{"id":1158,"date":"2016-09-20T18:35:22","date_gmt":"2016-09-20T18:35:22","guid":{"rendered":"http:\/\/matematika.besaba.com\/?p=1158"},"modified":"2023-11-02T20:23:39","modified_gmt":"2023-11-02T20:23:39","slug":"sutazne-ulohy-kategorii-z5-z9-5","status":"publish","type":"post","link":"https:\/\/matematika.besaba.com\/?p=1158","title":{"rendered":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9"},"content":{"rendered":"<p class=\"hh2\"><a name=\"top\"><\/a>66. ro\u010dn\u00edk matematickej olympi\u00e1dy 2016-2017<br \/> S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9<\/p>\n<p><span class=\"povinne\">aktualizovan\u00e9 4.7.2017 14:45<\/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=\"\/66\/66_z59.pdf\" target=\"_blank\" rel=\"noopener\"><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=2045\" target=\"_blank\" rel=\"noopener\"><input name=\"butt51z\" type=\"button\" value=\"Zadanie\" \/><\/a> <a href=\"http:\/\/skmo.sk\/dokument.php?id=2204\" target=\"_blank\" rel=\"noopener\"><input name=\"butt51r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z5-I-1<\/strong><br \/>Zvonkohra na n\u00e1dvor\u00ed hr\u00e1 o ka\u017edej celej hodine kr\u00e1tku skladbu, a to po\u010d\u00ednaj\u00fac 8. a kon\u010diac 22. hodinou. Skladieb je celkom osemn\u00e1s\u0165, o celej hodine sa hr\u00e1 v\u017edy iba jedna a po odohran\u00ed v\u0161etk\u00fdch osemn\u00e1stich sa za\u010d\u00edna v rovnakom porad\u00ed znova. O\u013ega a \u013dubo\u0161 boli na n\u00e1dvor\u00ed v pondelok o 15. hodine. Ten ist\u00fd t\u00fd\u017ede\u0148 si pri\u0161li zvonkohru vypo\u010du\u0165 e\u0161te raz na poludnie, na ich sklamanie v\u0161ak hrala t\u00e1 ist\u00e1 mel\u00f3dia, ktor\u00fa po\u010duli v pondelok. Ktor\u00fd de\u0148 bola O\u013ega s \u013dubo\u0161om na n\u00e1dvor\u00ed druh\u00fd raz?<\/p>\n<div class=\"pri\">(Libor \u0160im\u016fnek)<\/div>\n<p><strong>Z5-I-2<\/strong><br \/>V ka\u017edom z rohov\u00fdch pol\u00ed\u010dok vonkaj\u0161ieho \u0161tvorca m\u00e1 by\u0165 nap\u00edsan\u00e9 jedno z \u010d\u00edsel 2, 4, 6 a 8, pri\u010dom v r\u00f4znych pol\u00ed\u010dkach maj\u00fa by\u0165 r\u00f4zne \u010d\u00edsla. V \u0161tyroch pol\u00ed\u010dkach vn\u00fatorn\u00e9ho \u0161tvorca maj\u00fa by\u0165 s\u00fa\u010diny \u010d\u00edsel zo susediacich pol\u00ed\u010dok vonkaj\u0161ieho \u0161tvorca. V kruhu m\u00e1 by\u0165 s\u00fa\u010det \u010d\u00edsel zo susediacich pol\u00ed\u010dok vn\u00fatorn\u00e9ho \u0161tvorca. Ktor\u00e9 \u010d\u00edsla m\u00f4\u017eu by\u0165 nap\u00edsan\u00e9 v kruhu? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"66\/66z5a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Monika Dillingerov\u00e1)<\/div>\n<p><strong>Z5-I-3<\/strong><br \/>Na obr\u00e1zku je \u0161tvorcov\u00e1 dla\u017edica so stranou d\u013a\u017eky 10 dm, ktor\u00e1 je zlo\u017een\u00e1 zo \u0161tyroch zhodn\u00fdch obd\u013a\u017enikov a mal\u00e9ho \u0161tvorca. Obvod mal\u00e9ho \u0161tvorca je p\u00e4\u0165kr\u00e1t men\u0161\u00ed ako obvod celej dla\u017edice. Ur\u010dte rozmery obd\u013a\u017enikov.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"66\/66z5a2.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Karel Pazourek)<\/div>\n<p><strong>Z5-I-4<\/strong><br \/>Predava\u010d viano\u010dn\u00fdch strom\u010dekov pred\u00e1val smrie\u010dky po 22\u20ac, borovi\u010dky po 25\u20ac a jedli\u010dky po 33\u20ac. R\u00e1no mal rovnak\u00fd po\u010det smrie\u010dkov, jedli\u010diek a borov\u00edc. Ve\u010der mal v\u0161etky strom\u010deky predan\u00e9 a celkom za ne ut\u0155\u017eil 3 600\u20ac. Ko\u013eko strom\u010dekov v ten de\u0148 predava\u010d predal?<\/p>\n<div class=\"pri\">(Marie Krej\u010dov\u00e1)<\/div>\n<p><strong>Z5-I-5<\/strong><br \/>Nap\u00ed\u0161te namiesto hviezdi\u010diek cifry tak, aby s\u00fa\u010det doplnen\u00fdch cifer bol nep\u00e1rny a aby platila uveden\u00e1 rovnos\u0165:<\/p>\n<div class=\"r\">42 . *8 = 2***<\/div>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z5-I-6<\/strong><br \/>Jarka zostrojila dva zhodn\u00e9 rovnostrann\u00e9 trojuholn\u00edky ako na obr\u00e1zku. \u010ealej chce zostroji\u0165 v\u0161etky kru\u017enice, ktor\u00e9 bud\u00fa ma\u0165 stred v niektorom z vrcholov a bud\u00fa prech\u00e1dza\u0165 niektor\u00fdm in\u00fdm vrcholom niektor\u00e9ho z trojuholn\u00edkov. Zostrojte a spo\u010d\u00edtajte v\u0161etky kru\u017enice vyhovuj\u00face Jarkin\u00fdm po\u017eiadavk\u00e1m.