s\u00fa v centimetroch vyjadren\u00e9 cel\u00fdmi \u010d\u00edslami. Ur\u010dte d\u013a\u017eky str\u00e1n obd\u013a\u017enika <em>ABGH<\/em>.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"65\/65z6b1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Libor \u0160im\u016fnek)<\/div>\n<p><strong>Z6-II-3<\/strong><br \/>Na \u010cern\u00ed\u010dkovom statku mali tri sliepky. Prv\u00e1 zn\u00e1\u0161ala ka\u017ed\u00fd de\u0148 jedno vajce, druh\u00e1 zn\u00e1\u0161ala ka\u017ed\u00fd druh\u00fd de\u0148 jedno vajce a tretia ka\u017ed\u00fd tret\u00ed de\u0148 jedno vajce. P\u00e1n \u010cern\u00ed\u010dek prik\u00fapil na trhu dve nov\u00e9 sliepky, ktor\u00e9 tie\u017e zn\u00e1\u0161ali vaj\u00ed\u010dka \u00faplne pravidelne \u2013 v\u017edy jedno po nieko\u013ek\u00fdch d\u0148och, pri\u010dom jedna z nich zn\u00e1\u0161ala dvakr\u00e1t viac ako druh\u00e1. Pani \u010cern\u00ed\u010dkov\u00e1 spo\u010d\u00edtala, \u017ee v\u0161etk\u00fdch p\u00e4\u0165 sliepok znieslo za 60 dn\u00ed spolu 155 vajec. Ako \u010dasto zn\u00e1\u0161ali nov\u00e9 sliepky?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z71\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><center><script type=\"text\/javascript\" async=\"true\" charset=\"utf-8\" src=\"http:\/\/sk.search.etargetnet.com\/generic\/advert.php?g=ref:48528,area:600x70,tabl:4,divid:,design_name:green,border_color:ffffff,border_style:none,background_opacity:100,background_color:ffffff,title_color:426000,text_color:000000,url_color:426000,h_title_color:426000,h_text_color:000000,h_url_color:426000,freespace:0,logo_type:3,logo:1,title_underline:0,url_underline:0,h_title_underline:1,h_url_underline:1,nourl:,fsi:11,font:verdana,require_login_pass:,login_pass:,show_in_fb:,show_in_fb_text:,allow_youtube_ads:1\" ><\/script><\/center><\/p>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=1497\" target=\"_blank\"><input name=\"butt71z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=1863\" target=\"_blank\"><input name=\"butt71r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z7-I-1<\/strong><br \/>My\u0161ka Hryzka na\u0161la 27 rovnak\u00fdch koc\u00f4\u010dok syra. Najsk\u00f4r si z nich poskladala ve\u013ek\u00fa kocku a chv\u00ed\u013eu po\u010dkala, k\u00fdm sa syrov\u00e9 koc\u00f4\u010dky k sebe prilepili. Potom z ka\u017edej steny ve\u013ekej kocky vyhr\u00fdzla stredn\u00fa koc\u00f4\u010dku. Napokon zjedla aj koc\u00f4\u010dku, ktor\u00e1 bola v strede ve\u013ekej kocky. Zvy\u0161ok syra chce Hryzka spravodlivo rozdeli\u0165 svojim \u0161tyrom ml\u00e1\u010fat\u00e1m, a preto ho chce rozreza\u0165 na \u0161tyri kusy rovnak\u00e9ho tvaru aj ve\u013ekosti. Reza\u0165 bude iba pozd\u013a\u017e stien koc\u00f4\u010dok a ni\u010d k sebe u\u017e lepi\u0165 nebude. Ak\u00fd tvar m\u00f4\u017eu ma\u0165 kusy syra pre ml\u00e1\u010fat\u00e1? N\u00e1jdite aspo\u0148 dve mo\u017enosti.<\/p>\n<div class=\"pri\">(Veronika Huc\u00edkov\u00e1)<\/div>\n<p><strong>Z7-I-2<\/strong><br \/>Vl\u010dkovci maj\u00fa 4 deti. Ondrej je o 3 roky star\u0161\u00ed ako Matej a Jakub o 5 rokov star\u0161\u00ed ako najmlad\u0161ia Jana. Vieme, \u017ee maj\u00fa dokopy 30 rokov a pred 3 rokmi mali dokopy 19 rokov. Ur\u010dte, ko\u013eko m\u00e1 ktor\u00e9 die\u0165a rokov.<\/p>\n<div class=\"pri\">Marta Volfov\u00e1<\/div>\n<p><strong>Z7-I-3<\/strong><br \/>Vn\u00fatri pravideln\u00e9ho p\u00e4\u0165uholn\u00edka <em>ABCDE<\/em> je bod <em>P<\/em> tak\u00fd, \u017ee trojuholn\u00edk <em>ABP<\/em> je rovnostrann\u00fd. Ak\u00fd ve\u013ek\u00fd je uhol <em>BCP<\/em>?