{"id":1540,"date":"2022-09-20T20:00:13","date_gmt":"2022-09-20T20:00:13","guid":{"rendered":"http:\/\/matematika.besaba.com\/?p=1540"},"modified":"2023-01-25T21:05:02","modified_gmt":"2023-01-25T21:05:02","slug":"sutazne-ulohy-kategorii-z5-z9-10","status":"publish","type":"post","link":"https:\/\/matematika.besaba.com\/?p=1540","title":{"rendered":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9"},"content":{"rendered":"<p class=\"hh2\"><a name=\"top\"><\/a>72. ro\u010dn\u00edk matematickej olympi\u00e1dy 2022-2023<br \/>\nS\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9<\/p>\n<p><span class=\"povinne\">aktualizovan\u00e9 12.11.2022 18:30<\/span><\/p>\n\n<p><!--more--><\/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\">okresn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z62\">okresn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z72\">okresn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z82\">okresn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z92\">okresn\u00e9<\/a><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"4\" bgcolor=\"#c0c0c0\"><\/td>\n<td align=\"center\"><span class=\"a\"><a href=\"#z93\">krajsk\u00e9<\/a><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"z51\"><\/a><\/p>\n<form><a href=\"https:\/\/skmo.sk\/dokument.php?id=4166\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt51z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=4369\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt51r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<div class=\"at\"><strong>Term\u00edny odovzdania ries\u030cen\u00ed:<\/strong><\/p>\n<ul>\n<li>\u00falohy 1, 2, 3: <strong><em>16. 11. 2022<\/em><\/strong><\/li>\n<li>\u00falohy 4, 5, 6: <strong><em>16. 12. 2022<\/em><\/strong><\/li>\n<\/ul>\n<\/div>\n<p><strong>Z5-I-1<\/strong><br \/>\nNa l\u00fake bolo 45 oviec a nieko\u013eko pastierov. Potom ako z l\u00faky odi\u0161la polovica pastierov a tretina oviec, mali zvy\u0161n\u00ed pastieri a ovce spolu 126 n\u00f4h. Pritom v\u0161etky ovce a v\u0161etci pastieri mali obvykl\u00e9 po\u010dty n\u00f4h. Ko\u013eko pastierov bolo p\u00f4vodne na l\u00fake?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z5-I-2<\/strong><br \/>\nMarta hr\u00e1 hru, v ktorej h\u00e1da p\u00e4\u0165cifern\u00e9 \u010d\u00edslo tvoren\u00e9 navz\u00e1jom r\u00f4znymi \u010d\u00edslicami. Priebeh prv\u00fdch troch k\u00f4l vyzer\u00e1 takto:<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"72\/72z5a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<p>Farba pol\u00ed\u010dka prezr\u00e1dza nie\u010do o \u010d\u00edslici v \u0148om nap\u00edsanej:<\/p>\n<ul>\n<li>Zelen\u00e9 pol\u00ed\u010dko znamen\u00e1, \u017ee \u010d\u00edslica sa v h\u00e1danom \u010d\u00edsle vyskytuje, a to presne na tom mieste.<\/li>\n<li>\u017dlt\u00e9 pol\u00ed\u010dko znamen\u00e1, \u017ee \u010d\u00edslica sa v h\u00e1danom \u010d\u00edsle vyskytuje, ale na inom mieste.<\/li>\n<li>\u0160ed\u00e9 pol\u00ed\u010dko znamen\u00e1, \u017ee \u010d\u00edslica sa v h\u00e1danom \u010d\u00edsle nevyskytuje.