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"66\/66z5a3.png\" alt=\"\" border=\"0\" \/><\/div>\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=\"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=2205\" target=\"_blank\" rel=\"noopener\"><input name=\"butt52z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2206\" target=\"_blank\" rel=\"noopener\"><input name=\"butt52r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z5-II-1<\/strong><br \/>Otec hral so str\u00fdkom \u0161achy. Za vyhran\u00fa partiu dostal v\u00ed\u0165az od s\u00fapera 8 eur, za rem\u00edzu nikto ni\u010d. Str\u00fdko vyhral \u0161tyrikr\u00e1t, rem\u00edz bolo p\u00e4\u0165 a otec nakoniec z\u00edskal 24 eur. Ko\u013eko parti\u00ed otec so str\u00fdkom zohrali?<\/p>\n<div class=\"pri\">(Marta Volfov\u00e1)<\/div>\n<p><strong>Z5-II-2<\/strong><br \/>Veveri\u010dka Hryzka ujedala orie\u0161ky zo svojich z\u00e1sob nasleduj\u00facim sp\u00f4sobom:<\/p>\n<ul>\n<li>v di\u00e9tny de\u0148 zjedla jeden orie\u0161ok,<\/li>\n<li>v norm\u00e1lny de\u0148 zjedla o dva orie\u0161ky viac ako v di\u00e9tny de\u0148.<\/li>\n<\/ul>\n<p>Ist\u00fdch 19 po sebe id\u00facich dn\u00ed sa striedali dni di\u00e9tne s d\u0148ami norm\u00e1lnymi. Zistite, ko\u013eko najviac a ko\u013eko najmenej orie\u0161kov mohla Hryzka po\u010das t\u00fdchto 19 dn\u00ed zjes\u0165.\u00a0<\/p>\n<div class=\"pri\">(Erika Novotn\u00e1)<\/div>\n<p><strong>Z5-II-3<\/strong><br \/>Ema chce zostroji\u0165 trojuholn\u00edk ABC so stranami |<em>AB<\/em>| = 3 cm a |<em>BC<\/em>| = 4 cm. \u010ealej chce zostroji\u0165 v\u0161etky kru\u017enice, z ktor\u00fdch ka\u017ed\u00e1 bude ma\u0165 stred v niektorom z vrcholov trojuholn\u00edka a bude prech\u00e1dza\u0165 niektor\u00fdm jeho in\u00fdm vrcholom. Ak\u00e1 dlh\u00e1 mus\u00ed by\u0165 strana <em>AC<\/em>, aby tak\u00fdch kru\u017en\u00edc bolo pr\u00e1ve p\u00e4\u0165?<\/p>\n<div class=\"pri\">(Veronika Huc\u00edkov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z61\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><center><script type=\"text\/javascript\" async=\"true\" charset=\"utf-8\" src=\"http:\/\/sk.search.etargetnet.com\/generic\/advert.php?g=ref:48528,area:600x70,tabl:4,divid:,design_name:green,border_color:ffffff,border_style:none,background_opacity:100,background_color:ffffff,title_color:426000,text_color:000000,url_color:426000,h_title_color:426000,h_text_color:000000,h_url_color:426000,freespace:0,logo_type:3,logo:1,title_underline:0,url_underline:0,h_title_underline:1,h_url_underline:1,nourl:,fsi:11,font:verdana,require_login_pass:,login_pass:,show_in_fb:,show_in_fb_text:,allow_youtube_ads:1\" ><\/script><\/center><\/p>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=2044\" target=\"_blank\" rel=\"noopener\"><input name=\"butt61z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2317\" target=\"_blank\" rel=\"noopener\"><input name=\"butt61r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z6-I-1<\/strong><br \/>Jana a D\u00e1vid tr\u00e9nuj\u00fa s\u010d\u00edtanie desatinn\u00fdch \u010d\u00edsel tak, \u017ee ka\u017ed\u00fd z nich nap\u00ed\u0161e jedno \u010d\u00edslo, a tieto dve \u010d\u00edsla potom s\u010d\u00edtaj\u00fa. Posledn\u00fd pr\u00edklad im vy\u0161iel 11,11. D\u00e1vidovo \u010d\u00edslo malo pred desatinnou \u010diarkou rovnak\u00fd po\u010det cifier ako za \u0148ou, Janino \u010d\u00edslo tie\u017e. D\u00e1vidovo \u010d\u00edslo bolo zap\u00edsan\u00e9 navz\u00e1jom r\u00f4znymi ciframi, Janino \u010d\u00edslo malo pr\u00e1ve dve cifry rovnak\u00e9. Ur\u010dte najv\u00e4\u010d\u0161ie mo\u017en\u00e9 \u010d\u00edslo, ktor\u00e9 mohol nap\u00edsa\u0165 D\u00e1vid.<\/p>\n<div class=\"pri\">(Michaela Petrov\u00e1)<\/div>\n<p><strong>Z6-I-2<\/strong><br \/>P\u00e1n Kockor\u00e1d vlastnil z\u00e1hradu obd\u013a\u017enikov\u00e9ho tvaru, na ktorej postupne dl\u00e1\u017edil chodn\u00edky z jednej strany na druh\u00fa. Chodn\u00edky boli rovnako \u0161irok\u00e9, kri\u017eovali sa na dvoch miestach a u\u017e vydl\u00e1\u017eden\u00e1 plocha sa pri \u010fal\u0161om dl\u00e1\u017eden\u00ed preskakovala. Ke\u010f p\u00e1n Kockor\u00e1d vydl\u00e1\u017edil chodn\u00edk rovnobe\u017en\u00fd s dlh\u0161ou stranou, spotreboval 228 m<sup>2<\/sup> dla\u017eby. Potom vydl\u00e1\u017edil chodn\u00edk rovnobe\u017en\u00fd s krat\u0161ou stranou a spotreboval 117 m<sup>2<\/sup> dla\u017eby. Nakoniec vydl\u00e1\u017edil e\u0161te jeden chodn\u00edk rovnobe\u017en\u00fd s prv\u00fdm chodn\u00edkom, tentoraz spotreboval len 219 m<sup>2<\/sup> dla\u017eby. Ur\u010dte rozmery Kockor\u00e1dovej z\u00e1hrady.