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z7-I-4<\/strong><br \/>V \u0161kole pre robotov do jednej triedy chod\u00ed dvadsa\u0165 robotov Robertov, ktor\u00ed s\u00fa o\u010d\u00edslovan\u00ed Robert 1 a\u017e Robert 20. V triede je pr\u00e1ve nap\u00e4t\u00e1 atmosf\u00e9ra, rozpr\u00e1vaj \u00fa sa spolu iba niektor\u00ed roboti. Roboti s nep\u00e1rnym \u010d\u00edslom sa nerozpr\u00e1vaj\u00fa s robotmi s p\u00e1rnym \u010d\u00edslom. Medzi Robertmi s nep\u00e1rnym \u010d\u00edslom sa spolu rozpr\u00e1vaj\u00fa iba roboti, ktor\u00ed maj\u00fa \u010d\u00edslo s rovnak\u00fdm po\u010dtom cifier. Roberti s p\u00e1rnym \u010d\u00edslom sa rozpr\u00e1vaj\u00fa iba s t\u00fdmi, ktor\u00fdch \u010d\u00edslo za\u010d\u00edna rovnakou cifrou. Ko\u013eko dvoj\u00edc robotov Robertov sa m\u00f4\u017ee spolu navz\u00e1jom rozpr\u00e1va\u0165?<\/p>\n<div class=\"pri\">(Karel Pazourek)<\/div>\n<p><strong>Z7-I-5<\/strong><br \/>V koc\u00farkovskej \u0161kole pou\u017e\u00edvaj\u00fa zvl\u00e1\u0161tnu \u010d\u00edseln\u00fa os. Vzdialenos\u0165 medzi \u010d\u00edslami 1 a 2 je 1 cm, vzdialenos\u0165 medzi \u010d\u00edslami 2 a 3 je 3 cm, medzi \u010d\u00edslami 3 a 4 je 5 cm a tak \u010falej: vzdialenos\u0165 medzi ka\u017edou nasleduj\u00facou dvojicou prirodzen\u00fdch \u010d\u00edsel sa v\u017edy zv\u00e4\u010d\u0161\u00ed o 2 cm. Medzi ktor\u00fdmi dvoma prirodzen\u00fdmi \u010d\u00edslami je na koc\u00farkovskej \u010d\u00edselnej osi vzdialenos\u0165 39 cm? N\u00e1jdite v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Karel Pazourek)<\/div>\n<p><strong>Z7-I-6<\/strong><br \/>Na v\u00fdstave dlhosrst\u00fdch ma\u010diek sa zi\u0161lo spolu desa\u0165 vystavuj\u00facich. Vystavovalo sa v obd\u013a\u017enikovej miestnosti, v ktorej boli dva rady stolov ako na obr\u00e1zku. Ma\u010dky boli ozna\u010den\u00e9 navz\u00e1jom r\u00f4znymi \u010d\u00edslami v rozsahu 1 a\u017e 10 a na ka\u017edom stole sedela jedna ma\u010dka. Ur\u010dte \u010d\u00edslo ma\u010dky, ktor\u00e1 bola na v\u00fdstave hodnoten\u00e1 najlep\u0161ie, ak viete, \u017ee:<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"65\/65z7a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<ul type=\"circle\">\n<li>s\u00fa\u010det \u010d\u00edsel ma\u010diek sediacich oproti sebe bol v\u017edy rovnak\u00fd, <\/li>\n<li>s\u00fa\u010det \u010d\u00edsel ka\u017ed\u00fdch dvoch ma\u010diek sediacich ved\u013ea seba bol p\u00e1rny,<\/li>\n<li>s\u00fa\u010din \u010d\u00edsel ka\u017ed\u00fdch dvoch ma\u010diek sediacich ved\u013ea seba v dolnom rade bol n\u00e1sobok \u010d\u00edsla 8,<\/li>\n<li>ma\u010dka \u010d\u00edslo 1 nebola na kraji a bola viac vpravo ako ma\u010dka \u010d\u00edslo 6,<\/li>\n<li>vyhrala ma\u010dka sediaca v pravom dolnom rohu.<\/li>\n<\/ul>\n<div class=\"pri\">(Martin Mach)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z72\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=1866\" target=\"_blank\"><input name=\"butt72z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=1869\" target=\"_blank\"><input name=\"butt72r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z7-II-1<\/strong><br \/>V roku 1966 \u017eilo v obci Bezd\u00edkov o 30 \u017eien viac ako mu\u017eov. Do s\u00fa\u010dasnosti sa po\u010det \u017eien \u017eij\u00facich v obci zmen\u0161il \u0161tyrikr\u00e1t a po\u010det mu\u017eov \u017eij\u00facich v obci klesol o 196. Teraz je v Bezd\u00edkove o 10 \u017eien viac ako mu\u017eov. Ko\u013eko \u017eien a mu\u017eov dokopy \u017eije v s\u00fa\u010dasnosti v Bezd\u00edkove?