<\/li>\n<\/ul>\n<p>Vysvetlite, \u010di Marta m\u00f4\u017ee alebo nem\u00f4\u017ee p\u00e4\u0165cifern\u00e9 \u010d\u00edslo s istotou uh\u00e1dnu\u0165 v nasleduj\u00facom kole.<\/p>\n<div class=\"pri\">(Josef Tkadlec)<\/div>\n<p><strong>Z5-I-3<\/strong><br \/>\nVo \u0161tvorcovej sieti je dan\u00fd trojuholn\u00edk, ktor\u00e9ho vrcholy s\u00fa uzlov\u00fdmi bodmi siete:<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"72\/72z5a2.png\" alt=\"\" border=\"0\" \/><\/div>\n<p>V dostato\u010dne roz\u0161\u00edrenej sieti nakreslite \u0161tyri rozdielne (navz\u00e1jom nezhodne) mnohouholn\u00edky tak\u00e9, \u017ee ich vrcholy s\u00fa tie\u017e jej uzlov\u00fdmi bodmi a ka\u017ed\u00fd z nich ma dvojn\u00e1sobn\u00fd obvod, ako ma dan\u00fd trojuholn\u00edk.<\/p>\n<div class=\"pri\">(Erika Novotn\u00e1)<\/div>\n<p><strong>Z5-I-4<\/strong><br \/>\nNikola mala v zo\u0161ite nap\u00edsan\u00e9 jedno trojcifern\u00e9 a jedno dvojcifern\u00e9 \u010d\u00edslo. Ka\u017ed\u00e9 z t\u00fdchto \u010d\u00edsel bolo tvoren\u00e1 navz\u00e1jom r\u00f4znymi \u010d\u00edslicami. Rozdiel Nikolin\u00fdch \u010d\u00edsel bol 976. Ak\u00fd bol ich s\u00fa\u010det?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z5-I-5<\/strong><br \/>\nTri \u017eaby sa nau\u010dili sk\u00e1ka\u0165 po rebr\u00edku. Ka\u017ed\u00e1 dok\u00e1\u017ee sk\u00e1ka\u0165 smerom hore aj smerom dole, ale len o ur\u010dit\u00e9 po\u010dty prie\u010dok. \u017daby za\u010d\u00ednaj\u00fa na zemi a ka\u017ed\u00e1 by sa rada dostala na svoju ob\u013e\u00faben\u00fa prie\u010dku:<\/p>\n<ul>\n<li>Mal\u00e1 \u017eaba vie sk\u00e1ka\u0165 o 2 alebo o 3 prie\u010dky a chce sa dosta\u0165 na siedmu prie\u010dku.<\/li>\n<li>Stredn\u00e1 \u017eaba vie sk\u00e1ka\u0165 o 2 alebo o 4 prie\u010dky a chce sa dosta\u0165 na prv\u00fa prie\u010dku.<\/li>\n<li>Ve\u013ek\u00e1 \u017eaba vie sk\u00e1ka\u0165 o 6 alebo o 9 prie\u010dok a chce sa dosta\u0165 na tretiu prie\u010dku.<\/li>\n<\/ul>\n<p>Pre jednotliv\u00e1 \u017eaby rozhodnite, \u010di vedia dosko\u010di\u0165 na svoju ob\u013e\u00faben\u00fa prie\u010dku. Ak \u00e1no, pop\u00ed\u0161te ako. Ak nie, vysvetlite pre\u010do.<\/p>\n<div class=\"pri\">(Veronika Bachrat\u00e1)<\/div>\n<p><strong>Z5-I-6<\/strong><br \/>\nJakub zbiera hracie kocky, v\u0161etky rovnakej ve\u013ekosti. V\u010dera na\u0161iel \u0161katu\u013eku, do ktorej za\u010dal kocky uklada\u0165. Prv\u00e1 vrstva kociek pokryla presne \u0161tvorcov\u00e9 dno \u0161katu\u013eky. Podobne vyskladal p\u00e4\u0165 \u010fal\u0161\u00edch vrstiev, av\u0161ak v polovici nasleduj\u00facej vrstvy mu do\u0161li kocky. Dnes dostal Jakub od babi\u010dky 18 kociek a s prekvapen\u00edm zistil, \u017ee mu presne ch\u00fdbali na dokon\u010denie ne\u00faplnej vrstvy v \u0161katu\u013eke. Ko\u013eko kociek mal Jakub v\u010dera?