<\/p>\n<div class=\"pri\">(Michaela Petrov\u00e1)<\/div>\n<p><strong>Z6-I-3<\/strong><br \/>Mnohono\u017eka Mirka pozost\u00e1va z hlavy a nieko\u013ek\u00fdch \u010dl\u00e1nkov, na ka\u017edom \u010dl\u00e1nku m\u00e1 jeden p\u00e1r n\u00f4h. Ke\u010f sa ochladilo, rozhodla sa, \u017ee sa oble\u010die. Preto si na tre\u0165om \u010dl\u00e1nku od konca a potom na ka\u017edom \u010fal\u0161om tre\u0165om \u010dl\u00e1nku obliekla pono\u017eku na \u013eav\u00fa n\u00f4\u017eku. Podobne si na piatom \u010dl\u00e1nku od konca a potom na ka\u017edom \u010fal\u0161om piatom \u010dl\u00e1nku obliekla pono\u017eku na prav\u00fa n\u00f4\u017eku. Napokon zistila, \u017ee na 14 \u010dl\u00e1nkoch jej zostali obe nohy bos\u00e9. Zistite, ko\u013eko celkom n\u00f4h mohla ma\u0165 mnohono\u017eka Mirka; ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Erika Novotn\u00e1)<\/div>\n<p><strong>Z6-I-4<\/strong><br \/>\u0160tyri rodiny boli na spolo\u010dnom v\u00fdlete. V prvej rodine boli traja s\u00farodenci, a to Alica, Betka a Cyril. V druhej rodine boli \u0161tyria s\u00farodenci, a to D\u00e1vid, Erika, Filipa Gabika. V tretej rodine boli dvaja s\u00farodenci, a to Hugo a Iveta. Vo \u0161tvrtej rodine boli traja s\u00farodenci, a to J\u00e1n, Karol a Luk\u00e1\u0161. Cestou sa deti rozdelili na skupiny tak, \u017ee v ka\u017edej skupine boli v\u0161etky deti s rovnak\u00fdm po\u010dtom bratov a nikto in\u00fd. Ako sa mohli deti rozdeli\u0165? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Veronika Huc\u00edkov\u00e1)<\/div>\n<p><strong>Z6-I-5<\/strong><br \/>Juro si nakreslil \u0161tvorcov\u00fa sie\u0165 s 25 \u0161tvor\u010dekmi ako na obr\u00e1zku. Potom chcel ka\u017ed\u00fd \u0161tvor\u010dek vyfarbi\u0165 tak, aby rovnako vyfarben\u00e9 \u0161tvor\u010deky nemali spolo\u010dn\u00fd \u017eiadny vrchol. Ko\u013eko najmenej farieb musel Juro pou\u017ei\u0165?<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"66\/66z6a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Monika Dillingerov\u00e1)<\/div>\n<p><strong>Z6-I-6<\/strong><br \/>Do pr\u00e1zdnych pol\u00ed\u010dok v nasleduj\u00facom obr\u00e1zku dopl\u0148te cel\u00e9 \u010d\u00edsla v\u00e4\u010d\u0161ie ako 1 tak, aby v ka\u017edom tmav\u0161om pol\u00ed\u010dku bol s\u00fa\u010din \u010d\u00edsel zo susedn\u00fdch svetlej\u0161\u00edch pol\u00ed\u010dok.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"66\/66z6a2.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(T. Sal\u010d\u00e1k)<\/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=2320\" target=\"_blank\" rel=\"noopener\"><input name=\"butt62z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2323\" target=\"_blank\" rel=\"noopener\"><input name=\"butt62r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z6-II-1<\/strong><br \/> Na l\u00fake sa zi\u0161li dve rodiny s celkom \u0161iestimi de\u0165mi, ktor\u00e9 boli star\u00e9 2, 3, 4, 5, 6 a 8 rokov. S\u00fa\u010det vekov det\u00ed z jednej rodiny bol rovnak\u00fd ako s\u00fa\u010det vekov det\u00ed z druhej rodiny. Ak\u00e9 star\u00e9 mohli by\u0165 deti z ka\u017edej rodiny? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Erika Novotn\u00e1)<\/div>\n<p><strong>Z6-II-2<\/strong><br \/> Pat a Mat kopali stud\u0148u. Prv\u00fd de\u0148 vykopal Pat jamu hlbok\u00fa 40 cm. Druh\u00fd de\u0148 pokra\u010doval Mat a dokopal sa do trojn\u00e1sobnej h\u013abky. Tret\u00ed de\u0148 vykopal Pat to\u013eko, ko\u013eko predch\u00e1dzaj\u00faci de\u0148 vykopal Mat, a narazil na vodu. V tom okamihu bol povrch zeme 50 cm nad vrchom jeho hlavy. Ur\u010dte, ko\u013eko cm meral Pat.<\/p>\n<div class=\"pri\">(Monika Dillingerov\u00e1)<\/div>\n<p><strong>Z6-II-3<\/strong><br \/> Traja z\u00e1hradk\u00e1ri si pozvali or\u00e1\u010da, aby im zoral ich pozemky. Or\u00e1\u010d \u00fa\u010dtoval za ka\u017ed\u00fd zoran\u00fd meter \u0161tvorcov\u00fd v\u0161etk\u00fdm rovnak\u00fa cenu. Dokopy z\u00e1hradk\u00e1ri zaplatili 57 \u20ac. Dva z pozemkov boli \u0161tvorcov\u00e9, jeden bol obd\u013a\u017enikov\u00fd. Plo\u0161n\u00fd obsah obd\u013a\u017enikov\u00e9ho pozemku bol 6 \u00e1rov a jedna jeho strana merala 20 m. D\u013a\u017eka strany jedn\u00e9ho \u0161tvorcov\u00e9ho pozemku bola rovnak\u00e1 ako d\u013a\u017eka obd\u013a\u017enikov\u00e9ho pozemku, d\u013a\u017eka strany druh\u00e9ho \u0161tvorcov\u00e9ho pozemku bola rovnak\u00e1 ako \u0161\u00edrka obd\u013a\u017enikov\u00e9ho pozemku. Ko\u013eko zaplatil za zoranie ka\u017ed\u00fd zo z\u00e1hradk\u00e1rov?<\/p>\n<div class=\"pri\">(Marie Krej\u010dov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a 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=2043\" target=\"_blank\" rel=\"noopener\"><input name=\"butt71z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2318\" target=\"_blank\" rel=\"noopener\"><input name=\"butt71r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z7-I-1<\/strong><br \/>\u0160tvorec so stranou 4 cm je rozdelen\u00fd na \u0161tvor\u010deky so stranou 1 cm ako na obr\u00e1zku. Rozde\u013ete \u0161tvorec pozd\u013a\u017e vyzna\u010den\u00fdch \u010diar na dva \u00fatvary s obvodom 16 cm. N\u00e1jdite aspo\u0148 tri r\u00f4zne rie\u0161enia (tzn. tak\u00e9 tri rie\u0161enia, aby \u017eiadny \u00fatvar jedn\u00e9ho rie\u0161enia nebol zhodn\u00fd so \u017eiadnym \u00fatvarom in\u00e9ho rie\u0161enia).