<\/p>\n<div class=\"pri\">(Libor \u0160im\u016fnek)<\/div>\n<p><strong>Z7-II-2<\/strong><br \/>Pani u\u010dite\u013eka nap\u00edsala na tabu\u013eu dve \u010d\u00edsla pod seba a vyvolala Kl\u00e1ru, aby ich s\u010d\u00edtala. Kl\u00e1ra spr\u00e1vny v\u00fdsledok zap\u00edsala pod zadan\u00e9 \u010d\u00edsla. Pani u\u010dite\u013eka zotrela najvrchnej\u0161ie \u010d\u00edslo, a tak zvy\u0161n\u00e9 dve \u010d\u00edsla vytvorili nov\u00fd pr\u00edklad na s\u010d\u00edtanie. Tentoraz spr\u00e1vny v\u00fdsledok zap\u00edsal pod \u010d\u00edsla Luk\u00e1\u0161. Pani u\u010dite\u013eka op\u00e4\u0165 zotrela najvrchnej\u0161ie \u010d\u00edslo, novo vzniknut\u00fd pr\u00edklad na s\u010d\u00edtanie spr\u00e1vne vypo\u010d\u00edtala Magda a vy\u0161lo jej 94. Jedno z dvoch \u010d\u00edsel, ktor\u00e9 pani u\u010dite\u013eka p\u00f4vodne nap\u00edsala na tabu\u013eu, bolo 14, ale neprezrad\u00edme, ktor\u00e9. Ak\u00e9 mohlo by\u0165 druh\u00e9 z p\u00f4vodne nap\u00edsan\u00fdch \u010d\u00edsel? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Libor \u0160im\u016fnek)<\/div>\n<p><strong>Z7-II-3<\/strong><br \/>V obd\u013a\u017eniku <em>ABCD<\/em> so stranou <em>AD<\/em> maj\u00facou d\u013a\u017eku 5 cm le\u017e\u00ed bod <em>P<\/em> tak, \u017ee trojuholn\u00edk <em>APD<\/em> je rovnostrann\u00fd. Polpriamka <em>AP<\/em> pret\u00edna stranu <em>CD<\/em> v bode <em>E<\/em>, \u00fase\u010dka <em>CE<\/em> meria 5 cm. Ak\u00e1 dlh\u00e1 je \u00fase\u010dka <em>AE<\/em> a ak\u00e1 je ve\u013ekos\u0165 uhla <em>AEB<\/em>?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z81\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><center><script type=\"text\/javascript\" async=\"true\" charset=\"utf-8\" src=\"http:\/\/sk.search.etargetnet.com\/generic\/advert.php?g=ref:48528,area:600x70,tabl:4,divid:,design_name:green,border_color:ffffff,border_style:none,background_opacity:100,background_color:ffffff,title_color:426000,text_color:000000,url_color:426000,h_title_color:426000,h_text_color:000000,h_url_color:426000,freespace:0,logo_type:3,logo:1,title_underline:0,url_underline:0,h_title_underline:1,h_url_underline:1,nourl:,fsi:11,font:verdana,require_login_pass:,login_pass:,show_in_fb:,show_in_fb_text:,allow_youtube_ads:1\" ><\/script><\/center><\/p>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=1496\" target=\"_blank\"><input name=\"butt81z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=1864\" target=\"_blank\"><input name=\"butt81r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z8-I-1<\/strong><br \/>Mi\u0161o mal na poli\u010dke mal\u00fa klaviat\u00faru, ktor\u00fa vid\u00edte na obr\u00e1zku. Na bielych kl\u00e1vesoch boli vyzna\u010den\u00e9 ich t\u00f3ny. Klaviat\u00faru na\u0161la mal\u00e1 Kl\u00e1ra. Ke\u010f ju brala z poli\u010dky, vypadla jej z ruky a v\u0161etky biele kl\u00e1vesy sa z nej vysypali. Aby sa brat nehneval, za\u010dala ich Kl\u00e1ra sklada\u0165 sp\u00e4\u0165. V\u0161imla si pritom, \u017ee sa dali vlo\u017ei\u0165 iba na niektor\u00e9 miesta, lebo im prek\u00e1\u017eali \u010dierne kl\u00e1vesy umiestnen\u00e9 presne doprostred medzi dva biele. Kl\u00e1re sa podarilo kl\u00e1vesy nejako zlo\u017ei\u0165, av\u0161ak t\u00f3ny na nich boli pomie\u0161an\u00e9, ke\u010f\u017ee e\u0161te nepoznala hudobn\u00fa stupnicu. Zistite, ko\u013ek\u00fdmi sp\u00f4sobmi mohla Kl\u00e1ra kl\u00e1vesy posklada\u0165.<\/p>\n<p class=\"pc\"><img decoding=\"async\" src=\"65\/65z8a1.png\" alt=\"\" border=\"0\" \/><\/p>\n<div class=\"pri\">(Erika Novotn\u00e1)<\/div>\n<p><strong>Z8-I-2<\/strong><br \/>Na l\u00fake sa pas\u00fa kone, kravy a ovce, spolu ich je menej ako 200. Keby bolo kr\u00e1v 45-kr\u00e1t viac, kon\u00ed 60-kr\u00e1t viac a oviec 35-kr\u00e1t viac ako ich je teraz, ich po\u010dty by sa rovnali. Ko\u013eko sa spolu na l\u00fake pasie kon\u00ed, kr\u00e1v a oviec?<\/p>\n<div class=\"pri\">(Marie Krej\u010dov\u00e1)<\/div>\n<p><strong>Z8-I-3<\/strong><br \/>Dan\u00fd je rovnoramenn\u00fd lichobe\u017en\u00edk <em>ABCD<\/em>, v ktorom plat\u00ed<\/p>\n<p style=\"text-align: center;\">|<em>AB<\/em>|= 2|<em>BC<\/em>| = 2|<em>CD<\/em>| = 2|<em>DA<\/em>|.