<\/p>\n<div class=\"pri\">(Michaela Petrova\u0301)<\/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=\"https:\/\/skmo.sk\/dokument.php?id=4371\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt52z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=4373\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt52r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z5-II-1<\/strong><br \/>\nKanva plna\u0301 mlieka ma\u0301 hmotnos\u0165 35 kg. Ta\u0301 ista\u0301 kanva s polovic\u030cny\u0301m mnoz\u030cstvom mlieka ma\u0301 hmotnos\u0165 18 kg.<br \/>Ko\u013eko kilogramov va\u0301z\u030ci pra\u0301zdna kanva?<\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z5-II-2<\/strong><br \/>\nDvaja stra\u0301z\u030ccovia dohliadaju\u0301 na miestnos\u0165, ktorej tvar a rozmery su\u0301 zna\u0301zornene\u0301 na obra\u0301zku. Kaz\u030cde\u0301 dve susedne\u0301 steny su\u0301 navza\u0301jom kolm\u00e9, rozmery su\u0301 uvedene\u0301 v metroch. Stra\u0301z\u030ccovia stoja tesne pri stene v miestach oznac\u030ceny\u0301ch kru\u0301z\u030ckami. Spoloc\u030cne takto maju\u0301 pod doh\u013eadom celu\u0301 miestnos\u0165, pritom len na c\u030cas\u0165 miestnosti mo\u0302z\u030cu dohliadnu\u0165 obaja su\u0301c\u030casne.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"72\/72z5b1.png\" alt=\"\" border=\"0\" \/><\/div>\n<ol style=\"list-style-type:lower-alpha\">\n<li>Vyznac\u030cte v obra\u0301zku c\u030cas\u0165 miestnosti, kam mo\u0302z\u030cu dohliadnu\u0165 obaja stra\u0301z\u030ccovia.<\/li>\n<li>Ko\u013ekokra\u0301t je cela\u0301 miestnos\u0165 va\u0308c\u030cs\u030cia ako c\u030cas\u0165, kam mo\u0302z\u030cu dohliadnu\u0165 obaja stra\u0301z\u030ccovia?<\/li>\n<\/ol>\n<div class=\"pri\">()<\/div>\n<p><strong>Z5-II-3<\/strong><br \/>\nNa\u0301jdite vs\u030cetky \u010d\u00edsla s nasleduju\u0301cimi vlastnos\u0165ami:<\/p>\n<ul>\n<li>\u010d\u00edslo je pa\u0301rne,<\/li>\n<li>\u010d\u00edslo je dvojciferne\u0301,<\/li>\n<li>su\u0301c\u030cet jeho \u010d\u00edslic je va\u0308\u010d\u0161\u00ed ako 6,<\/li>\n<li>\u010d\u00edslica na mieste desiatok je aspon\u030c o 4 va\u0308c\u030cs\u030cia ako \u010d\u00edslica na mieste jednotiek.<\/li>\n<\/ul>\n<div class=\"pri\">()<\/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<form><a href=\"https:\/\/skmo.sk\/dokument.php?id=4167\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt61z\" type=\"button\" value=\"Zadanie\" \/><\/a><input name=\"butt61r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z6-I-1<\/strong><br \/>\nP\u00e1n \u0160kovr\u00e1nok bol zn\u00e1mym chovate\u013eom vt\u00e1kov. Celkovo ich mal viac ako 50 a menej ako 100. Andulky tvorili dev\u00e4tinu a kan\u00e1riky \u0161tvrtinu celkov\u00e9ho mno\u017estva vt\u00e1kov. Ko\u013eko vt\u00e1kov choval p\u00e1n \u0160kovr\u00e1nok?