<\/p>\n<table class=\"ttt\">\n<tbody>\n<tr>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<\/tr>\n<tr>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<\/tr>\n<tr>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<\/tr>\n<tr>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"pri\">(Veronika Huc\u00edkov\u00e1)<\/div>\n<p><strong>Z7-I-2<\/strong><br \/>Na ly\u017eiarske s\u00fastredenie pri\u0161li 4 kamar\u00e1ti zo 4 svetov\u00fdch str\u00e1n a viedli nasleduj\u00faci rozhovor.<br \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Karol: &#8222;Nepri\u0161iel som zo severu ani z juhu.&#8220;<br \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Mojm\u00edr: &#8222;Zato ja som pri\u0161iel z juhu.&#8220;<br \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Jozef: &#8222;Pri\u0161iel som zo severu.&#8220;<br \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Zdeno: &#8222;Ja som z juhu nepri\u0161iel.&#8220;<br \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Vieme, \u017ee jedna v\u00fdpove\u010f nie je pravdiv\u00e1. Ur\u010dte, ktor\u00e1 to je. Kto teda pri\u0161iel zo severu a kto z juhu?<\/p>\n<div class=\"pri\">(Marta Volfov\u00e1)<\/div>\n<p><strong>Z7-I-3<\/strong><br \/>Ani\u010dka m\u00e1 5\u20ac, Ane\u017eka m\u00e1 4,60\u20ac a za v\u0161etky peniaze chc\u00fa k\u00fapi\u0165 z\u00e1kusky na rodinn\u00fa oslavu. Rozhoduj\u00fa sa medzi torti\u010dkami a vetern\u00edkmi: vetern\u00edk je o 0,40\u20ac drah\u0161\u00ed ako torti\u010dka a torti\u010diek by sa dalo za v\u0161etky peniaze k\u00fapi\u0165 o tretinu viac ako vetern\u00edkov. Ko\u013eko stoj\u00ed ka\u017ed\u00fd zo z\u00e1kuskov?<\/p>\n<div class=\"pri\">(Marta Volfov\u00e1)<\/div>\n<p><strong>Z7-I-4<\/strong><br \/>Nap\u00ed\u0161te namiesto hviezdi\u010diek cifry tak, aby nasleduj\u00faci z\u00e1pis s\u00fa\u010dinu dvoch \u010d\u00edsel bol platn\u00fd:<\/p>\n<table class=\"tt\">\n<tbody>\n<tr>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>*<\/td>\n<td>*<\/td>\n<td>*<\/td>\n<\/tr>\n<tr class=\"pod\">\n<td>\u00a0<\/td>\n<td>.<\/td>\n<td>\u00a0<\/td>\n<td>*<\/td>\n<td>*<\/td>\n<td>*<\/td>\n<\/tr>\n<tr>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>*<\/td>\n<td>*<\/td>\n<td>*<\/td>\n<td>*<\/td>\n<\/tr>\n<tr>\n<td>\u00a0<\/td>\n<td>4<\/td>\n<td>9<\/td>\n<td>4<\/td>\n<td>9<\/td>\n<td>\u00a0<\/td>\n<\/tr>\n<tr class=\"pod\">\n<td>\u00a0<\/td>\n<td>*<\/td>\n<td>*<\/td>\n<td>*<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<\/tr>\n<tr>\n<td>*<\/td>\n<td>*<\/td>\n<td>*<\/td>\n<td>4<\/td>\n<td>*<\/td>\n<td>*<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z7-I-5<\/strong><br \/>Dan\u00fd je trojuholn\u00edk <em>ABC<\/em> so stranami |<em>AB<\/em>|= 3 cm, |<em>BC<\/em>|= 10 cm a s uhlom <em>ABC<\/em> o ve\u013ekosti 120\u00b0. Narysujte v\u0161etky body <em>X<\/em> tak\u00e9, aby platilo, \u017ee trojuholn\u00edk <em>BCX<\/em> je rovnoramenn\u00fd a s\u00fa\u010dasne trojuholn\u00edk <em>ABX<\/em> je rovnoramenn\u00fd so z\u00e1klad\u0148ou <em>AB<\/em>.<\/p>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z7-I-6<\/strong><br \/>Ur\u010dte, pre ko\u013eko prirodzen\u00fdch \u010d\u00edsel v\u00e4\u010d\u0161\u00edch ako 900 a men\u0161\u00edch ako 1 001 plat\u00ed, \u017ee cifern\u00fd s\u00fa\u010det cifern\u00e9ho s\u00fa\u010dtu ich cifern\u00e9ho s\u00fa\u010dtu je rovn\u00fd 1.<\/p>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"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=2321\" target=\"_blank\" rel=\"noopener\"><input name=\"butt72z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2324\" target=\"_blank\" rel=\"noopener\"><input name=\"butt72r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z7-II-1<\/strong><br \/> Marienka dostala od babi\u010dky z\u00e1zra\u010dn\u00fd me\u0161ec, ktor\u00fd v\u017edy cez noc zdvojn\u00e1soboval mno\u017estvo duk\u00e1tov, ktor\u00e9 obsahoval. V pondelok Marienka vlo\u017eila do pr\u00e1zdneho me\u0161ca nejak\u00e9 duk\u00e1ty. V utorok a v stredu si z me\u0161ca vybrala v\u017edy 40 duk\u00e1tov a ni\u010d do neho nevkladala. Vo \u0161tvrtok si op\u00e4\u0165 vybrala 40 duk\u00e1tov a me\u0161ec zostal pr\u00e1zdny. Ko\u013eko duk\u00e1tov vlo\u017eila Marienka v pondelok do me\u0161ca? Ko\u013eko duk\u00e1tov mala do pr\u00e1zdneho me\u0161ca vlo\u017ei\u0165, aby mohla opakovane ka\u017ed\u00fd de\u0148 vybera\u0165 40 duk\u00e1tov, nemusela ni\u010d vklada\u0165 a aby ka\u017ed\u00fd de\u0148 pred v\u00fdberom bol po\u010det duk\u00e1tov v me\u0161ci rovnak\u00fd?