<\/p>\n<p>Na jeho strane <em>BC<\/em> je bod <em>K<\/em> tak\u00fd, \u017ee |<em>BK<\/em>| = 2|<em>KC<\/em>|, na jeho strane <em>CD<\/em> je bod <em>L<\/em> tak\u00fd, \u017ee |<em>CL<\/em>| = 2|<em>LD<\/em>|, a na jeho strane <em>DA<\/em> je bod <em>M<\/em> tak\u00fd, \u017ee |<em>DM<\/em>| = 2|<em>MA<\/em>|. Ur\u010dte ve\u013ekosti vn\u00fatorn\u00fdch uhlov trojuholn\u00edka <em>KLM<\/em>.<\/p>\n<div class=\"pri\">(Jaroslav Zhouf)<\/div>\n<p><strong>Z8-I-4<\/strong><br \/>V komore, kde sa rozbilo svetlo a v\u0161etko z nej mus\u00edme bra\u0165 naslepo, m\u00e1me pono\u017eky \u0161tyroch r\u00f4znych farieb. Ak si chceme by\u0165 ist\u00ed, \u017ee vytiahneme aspo\u0148 dve biele pono\u017eky, mus\u00edme ich z komory prinies\u0165 28. Aby sme mali tak \u00fa istotu pre siv\u00e9 pono\u017eky, mus\u00edme ich prinies\u0165 tie\u017e 28, pre \u010dierne pono\u017eky sta\u010d\u00ed 26 a pre modr\u00e9 pono\u017eky 34. Ko\u013eko je spolu v komore pono\u017eiek?<\/p>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z8-I-5<\/strong><br \/>\u010c\u00edslo d\u0148a je poradov\u00e9 \u010d\u00edslo dan\u00e9ho d\u0148a v pr\u00edslu\u0161nom mesiaci (teda napr. \u010d\u00edslo d\u0148a 5. augusta 2016 je 5). Cifern\u00fd s\u00fa\u010det d\u0148a je s\u00fa\u010det hodn\u00f4t v\u0161etk\u00fdch cifer v d\u00e1tume tohto d\u0148a (teda napr. cifern\u00fd s \u00fa\u010det d\u0148a 5. augusta 2016 je 5 + 8 + 2 + 0 + 1 + 6 = 22). \u0160\u0165astn\u00fd de\u0148 je tak\u00fd de\u0148, ktor\u00e9ho \u010d\u00edslo d\u0148a je rovn\u00e9 cifern\u00e9mu s\u00fa\u010dtu d\u0148a. Ur\u010dte, ko\u013eko \u0161\u0165astn\u00fdch dn\u00ed je v roku 2016 a ktor\u00e9 dni to s\u00fa.<\/p>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\u00e1)<\/div>\n<p><strong>Z8-I-6<\/strong><br \/>Katka narysovala trojuholn\u00edk <em>ABC<\/em>. Stred strany <em>AB<\/em> ozna\u010dila <em>X<\/em> a stred strany <em>AC<\/em> ozna\u010dila <em>Y<\/em>. Na strane <em>BC<\/em> chce n\u00e1js\u0165 tak\u00fd bod <em>Z<\/em>, aby obsah \u0161tvoruholn\u00edka <em>AXZY<\/em> bol \u010do najv\u00e4\u010d\u0161\u00ed. Ak\u00fa \u010das\u0165 trojuholn\u00edka <em>ABC<\/em> m\u00f4\u017ee maxim\u00e1lne zabera\u0165 \u0161tvoruholn\u00edk <em>AXZY<\/em>?<\/p>\n<div class=\"pri\">(Al\u017ebeta Bohinikov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z82\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><!--<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=1379\" target=\"_blank\"><input name=\"butt82z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=1382\" target=\"_blank\"><input name=\"butt82r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n<p>--><\/p>\n<p><strong>Z8-II-1<\/strong><br \/>Janko m\u00e1 v komore na chalupe krabicu s pastelkami, v ktorej je 5 modr\u00fdch, 9 \u010derven\u00fdch, 6 zelen\u00fdch a 4 \u017elt\u00e9 pastelky. Je tma, svetlo v komore nesvieti a Janko pri sebe nem\u00e1 \u017eiadny zdroj osvetlenia. Farbu pasteliek teda nedok\u00e1\u017ee rozl\u00ed\u0161i\u0165.<\/p>\n<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;a) Ko\u013eko najmenej pasteliek mus\u00ed vzia\u0165, aby mal istotu, \u017ee prinesie aspo\u0148 jednu pastelku z ka\u017edej farby?<\/p>\n<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;b) Ko\u013eko najviac pasteliek m\u00f4\u017ee vzia\u0165, aby mal istotu, \u017ee v krabici zostane z ka\u017edej farby aspo\u0148 jedna pastelka?<\/p>\n<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;c) Ko\u013eko najviac pasteliek m\u00f4\u017ee vzia\u0165, aby mal istotu, \u017ee v krabici zostane aspo\u0148 p\u00e4\u0165 \u010derven\u00fdch pasteliek?