<\/p>\n<div class=\"pri\">(Libus\u030ce Hozova\u0301)<\/div>\n<p><strong>Z6-I-2<\/strong><br \/>\nV\u00e1clav n\u00e1sobil dve trojciferne \u010d\u00edsla obvykl\u00fd p\u00edsomnom sp\u00f4sobom. Overil si, \u017ee v\u00fdsledok je naozaj spr\u00e1vny a svoj v\u00fdpo\u010det niekam zalo\u017eil. Po \u010dase potreboval v\u00fdsledok uk\u00e1za\u0165 mami\u010dke. Na\u0161iel s\u00edce svoj predch\u00e1dzaj\u00faci v\u00fdpo\u010det, ale mnoho \u010d\u00edslic bolo rozmazan\u00fdch, tak\u017ee sa nedali v\u00f4bec precita\u0165 (hviezdi\u010dky nahr\u00e1dzaj\u00fa ne\u010ditate\u013en\u00e9 \u010d\u00edslice):<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"72\/72z6a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<p>V\u00e1clav si u\u017e nepam\u00e4tal, ktor\u00e9 \u010d\u00edsla n\u00e1sobil, napriek tomu bol schopn\u00fd ur\u010di\u0165 ich s\u00fa\u010din. Ak\u00fd bol tento s\u00fa\u010din?<\/p>\n<div class=\"pri\">(Libus\u030ce Hozova\u0301)<\/div>\n<p><strong>Z6-I-3<\/strong><br \/>\nMagda si z papiera vystrihla dva rovnak\u00e9 rovnoramenne trojuholn\u00edky, z ktor\u00fdch ka\u017ed\u00fd mal obvod 100 cm. Najprv z t\u00fdchto trojuholn\u00edkov zlozila \u0161tvoruholn\u00edk tak, \u017ee ich k sebe prilo\u017eila ramenami. Potom z nich zlozila \u0161tvoruholn\u00edk tak, \u017ee ich k sebe prilo\u017eila z\u00e1klad\u0148ami.<br \/>\n(Ak by mali z\u00e1kladne 1 cm a ramena 2 cm, vzniknut\u00e1 \u0161tvoruholn\u00edky by teda vyzerali ako na obr\u00e1zkoch.)<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"72\/72z6a2.png\" alt=\"\" border=\"0\" \/><\/div>\n<p>V prvom pr\u00edpade jej vy\u0161iel \u0161tvoruholn\u00edk s obvodom o 4 cm krat\u0161\u00edm ako v druhom pr\u00edpade. Ur\u010dte d\u013a\u017eky str\u00e1n Magdin\u00fdch trojuholn\u00edkov.<\/p>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z6-I-4<\/strong><br \/>\nSedem trpasl\u00edkov sa narodilo v rovnak\u00fd de\u0148 v siedmich po sebe id\u00facich rokoch. S\u00fa\u010det vekov troch najmlad\u0161\u00edch trpasl\u00edkov je 42 rokov. Ke\u010f jeden trpasl\u00edk odi\u0161iel so Snehulienkou po vodu, zistili zvy\u0161n\u00ed trpasl\u00edci, \u017ee ich priemern\u00fd vek je rovnak\u00fd ako priemern\u00fd vek v\u0161etk\u00fdch siedmich. Ko\u013eko rokov mal trpasl\u00edk, ktor\u00fd \u0161iel so Snehulienkou po vodu?<\/p>\n<div class=\"pri\">(Libus\u030ce Hozova\u0301)<\/div>\n<p><strong>Z6-I-5<\/strong><br \/>\nPat a Mat si precvi\u010dovali po\u010d\u00edtanie. Vo \u0161tvorcovej sieti orientovanej pod\u013ea svetov\u00fdch str\u00e1n priradili posunu o jedno pol\u00ed\u010dko nasleduj\u00face matematick\u00e1 oper\u00e1cie:<\/p>\n<ul>\n<li>Pri posune na sever (S) pripo\u010d\u00edtali sedem.<\/li>\n<li>Pri posune na v\u00fdchod (V) odpo\u010d\u00edtali \u0161tyri.<\/li>\n<li>Pri posune na juh (J) vydelili dvoma.<\/li>\n<li>Pri posune na z\u00e1pad (Z) vyn\u00e1sobili tromi.<\/li>\n<\/ul>\n<p>(Napr\u00edklad ke\u010f Mat zadal Patovi \u010d\u00edslo 5 a cestu S-V-J, vy\u0161lo im pri spr\u00e1vnom po\u010d\u00edtan\u00ed \u010d\u00edslo 4.)<br \/>\nKtor\u00e9 \u010d\u00edslo zadal Pat Matovi, ak pri ceste S-V-J-Z-Z-J-V-S pri spr\u00e1vnom po\u010d\u00edtan\u00ed vy\u0161lo na konci \u010d\u00edslo 57?