<\/p>\n<div class=\"pri\">(Marta Volfov\u00e1)<\/div>\n<p><strong>Z7-II-2<\/strong><br \/> Mrie\u017eka s deviatimi pol\u00ed\u010dkami ako na obr\u00e1zku je vyplnen\u00e1 deviatimi bezprostredne po sebe id\u00facimi prirodzen\u00fdmi \u010d\u00edslami. Tie s\u00fa zoraden\u00e9 pod\u013ea ve\u013ekosti z\u013eava doprava a zhora nadol (t. j. najmen\u0161ie \u010d\u00edslo je v\u013eavo hore, najv\u00e4\u010d\u0161ie je vpravo dole). Tu\u010dn\u00e1 lomen\u00e1 \u010diara rozde\u013euje mrie\u017eku na dve \u010dasti. S\u00fa\u010det \u010d\u00edsel v \u013eavej \u010dasti je o 100 men\u0161\u00ed ako s\u00fa\u010det \u010d\u00edsel v pravej. Ktor\u00e9 \u010d\u00edslo je v prostrednom pol\u00ed\u010dku?<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"66\/66z7b1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Libor \u0160im\u016fnek)<\/div>\n<p><strong>Z7-II-3<\/strong><br \/> Adam m\u00e1 dva kv\u00e1dre s objemami 12 cm<sup>3<\/sup> a 30 cm<sup>3<\/sup>, pri\u010dom rozmery ka\u017ed\u00e9ho z nich s\u00fa v centimetroch vyjadren\u00e9 navz\u00e1jom r\u00f4znymi cel\u00fdmi \u010d\u00edslami. Adam zistil, \u017ee kv\u00e1dre sa daj\u00fa zlepi\u0165 k sebe tak, aby lepen\u00e9 steny spl\u00fdvali, a z\u00edska\u0165 tak nov\u00fd kv\u00e1der. Ak\u00e9 rozmery m\u00f4\u017ee ma\u0165 nov\u00fd kv\u00e1der? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"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=2042\" target=\"_blank\" rel=\"noopener\"><input name=\"butt81z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2319\" target=\"_blank\" rel=\"noopener\"><input name=\"butt81r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z8-I-1<\/strong><br \/>Tri kamar\u00e1tky veveri\u010dky spolu vyrazili na zber lieskov\u00fdch orie\u0161kov. Ry\u0161avka ich na\u0161la dvakr\u00e1t viac ako Pizizubka a U\u0161ka dokonca trikr\u00e1t viac ako Pizizubka. Cestou domov sa zhov\u00e1rali a pritom l\u00faskali a jedli svoje orie\u0161ky. Pizizubka zjedla polovicu v\u0161etk\u00fdch orie\u0161kov, ktor\u00e9 nazbierala, Ry\u0161avka tretinu v\u0161etk\u00fdch svojich orie\u0161kov a U\u0161ka \u0161tvrtinu t\u00fdch svojich. Doma veveri\u010dky zistili, \u017ee im dokopy zv\u00fd\u0161ilo 196 orie\u0161kov. Ko\u013eko orie\u0161kov na\u0161la ka\u017ed\u00e1 z veveri\u010diek?<\/p>\n<div class=\"pri\">(Michaela Petrov\u00e1)<\/div>\n<p><strong>Z8-I-2<\/strong><br \/>Na ka\u017edej stene pravideln\u00e9ho osemstena je nap\u00edsan\u00e9 jedno z \u010d\u00edsel 1, 2, 3, 4, 5, 6, 7 a 8, pri\u010dom na r\u00f4znych sten\u00e1ch s\u00fa r\u00f4zne \u010d\u00edsla. Pre ka\u017ed\u00fa stenu Jaro ur\u010dil s\u00fa\u010det \u010d\u00edsla na nej nap\u00edsan\u00e9ho s \u010d\u00edslami troch susedn\u00fdch stien. Takto dostal osem s\u00fa\u010dtov, ktor\u00e9 tie\u017e s\u010d\u00edtal. Ak\u00e9 hodnoty m\u00f4\u017ee tento v\u00fdsledn\u00fd s\u00fa\u010det nadob\u00fada\u0165?<\/p>\n<div class=\"pri\">(Jaroslav Zhouf)<\/div>\n<p><strong>Z8-I-3<\/strong><br \/>Pri stre\u013ebe z luku sa okrem in\u00e9ho sleduje v\u00fdkonnos\u0165 strelca. T\u00e1 sa po\u010d\u00edta tak, \u017ee sa zo v\u0161etk\u00fdch pokusov odoberie jeden najlep\u0161\u00ed a jeden najhor\u0161\u00ed a z hodnotenia zvy\u0161n\u00fdch sa spo\u010d\u00edta aritmetick\u00fd priemer. Kamar\u00e1ti Peter, Juraj, Michal a Zdeno strie\u013eali po jednom \u0161\u00edpe v \u0161tyroch kol\u00e1ch. Ka\u017ed\u00e1 strela bola hodnoten\u00e1 cel\u00fdm \u010d\u00edslom od 0 do 10. V ka\u017edom kole bol s\u00fa\u010det hodnoten\u00ed v\u0161etk\u00fdch chlapcov 32 bodov, ale ani v jednom kole nemali \u017eiadni dvaja chlapci rovnak\u00e9 hodnotenie. V nasleduj\u00facej tabu\u013eke s\u00fa vyplnen\u00e9 iba niektor\u00e9 \u00fadaje z uveden\u00e9ho z\u00e1pasu, dopl\u0148te tie ch\u00fdbaj\u00face.<\/p>\n<table class=\"r\" border=\"2\" width=\"50%\" cellspacing=\"2\" cellpadding=\"2\" align=\"center\">\n<tbody>\n<tr>\n<td>\u00a0<\/td>\n<td>1. kolo<\/td>\n<td>2. kolo<\/td>\n<td>3. kolo<\/td>\n<td>4. kolo<\/td>\n<td>v\u00fdkonnos\u0165<\/td>\n<\/tr>\n<tr>\n<td>Peter<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>5<\/td>\n<td>10<\/td>\n<\/tr>\n<tr>\n<td>Juraj<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>9<\/td>\n<td>10<\/td>\n<td>7,5<\/td>\n<\/tr>\n<tr>\n<td>Michal<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>5<\/td>\n<td>\u00a0<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>Zdeno<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>8,5<\/td>\n<\/tr>\n<tr>\n<td>celkom<\/td>\n<td>32<\/td>\n<td>32<\/td>\n<td>32<\/td>\n<td>32<\/td>\n<td>&#8211;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"pri\">(Monika Dillingerov\u00e1)<\/div>\n<p><strong>Z8-I-4<\/strong><br \/>Lichobe\u017en\u00edk <em>ABCD<\/em> je \u00fase\u010dkou <em>CE<\/em> rozdelen\u00fd na trojuholn\u00edk a rovnobe\u017en\u00edk, pozri obr\u00e1zok. Bod <em>F<\/em> je stredom \u00fase\u010dky <em>CE<\/em>, priamka <em>DF<\/em> prech\u00e1dza stredom \u00fase\u010dky <em>BE<\/em> a obsah trojuholn\u00edka <em>CDE<\/em> je 3 cm<sup>2<\/sup>. Ur\u010dte obsah lichobe\u017en\u00edka <em>ABCD<\/em>.