<\/p>\n<div class=\"pri\">(Marta Volfov\u00e1)<\/div>\n<p><strong>Z8-II-2<\/strong><br \/>Dedo chov\u00e1 husi, prasat\u00e1, kozy a sliepky &#8211; celkom 40 kusov. Na ka\u017ed\u00fa kozu pripadaj\u00fa 3 husi. Keby bolo sliepok o 8 menej, bolo by ich rovnako ako hus\u00ed a prasiat dokopy. Keby dedo vymenil \u0161tvrtinu hus\u00ed za sliepky v pomere 3 sliepky za 1 hus, mal by celkom 46 kusov zvierat. Ko\u013eko ktor\u00fdch zvierat dedo chov\u00e1?<\/p>\n<div class=\"pri\">(Marta Volfov\u00e1)<\/div>\n<p><strong>Z8-II-3<\/strong><br \/>Dobr\u00e1kovci pestovali tulip\u00e1ny na \u0161tvorcovom z\u00e1hone so stranou 6 metrov. Nesk\u00f4r pristavali k svojmu domu \u0161tvorcov\u00fa terasu so stranou 7 metrov. Jeden vrchol terasy le\u017eal presne uprostred tulip\u00e1nov\u00e9ho z\u00e1hona a jedna strana terasy delila stranu tulip\u00e1nov\u00e9ho z\u00e1hona v pomere 1 : 5. V akom pomere delila druh\u00e1 strana terasy druh\u00fa stranu z\u00e1hona? O ko\u013eko metrov \u0161tvorcov\u00fdch sa stavbou terasy zmen\u0161il z\u00e1hon tulip\u00e1nov?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z91\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><center><script type=\"text\/javascript\" async=\"true\" charset=\"utf-8\" src=\"http:\/\/sk.search.etargetnet.com\/generic\/advert.php?g=ref:48528,area:600x70,tabl:4,divid:,design_name:green,border_color:ffffff,border_style:none,background_opacity:100,background_color:ffffff,title_color:426000,text_color:000000,url_color:426000,h_title_color:426000,h_text_color:000000,h_url_color:426000,freespace:0,logo_type:3,logo:1,title_underline:0,url_underline:0,h_title_underline:1,h_url_underline:1,nourl:,fsi:11,font:verdana,require_login_pass:,login_pass:,show_in_fb:,show_in_fb_text:,allow_youtube_ads:1\" ><\/script><\/center><\/p>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=1734\" target=\"_blank\"><input name=\"butt91\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-I-1<\/strong><br \/>Objem vody v mestskom baz\u00e9ne s obd\u013a\u017enikov\u00fdm dnom je 6 998,4 hektolitrov. Propaga\u010dn\u00fd let\u00e1k uv\u00e1dza, \u017ee keby sme chceli v\u0161etku vodu z baz\u00e9na prelia\u0165 do pravideln\u00e9ho \u0161tvorbok\u00e9ho hranola s hranou podstavy rovnaj\u00facou sa priemernej h\u013abke baz\u00e9na, musel by by\u0165 hranol vysok\u00fd ako bl\u00edzky telev\u00edzny vysiela\u010d a potom by bol naplnen\u00fd a\u017e po okraj. Dod\u00e1vame, \u017ee keby sme chceli prepl\u00e1va\u0165 vzdialenos\u0165 rovnak\u00fa, ako je v\u00fd\u0161ka vysiela\u010da, museli by sme prepl\u00e1va\u0165 bu\u010f osem d\u013a\u017eok, alebo p\u00e4tn\u00e1s\u0165 \u0161\u00edrok baz\u00e9na. Ak\u00fd vysok\u00fd je vysiela\u010d?<\/p>\n<div class=\"pri\">(Libor \u0160im\u016fnek)<\/div>\n<p><strong>Z9-I-2<\/strong><br \/>\u00da\u017easn\u00fdm \u010d\u00edslom nazveme tak\u00e9 p\u00e1rne \u010d\u00edslo, ktor\u00e9ho rozklad na s\u00fa\u010din prvo\u010d\u00edsel m\u00e1 pr\u00e1ve tri nie nutne r\u00f4zne \u010dinitele a s\u00fa\u010det v\u0161etk\u00fdch jeho delite\u013eov je rovn\u00fd dvojn\u00e1sobku tohto \u010d\u00edsla. N\u00e1jdite v\u0161etky \u00fa\u017easn\u00e9 \u010d\u00edsla.<\/p>\n<div class=\"pri\">(Martin Mach)<\/div>\n<p><strong>Z9-I-3<\/strong><br \/>Juro zostrojil \u0161tvorec <em>ABCD<\/em> so stranou 12 cm. Do tohto \u0161tvorca narysoval \u0161tvr\u0165kru\u017enicu <em>k<\/em>, ktor\u00e1 mala stred v bode <em>B<\/em> a prech\u00e1dzala bodom <em>A<\/em>, a polkru\u017enicu <em>l<\/em>, ktor\u00e1 mala stred v strede strany <em>BC<\/em> a prech\u00e1dzala bodom <em>B<\/em>. R\u00e1d by e\u0161te zostrojil kru\u017enicu, ktor\u00e1 by le\u017eala vn\u00fatri \u0161tvorca a dot\u00fdkala sa \u0161tvr\u0165kru\u017enice <em>k<\/em>, polkru\u017enice <em>l<\/em> aj strany <em>AB<\/em>. Ur\u010dte polomer takej kru\u017enice.<\/p>\n<div class=\"pri\">(Marta Volfov\u00e1)<\/div>\n<p><strong>Z9-I-4<\/strong><br \/>V tabu\u013eke je kurzov\u00fd l\u00edstok zmen\u00e1rne, av\u0161ak niektor\u00e9 hodnoty s\u00fa v \u0148om nahraden\u00e9 ot\u00e1znikmi. Zmen\u00e1re\u0148 vymie\u0148a peniaze v uveden\u00fdch kurzoch a ne\u00fa\u010dtuje si in\u00e9 poplatky.<\/p>\n<table style=\"width: 200px; height: 60px;\" border=\"2\" cellspacing=\"2\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\">\u00a0<\/td>\n<td style=\"text-align: center;\">n\u00e1kup<\/td>\n<td style=\"text-align: center;\">predaj<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">1 EUR<\/td>\n<td style=\"text-align: center;\">26,20 CZK<\/td>\n<td style=\"text-align: center;\">28,00 CZK<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">1 GBP<\/td>\n<td style=\"text-align: center;\">? CZK<\/td>\n<td style=\"text-align: center;\">? CZK<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>1. Ko\u013eko eur dostane z\u00e1kazn\u00edk, ak tu zmen\u00ed 4 200 \u010desk\u00fdch kor\u00fan?<\/p>\n<p>Ak zmen\u00e1rnik vyk\u00fapi od z\u00e1kazn\u00edka 1 000 libier a potom ich v\u0161etky pred\u00e1, jeho celkov\u00fd zisk je 2 200 \u010desk\u00fdch kor\u00fan. Keby namiesto toho zmen\u00e1rnik predal 1 000 libier a potom by v\u0161etky ut\u0155\u017een\u00e9 \u010desk\u00e9 koruny zmenil s in\u00fdm z\u00e1kazn\u00edkom za libry, zarobil by na tom 68,75 libier.<\/p>\n<p>2. Za ko\u013eko \u010desk\u00fdch kor\u00fan zmen\u00e1rnik nakupuje a za ko\u013eko pred\u00e1va 1 libru?<\/p>\n<div class=\"pri\">(Libor \u0160im\u016fnek)<\/div>\n<p><strong>Z9-I-5<\/strong><br \/> Betka si myslela prirodzen\u00e9 \u010d\u00edslo s navz\u00e1jom r\u00f4znymi ciframi a nap\u00edsala ho na tabu\u013eu. Pode\u0148 zap\u00edsala cifry p\u00f4vodn\u00e9ho \u010d\u00edsla odzadu a tak z\u00edskala nov\u00e9 \u010d\u00edslo. S\u010d\u00edtan\u00edm t\u00fdchto dvoch \u010d\u00edsel dostala \u010d\u00edslo, ktor\u00e9 malo rovnak\u00fd po\u010det cifier ako myslen\u00e9 \u010d\u00edslo a skladalo sa iba z cifier myslen\u00e9ho \u010d\u00edsla (av\u0161ak nemuselo obsahova\u0165 v\u0161etky jeho cifry). Erike sa Betkino \u010d\u00edslo zap\u00e1\u010dilo a chcela n\u00e1js\u0165 in\u00e9 \u010d\u00edslo s rovnak\u00fdmi vlastnos\u0165ami. Zistila, \u017ee neexistuje men\u0161ie tak\u00e9 \u010d\u00edslo ako Betkino a v\u00e4\u010d\u0161ie sa jej h\u013eada\u0165 nechcelo. Ur\u010dte, ak\u00e9 \u010d\u00edslo si myslela Betka a ak\u00e9 \u010d\u00edslo by mohla n\u00e1js\u0165 Erika, keby mala viac trpezlivosti.<\/p>\n<div class=\"pri\">(Katar\u00edna Jasen\u010d\u00e1kov\u00e1)<\/div>\n<p><strong>Z9-I-6<\/strong><br \/>Na stran\u00e1ch <em>AB<\/em> a <em>AC<\/em> trojuholn\u00edka <em>ABC<\/em> le\u017eia postupne body <em>E<\/em> a <em>F<\/em>, na \u00fase\u010dke <em>EF<\/em> le\u017e\u00ed bod <em>D<\/em>. Priamky <em>EF<\/em> a <em>BC<\/em> s\u00fa rovnobe\u017en\u00e9 a s\u00fa\u010dasne plat\u00ed<\/p>\n<p style=\"text-align: center;\">|FD| : |DE| = |AE| : |EB| = 2 : 1.<\/p>\n<p>Trojuholn\u00edk <em>ABC<\/em> m\u00e1 obsah 27 hekt\u00e1rov a \u00fase\u010dkami <em>EF<\/em>, <em>AD<\/em> a <em>DB<\/em> je rozdelen\u00fd na \u0161tyri \u010dasti. Ur\u010dte obsahy t\u00fdchto \u0161tyroch \u010dast\u00ed.