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"66\/66z8a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z8-I-5<\/strong><br \/>Mami\u010dka doniesla 10 z\u00e1kuskov troch druhov: kokosiek bolo menej ako laskoniek a najviac bolo karamelov\u00fdch kociek. Jozef si vybral dva z\u00e1kusky r\u00f4znych druhov, Jakub urobil to ist\u00e9 a na J\u00e1na zv\u00fd\u0161ili iba z\u00e1kusky rovnak\u00e9ho druhu. Ko\u013eko kokosiek, laskoniek a karamelov\u00fdch kociek mami\u010dka doniesla?<\/p>\n<div class=\"pri\">(Veronika Huc\u00edkov\u00e1)<\/div>\n<p><strong>Z8-I-6<\/strong><br \/>Ka\u017ed\u00e1 tehli\u010dka nasleduj\u00facej pyram\u00eddy obsahuje jedno \u010d\u00edslo. Kedyko\u013evek to je mo\u017en\u00e9, je \u010d\u00edslo v ka\u017edej tehli\u010dke najmen\u0161\u00edm spolo\u010dn\u00fdm n\u00e1sobkom \u010d\u00edsel z dvoch tehli\u010diek le\u017eiacich priamo na nej. Ktor\u00e9 \u010d\u00edslo m\u00f4\u017ee by\u0165 v najspodnej\u0161ej tehli\u010dke? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"66\/66z8a2.png\" alt=\"\" border=\"0\" \/><\/div>\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<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=2322\" target=\"_blank\" rel=\"noopener\"><input name=\"butt82z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2325\" target=\"_blank\" rel=\"noopener\"><input name=\"butt82r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z8-II-1<\/strong><br \/>Monika prem\u00fd\u0161\u013ea o \u0161tvorcifernom \u010d\u00edsle, ktor\u00e9 m\u00e1 nasleduj\u00face vlastnosti:<\/p>\n<ul>\n<li>s\u00fa\u010din dvoch krajn\u00fdch cifier je 40,<\/li>\n<li>s\u00fa\u010din dvoch vn\u00fatorn\u00fdch cifier je 18,<\/li>\n<li>rozdiel dvoch krajn\u00fdch cifier je rovnak\u00fd ako rozdiel dvoch vn\u00fatorn\u00fdch cifier,<\/li>\n<li>rozdiel myslen\u00e9ho \u010d\u00edsla a opa\u010dne nap\u00edsan\u00e9ho \u010d\u00edsla (t. j. \u010d\u00edsla nap\u00edsan\u00e9ho rovnak\u00fdmi ciframi, ale v opa\u010dnom porad\u00ed) je najv\u00e4\u010d\u0161\u00ed mo\u017en\u00fd.<\/li>\n<\/ul>\n<p>Ur\u010dte Monikino myslen\u00e9 \u010d\u00edslo.<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z8-II-2<\/strong><br \/>Mat kopal jamu. Pat sa ho sp\u00fdtal, ak\u00e1 bude jama hlbok\u00e1. Mat odpovedal h\u00e1dankou:<\/p>\n<p style=\"margin-left:3em; margin-right:3em;\">&#8222;Meriam 90 cm a pr\u00e1ve m\u00e1m vykopan\u00fa polovicu jamy. Ke\u010f vykopem jamu cel\u00fa, bude vrch mojej hlavy pod povrchom zeme tak hlboko, ako je teraz nad povrchom zeme.&#8220;<\/p>\n<p>Ak\u00fa hlbok\u00fa jamu Mat kopal?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z8-II-3<\/strong><br \/>Pravouhl\u00fd trojuholn\u00edk <em>ABC<\/em> m\u00e1 odvesny |<em>AB<\/em>|= 5 cm a |<em>BC<\/em>|= 12 cm. Vrcholy trojuholn\u00edka <em>ABC<\/em> s\u00fa zobrazen\u00e9 tak, \u017ee<\/p>\n<ul>\n<li>bod <em>A<\/em> sa zobrazuje do bodu <em>A&#8216;<\/em> s\u00famerne pod\u013ea osi uhla <em>ABC<\/em>,<\/li>\n<li>bod <em>B<\/em> sa zobrazuje do bodu <em>B&#8216;<\/em> s\u00famerne pod\u013ea stredu \u00fase\u010dky <em>BC<\/em>,<\/li>\n<li>bod <em>C<\/em> sa zobrazuje do bodu <em>C&#8216;<\/em> s\u00famerne pod\u013ea osi \u00fase\u010dky <em>AB<\/em>.<\/li>\n<\/ul>\n<p>Ur\u010dte obsah trojuholn\u00edka <em>A&#8217;B&#8217;C<\/em>.\u00a0<\/p>\n<div class=\"pri\">(Monika Dillingerov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"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=2041\" target=\"_blank\" rel=\"noopener\"><input name=\"butt91z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2203\" target=\"_blank\" rel=\"noopener\"><input name=\"butt91r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-I-1<\/strong><br \/>Vo v\u0161etk\u00fdch deviatich pol\u00ed\u010dkach \u00fatvaru maj\u00fa by\u0165 vyplnen\u00e9 prirodzen\u00e9 \u010d\u00edsla tak, aby platilo:<\/p>\n<ul>\n<li>ka\u017ed\u00e9 z \u010d\u00edsel 2, 4, 6 a 8 je pou\u017eit\u00e9 aspo\u0148 raz,<\/li>\n<li>\u0161tyri pol\u00ed\u010dka vn\u00fatorn\u00e9ho \u0161tvorca obsahuj\u00fa s\u00fa\u010diny \u010d\u00edsel zo susediacich pol\u00ed\u010dok vonkaj\u0161ieho \u0161tvorca,<\/li>\n<li>v kruhu je s\u00fa\u010det \u010d\u00edsel zo susediacich pol\u00ed\u010dok vn\u00fatorn\u00e9ho \u0161tvorca.<\/li>\n<\/ul>\n<p>Zistite, ktor\u00e9 najmen\u0161ie a ktor\u00e9 najv\u00e4\u010d\u0161ie \u010d\u00edslo m\u00f4\u017ee by\u0165 nap\u00edsan\u00e9 v kruhu.