<\/p>\n<div class=\"pri\">(Vojt\u011bch \u017d\u00e1dn\u00edk)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z92\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=1738\" target=\"_blank\"><input name=\"butt52\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-II-1<\/strong><br \/>V Zverimexe vypred\u00e1vali rybi\u010dky z jedn\u00e9ho akv\u00e1ria. Ondrej chcel polovicu v\u0161etk\u00fdch rybi\u010diek, ale aby nemuseli \u017eiadnu rybi\u010dku deli\u0165, dostal o polovicu rybi\u010dky viac, ako po\u017eadoval. Matej si \u017eelal polovicu zvy\u0161n\u00fdch rybi\u010diek, ale rovnako ako Ondrej dostal o polovicu rybi\u010dky viac, ako po\u017eadoval. Nakoniec Petr\u00edk chcel polovicu zvy\u0161n\u00fdch rybi\u010diek, ale tie\u017e dostal o polovicu rybi\u010dky viac, ako po\u017eadoval. Potom bolo akv\u00e1rium bez rybi\u010diek. Ko\u013eko rybi\u010diek bolo p\u00f4vodne v akv\u00e1riu a ko\u013eko ich dostal Ondrej, ko\u013eko Matej a ko\u013eko Petr\u00edk?<\/p>\n<div class=\"pri\">(Marta Volfov\u00e1)<\/div>\n<p><strong>Z9-II-2<\/strong><br \/>Zuzka vp\u00edsala do deviatich pol\u00ed\u010dok na nasleduj\u00facom obr\u00e1zku cel\u00e9 \u010d\u00edsla od 1 do 9, ka\u017ed\u00e9 pr\u00e1ve raz. Pomer s\u00fa\u010dtov \u010d\u00edsel nap\u00edsan\u00fdch v kruhoch, trojuholn\u00edkoch a \u0161es\u0165uholn\u00edkoch bol 2 : 3 : 6. Zistite, ak\u00e9 \u010d\u00edslo mohlo by\u0165 nap\u00edsan\u00e9 v hornom trojuholn\u00edku; ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<p class=\"pc\"><img decoding=\"async\" src=\"65\/65z9b1.png\" alt=\"\" border=\"0\" \/><\/p>\n<div class=\"pri\">(Erika Novotn\u00e1)<\/div>\n<p><strong>Z9-II-3<\/strong><br \/>Dan\u00e9 s\u00fa kru\u017enice k<sub>1<\/sub>, k<sub>2<\/sub>, k<sub>3<\/sub> a k<sub>4<\/sub> so stredmi postupne S<sub>1<\/sub>, S<sub>2<\/sub>, S<sub>3<\/sub> a S<sub>4<\/sub>. Kru\u017enice k<sub>1<\/sub> a k<sub>3<\/sub> sa zvonka dot\u00fdkaj\u00fa v\u0161etk\u00fdch ostatn\u00fdch kru\u017en\u00edc, polomer kru\u017enice k<sub>1<\/sub> je 5 cm, vzdialenos\u0165 stredov S<sub>2<\/sub> a S<sub>4<\/sub> je 24 cm a \u0161tvoruholn\u00edk S<sub>1<\/sub>S<sub>2<\/sub>S<sub>3<\/sub>S<sub>4<\/sub> je koso\u0161tvorec. Ur\u010dte polomery kru\u017en\u00edc k<sub>2<\/sub>, k<sub>3<\/sub> a k<sub>4<\/sub>. Pozn\u00e1mka: obr\u00e1zok je len ilustra\u010dn\u00fd.<\/p>\n<p class=\"pc\"><img decoding=\"async\" src=\"65\/65z9b2.png\" alt=\"\" border=\"0\" \/><\/p>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z9-II-4<\/strong><br \/>Pred ka\u017ed\u00e9 z \u010d\u00edsel v nasleduj\u00facich dvoch zoznamoch dopl\u0148te bu\u010f znamienko plus, alebo m\u00ednus tak, aby hodnota takto zap\u00edsan\u00fdch v\u00fdrazov bola rovn\u00e1 nule:<\/p>\n<p>a)\u00a0\u00a0\u00a01\u00a0\u00a02\u00a0\u00a03\u00a0\u00a04\u00a0\u00a05\u00a0\u00a06\u00a0\u00a07\u00a0\u00a08\u00a0\u00a09\u00a0\u00a010,<br \/> b)\u00a0\u00a0\u00a01\u00a0\u00a02\u00a0\u00a03\u00a0\u00a04\u00a0\u00a05\u00a0\u00a06\u00a0\u00a07\u00a0\u00a08\u00a0\u00a09\u00a0\u00a010\u00a0\u00a011.<br \/> Pri oboch \u00faloh\u00e1ch uve\u010fte aspo\u0148 jedno rie\u0161enie alebo zd\u00f4vodnite, \u017ee \u00faloha rie\u0161enie nem\u00e1.<\/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 align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z93\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=1829\" target=\"_blank\"><input name=\"butt52\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-III-1<\/strong><br \/>Obd\u013a\u017enik m\u00e1 d\u013a\u017eky str\u00e1n v pomere 2 : 5. Ke\u010f pred\u013a\u017eime v\u0161etky jeho strany o 9\u00a0cm, dostaneme obd\u013a\u017enik, ktor\u00e9ho d\u013a\u017eky str\u00e1n s\u00fa v pomere 3 : 7. V akom pomere bud\u00fa d\u013a\u017eky str\u00e1n obd\u013a\u017enika, ktor\u00fd vznikne pred\u013a\u017een\u00edm v\u0161etk\u00fdch str\u00e1n o \u010fal\u0161\u00edch 9\u00a0cm?