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"66\/66z5a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Monika Dillingerov\u00e1)<\/div>\n<p><strong>Z9-I-2<\/strong><br \/>Z bodu A do bodu C vedie n\u00e1u\u010dn\u00fd chodn\u00edk prech\u00e1dzaj\u00faci bodom B a inakadia\u013e tie\u017e \u010derven\u00e1 turistick\u00e1 zna\u010dka, pozri obr\u00e1zok. Okrem toho sa d\u00e1 pou\u017ei\u0165 aj nezakreslen\u00e1 skratka dlh\u00e1 1 500metrov za\u010d\u00ednaj\u00faca v A a \u00fastiaca na n\u00e1u\u010dnom chodn\u00edku. Vojtech zistil, \u017ee v\u00fdlet z A po \u010dervenej do C a po n\u00e1u\u010dnom chodn\u00edku sp\u00e4\u0165 do A je dlh\u00fd 7 700 metrov, v\u00fdlet z B po n\u00e1u\u010dnom chodn\u00edku do C a potom po \u010dervenej do A je dlh\u00fd 5 800 metrov, s vyu\u017eit\u00edm skratky je cesta z A do B dlh\u00e1 1 700 metrov, v\u00fdlet z A po n\u00e1u\u010dnom chodn\u00edku do C a sp\u00e4\u0165 do A najsk\u00f4r po n\u00e1u\u010dnom chodn\u00edku a potom po skratke je dlh\u00fd 8 800 metrov. Ur\u010dte d\u013a\u017eku n\u00e1u\u010dn\u00e9ho chodn\u00edka z A do C. Pokia\u013e zadanie prip\u00fa\u0161\u0165a viac odpoved\u00ed, uve\u010fte v\u0161etky.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"66\/66z9a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Libor \u0160im\u016fnek)<\/div>\n<p><strong>Z9-I-3<\/strong><br \/>J\u00falii sa zakot\u00fa\u013eala lopti\u010dka do baz\u00e9na a pl\u00e1vala vo vode. Jej najvy\u0161\u0161\u00ed bod bol 2 cm nad hladinou. Priemer kru\u017enice, ktor\u00fa vyzna\u010dila hladina vody na povrchu lopti\u010dky, bol 8 cm. Ur\u010dte priemer J\u00faliinej lopti\u010dky.<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z9-I-4<\/strong><br \/>Katka si myslela p\u00e4\u0165cifern\u00e9 prirodzen\u00e9 \u010d\u00edslo. Do zo\u0161ita nap\u00edsala na prv\u00fd riadok s\u00fa\u010det myslen\u00e9ho \u010d\u00edsla a polovice myslen\u00e9ho \u010d\u00edsla. Na druh\u00fd riadok nap\u00edsala s\u00fa\u010det myslen\u00e9ho \u010d\u00edsla a p\u00e4tiny myslen\u00e9ho \u010d\u00edsla. Na tret\u00ed riadok nap\u00edsala s\u00fa\u010det myslen\u00e9ho \u010d\u00edsla a dev\u00e4tiny myslen\u00e9ho \u010d\u00edsla. Nakoniec s\u010d\u00edtala v\u0161etky tri zap\u00edsan\u00e9 \u010d\u00edsla a v\u00fdsledok nap\u00edsala na \u0161tvrt\u00fd riadok. Potom s \u00fa\u017easom zistila, \u017ee na \u0161tvrtom riadku m\u00e1 zap\u00edsan\u00fa tretiu mocninu ist\u00e9ho prirodzen\u00e9ho \u010d\u00edsla. Ur\u010dte najmen\u0161ie \u010d\u00edslo, ktor\u00e9 si Katka mohla myslie\u0165 na za\u010diatku.<\/p>\n<div class=\"pri\">(Lucia R\u016f\u017ei\u010dkov\u00e1)<\/div>\n<p><strong>Z9-I-5<\/strong><br \/>My\u0161ky si postavili podzemn\u00fd dom\u010dek pozost\u00e1vaj\u00faci z kom\u00f4rok a tunel\u010dekov:<\/p>\n<ul>\n<li>ka\u017ed\u00fd tunel\u010dek vedie z kom\u00f4rky do kom\u00f4rky (tzn. \u017eiadny nie je slep\u00fd),<\/li>\n<li>z ka\u017edej kom\u00f4rky ved\u00fa pr\u00e1ve tri tunel\u010deky do troch r\u00f4znych kom\u00f4rok,<\/li>\n<li>z ka\u017edej kom\u00f4rky sa d\u00e1 tunel\u010dekmi dosta\u0165 do ktorejko\u013evek inej kom\u00f4rky,<\/li>\n<li>v dom\u010deku je pr\u00e1ve jeden tunel\u010dek tak\u00fd, \u017ee jeho zasypan\u00edm sa dom\u010dek rozdel\u00ed na dve oddelen\u00e9 \u010dasti.<\/li>\n<\/ul>\n<p>Ko\u013eko najmenej kom\u00f4rok mohol ma\u0165 my\u0161\u00ed dom\u010dek? Na\u010drtnite, ako mohli by\u0165 kom\u00f4rky posp\u00e1jan\u00e9.<\/p>\n<div class=\"pri\">(Al\u017ebeta Bohinikov\u00e1)<\/div>\n<p><strong>Z9-I-6<\/strong><br \/>Dan\u00e1 je \u00fase\u010dka <em>AB<\/em> d\u013a\u017eky 12 cm, na ktorej je jednou stranou polo\u017een\u00fd \u0161tvorec <em>MRAK<\/em> so stranou d\u013a\u017eky 2 cm, pozri obr\u00e1zok. <em>MRAK<\/em> sa postupne prekl\u00e1pa po \u00fase\u010dke <em>AB<\/em>, pri\u010dom bod <em>R<\/em> zanech\u00e1va na papieri stopu. Narysujte cel\u00fa stopu bodu <em>R<\/em>, k\u00fdm \u0161tvorec neob\u00edde \u00fase\u010dku <em>AB<\/em> z oboch str\u00e1n a nevr\u00e1ti sa do svojej p\u00f4vodnej polohy.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"66\/66z9a2.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Monika Dillingerov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"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=2207\" target=\"_blank\" rel=\"noopener\"><input name=\"butt92z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2208\" target=\"_blank\" rel=\"noopener\"><input name=\"butt92r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-II-1<\/strong><br \/>Do pr\u00e1zdnych pol\u00ed\u010dok dopl\u0148te \u010d\u00edsla tak, aby v pol\u00ed\u010dkach uprostred ka\u017edej vyzna\u010denej<br \/> \u00fase\u010dky bol s\u00fa\u010det \u010d\u00edsel z jej krajn\u00fdch pol\u00ed\u010dok a aby s\u00fa\u010dty \u010d\u00edsel z pol\u00ed\u010dok na oboch uhloprie\u010dkach boli rovnak\u00e9.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"66\/66z9b1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Svetlana Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z9-II-2<\/strong><br \/>Pat s\u010d\u00edtal v\u0161etky \u0161tvorcifern\u00e9 \u010d\u00edsla, z ktor\u00fdch ka\u017ed\u00e9 obsahovalo v\u0161etky cifry 1, 2, 3 a 4, a dospel k s\u00fa\u010dtu 58 126. Mat mu potvrdil, \u017ee po\u010d\u00edtal spr\u00e1vne, ale \u017ee zabudol pripo\u010d\u00edta\u0165 dve z uva\u017eovan\u00fdch \u010d\u00edsel. Zistite, na ktor\u00e9 \u010d\u00edsla Pat zabudol.<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z9-II-3<\/strong><br \/>Vedci p\u00fa\u0161\u0165ali do bludiska potkany a sledovali, \u010di sa dostan\u00fa do cie\u013ea. Zistili, \u017ee \u010diernych potkanov do\u0161lo do cie\u013ea 56%, bielych 84%. V cieli bol pomer po\u010dtu \u010diernych a bielych potkanov 1 : 2. Ak\u00fd bol pomer po\u010dtu \u010diernych a bielych potkanov na \u0161tarte?<\/p>\n<div class=\"pri\">(Michaela Petrov\u00e1)<\/div>\n<p><strong>Z9-II-4<\/strong><br \/>Na \u00fase\u010dke <em>PQ<\/em> je jednou stranou polo\u017een\u00fd \u0161tvorec <em>MNOP<\/em> (pozri obr\u00e1zok). Priamka <em>PQ<\/em> sa postupne prekl\u00e1pa po stran\u00e1ch \u0161tvorca <em>MNOP<\/em>, pri\u010dom bod <em>Q<\/em> zanech\u00e1va na papieri stopu. Po prvom preklopen\u00ed je t\u00e1to stopa dlh\u00e1 5 cm, po piatich preklopeniach bod <em>Q<\/em> splynie s jedn\u00fdm z vrcholov \u0161tvorca. Ur\u010dte d\u013a\u017eku celej stopy bodu <em>Q<\/em>.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"66\/66z9b2.png\" alt=\"\" border=\"0\" \/><\/div>\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=\"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=2312\" target=\"_blank\" rel=\"noopener\"><input name=\"butt93z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=2313\" target=\"_blank\" rel=\"noopener\"><input name=\"butt93r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-III-1<\/strong><br \/>Michaela a Jana dnes obe maj\u00fa narodeniny, dokopy maj\u00fa 84 rokov. Pritom Michaela m\u00e1 dvakr\u00e1t viac rokov, ako mala Jana, ke\u010f Michaela mala to\u013eko rokov, ko\u013eko m\u00e1 Jana dnes. Ko\u013eko rokov m\u00e1 Michaela a ko\u013eko Jana?<\/p>\n<div class=\"pri\">(Marta Volfov\u00e1)<\/div>\n<p><strong>Z9-III-2<\/strong><br \/>Pred Janom sedeli tri zahalen\u00e9 princezn\u00e9 a Jano mal za \u00falohu zisti\u0165, ktor\u00e1 z nich je Zlatovl\u00e1ska.<\/p>\n<p style=\"margin-left:3em;\">Princezn\u00e1 v prvom kresle povedala:&nbsp;&nbsp;&nbsp;&#8222;V tre\u0165om kresle Zlatovl\u00e1ska nesed\u00ed.&#8220;<br \/>\nPrincezn\u00e1 v druhom kresle povedala:&nbsp;&nbsp;&nbsp;&#8222;Ja Zlatovl\u00e1ska nie som.&#8220;<br \/>\nPrincezn\u00e1 v tre\u0165om kresle povedala:&nbsp;&nbsp;&nbsp;&#8222;Ja som Zlatovl\u00e1ska.&#8220;<\/p>\n<p>Z\u00e1zra\u010dn\u00e1 mu\u0161ka Janovi prezradila, ko\u013eko princezien klamalo. A\u017e s touto radou dok\u00e1zal Jano odhali\u0165 prav\u00fa Zlatovl\u00e1sku. Ktor\u00e1 z princezien bola Zlatovl\u00e1ska?<\/p>\n<div class=\"pri\">(Marta Volfov\u00e1)<\/div>\n<p><strong>Z9-III-3<\/strong><br \/>Velite\u013e zvolal ostatn\u00fdch obrancov hradu a rozhodol, ako sa rozdelia o svoju odmenu:<\/p>\n<p style=\"margin-left:3em; margin-right:2em;\">&#8222;Prv\u00fd si vezme jeden duk\u00e1t a sedminu zvy\u0161ku, druh\u00fd si vezme dva duk\u00e1ty  a sedminu nov\u00e9ho zvy\u0161ku a tak \u010falej. Teda n-t\u00fd obranca si vezme n duk\u00e1tov a k tomu e\u0161te sedminu zo zvy\u0161n\u00e9ho mno\u017estva duk\u00e1tov, pokia\u013e nejak\u00e9 bud\u00fa.&#8220;<\/p>\n<p>Takto sa podarilo rozdeli\u0165 v\u0161etky duk\u00e1ty a pritom v\u0161etci obrancovia dostali rovnako. Ko\u013eko obrancov sa delilo o odmenu?<\/p>\n<div class=\"pri\">(Marta Volfov\u00e1)<\/div>\n<p><strong>Z9-III-4<\/strong><br \/>V rovnoramennom trojuholn\u00edku <em>ABC<\/em> je z\u00e1klad\u0148a <em>AB<\/em> dlh\u00e1 6 cm a uhol <em>BCA<\/em> m\u00e1 ve\u013ekos\u0165 45\u25e6. Vypo\u010d\u00edtajte polomer kru\u017enice op\u00edsanej tomuto trojuholn\u00edku.<\/p>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\u00e1)<\/div>\n<p>\u00a0<\/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 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>66. ro\u010dn\u00edk matematickej olympi\u00e1dy 2016-2017 S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9 aktualizovan\u00e9 4.7.2017 14:45<\/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":[24],"tags":[],"class_list":["post-1158","post","type-post","status-publish","format-standard","hentry","category-66-rocnik"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.6 - 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