<\/p>\n<div class=\"pri\">(Michaela Petrov\u00e1)<\/div>\n<p><strong>Z9-III-2<\/strong><br \/>Traja kamar\u00e1ti si mysleli tri navz\u00e1jom r\u00f4zne nenulov\u00e9 cifry, z ktor\u00fdch jedna bola 3. Z t\u00fdchto cifier vytvorili v\u0161etk\u00fdch \u0161es\u0165 mo\u017en\u00fdch trojcifern\u00fdch \u010d\u00edsel, ktor\u00e9 potom rozdelili do troch dvoj\u00edc. Rozdielom prvej dvojice \u010d\u00edsel bolo jednocifern\u00e9 \u010d\u00edslo, rozdielom druhej dvojice \u010d\u00edsel bolo dvojcifern\u00e9 \u010d\u00edslo a rozdielom tretej dvojice \u010d\u00edsel bolo trojcifern\u00e9 \u010d\u00edslo delite\u013en\u00e9 piatimi. Zistite, ak\u00e9 tri cifry si mohli kamar\u00e1ti myslie\u0165. Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Erika Novotn\u00e1)<\/div>\n<p><strong>Z9-III-3<\/strong><br \/>Na papieri bolo nap\u00edsan\u00fdch nieko\u013eko bezprostredne po sebe id\u00facich kladn\u00fdch n\u00e1sobkov ur\u010dit\u00e9ho prirodzen\u00e9ho \u010d\u00edsla v\u00e4\u010d\u0161ieho ako jedna. Rado uk\u00e1zal na jedno z nap\u00edsan\u00fdch \u010d\u00edsel: ke\u010f ho vyn\u00e1sobil \u010d\u00edslom, ktor\u00e9 s n\u00edm susedilo na\u013eavo, dostal s\u00fa\u010din o 216 men\u0161\u00ed, ako ke\u010f ho vyn\u00e1sobil \u010d\u00edslom, ktor\u00e9 s n\u00edm susedilo napravo. Na ktor\u00e9 \u010d\u00edslo mohol Rado uk\u00e1za\u0165? N\u00e1jdite v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Libor \u0160im\u016fnek)<\/div>\n<p><strong>Z9-III-4<\/strong><br \/>Eva vp\u00edsala do dan\u00e9ho trojuholn\u00edka kru\u017enicu. Potom dokreslila tri \u00fase\u010dky, ktor\u00e9 sa dot\u00fdkali vp\u00edsanej kru\u017enice a v p\u00f4vodnom trojuholn\u00edku vytv\u00e1rali tri men\u0161ie trojuholn\u00edky, pozri obr\u00e1zok. Obvody t\u00fdchto troch trojuholn\u00edkov boli 12cm, 14cm a 16cm. Ur\u010dte obvod p\u00f4vodn\u00e9ho trojuholn\u00edka.<\/p>\n<p class=\"pc\"><img decoding=\"async\" src=\"65\/65z9c1.png\" alt=\"\" border=\"0\" \/><\/p>\n<div class=\"pri\">(Erika Novotn\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><center><script type=\"text\/javascript\" async=\"true\" charset=\"utf-8\" src=\"http:\/\/sk.search.etargetnet.com\/generic\/advert.php?g=ref:48528,area:600x70,tabl:4,divid:,design_name:green,border_color:ffffff,border_style:none,background_opacity:100,background_color:ffffff,title_color:426000,text_color:000000,url_color:426000,h_title_color:426000,h_text_color:000000,h_url_color:426000,freespace:0,logo_type:3,logo:1,title_underline:0,url_underline:0,h_title_underline:1,h_url_underline:1,nourl:,fsi:11,font:verdana,require_login_pass:,login_pass:,show_in_fb:,show_in_fb_text:,allow_youtube_ads:1\" ><\/script><\/center><\/p>\n<p><script type=\"text\/javascript\"><!--\r\ngoogle_ad_client = \"pub-0508472359151885\";\r\n\/* 468x60, bola vytvoren\u00e1 12.11.2010 *\/\r\ngoogle_ad_slot = \"3127737040\";\r\ngoogle_ad_width = 468;\r\ngoogle_ad_height = 60;\r\n\/\/-->\r\n<\/script>\r\n<script type=\"text\/javascript\"\r\nsrc=\"http:\/\/pagead2.googlesyndication.com\/pagead\/show_ads.js\">\r\n<\/script><\/p>\n<!--CusAds0-->\n<div style=\"font-size: 0px; height: 0px; line-height: 0px; margin: 0; padding: 0; clear: both;\"><\/div>","protected":false},"excerpt":{"rendered":"<p>65. ro\u010dn\u00edk matematickej olympi\u00e1dy 2015-2016 S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9 aktualizovan\u00e9 15.8.2016 17:00<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[22],"tags":[],"class_list":["post-1089","post","type-post","status-publish","format-standard","hentry","category-65-rocnik"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.6 - 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