<\/p>\n<div class=\"pri\">(Michaela Petrov\u00e1)<\/div>\n<p><strong>Z6-I-6<\/strong><br \/>\nBoris m\u00e1 zvl\u00e1\u0161tne digit\u00e1lne hodiny. Id\u00fa s\u00edce presne, ale namiesto hod\u00edn a min\u00fat ukazuj\u00fa in\u00e9 dve \u010d\u00edsla: Prv\u00e9 je cifern\u00fdm s\u00fa\u010dtom \u010d\u00edsla, ktor\u00e9 by bolo na displeji be\u017en\u00fdch hod\u00edn, druh\u00e9 je s\u00fa\u010dtom hod\u00edn a min\u00fat (<em>napr\u00edklad o 7:30 ukazuj\u00fa Borisove hodiny 10:37<\/em>). Ak\u00fd m\u00f4\u017ee by\u0165 skuto\u010dn\u00fd \u010das, ke\u010f Borisove hodiny ukazuj\u00fa 6:15? Ur\u010dte v\u0161etky mo\u017enosti.<\/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=\"z62\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><input name=\"butt62z\" type=\"button\" value=\"Zadanie\" \/><input name=\"butt62r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z6-II-1<\/strong><\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z6-II-2<\/strong><\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z6-II-3<\/strong><\/p>\n<div class=\"pri\">()<\/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\n<form><a href=\"https:\/\/skmo.sk\/dokument.php?id=4168\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt71z\" type=\"button\" value=\"Zadanie\" \/><\/a><input name=\"butt71r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z7-I-1<\/strong><br \/>\nPriemern\u00fd vek dedka, babi\u010dky a ich piatich vn\u00fa\u010dat je 26 rokov. Priemern\u00fd vek samotn\u00fdch vn\u00fa\u010dat je 7 rokov. Babi\u010dka je o rok mlad\u0161ia ako dedko. Ko\u013eko rokov ma babi\u010dka?<\/p>\n<div class=\"pri\">(Libus\u030ce Hozova\u0301)<\/div>\n<p><strong>Z7-I-2<\/strong><br \/>\nS\u00fa dan\u00e9 dva zhodn\u00e9 rovnostrann\u00e9 trojuholn\u00edky <em>ABC<\/em> a <em>BDE<\/em> tak, \u017ee body <em>A, B, D<\/em> le\u017eia na jednej priamke a body <em>C, E<\/em> le\u017eia v rovnakej polrovine vymedzenej priamkou <em>AD<\/em>. Priese\u010dn\u00edk <em>CD<\/em> a <em>AE<\/em> je ozna\u010den\u00fd <em>F<\/em>. Ur\u010dte ve\u013ekos\u0165 uhla <em>AFD<\/em>.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"72\/72z7a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Iveta Janc\u030cigova\u0301)<\/div>\n<p><strong>Z7-I-3<\/strong><br \/>\n<em>Obkro\u010dn\u00e9 \u010d\u00edslo<\/em> je tak\u00e9 prirodzene \u010d\u00edslo, v ktor\u00e9ho z\u00e1pise<\/p>\n<ul>\n<li>je ka\u017ed\u00e1 nenulov\u00e1 \u010d\u00edslica pou\u017eit\u00e1 pr\u00e1ve dvakr\u00e1t,<\/li>\n<li>medzi dvoma rovnak\u00fdmi nenulov\u00fdmi \u010d\u00edslicami sa nach\u00e1dza pr\u00e1ve to\u013eko n\u00fal, ak\u00e1 je hodnota t\u00fdchto \u010d\u00edslic.<\/li>\n<\/ul>\n<p>(Obkro\u010dn\u00e9 \u010d\u00edsla s\u00fa napr\u00edklad 40001041 alebo 300103100.)<br \/>\nZistite, ko\u013eko existuje sedemcifern\u00fdch obkro\u010dn\u00fdch \u010d\u00edsel, v ktor\u00fdch z\u00e1pise sa vyskytuj\u00fa pr\u00e1ve jednotky, dvojky a nuly.<\/p>\n<div class=\"pri\">(Matu\u0301s\u030c Paps\u030co)<\/div>\n<p><strong>Z7-I-4<\/strong><br \/>\nJarko mal nap\u00edsan\u00fa postupnos\u0165 slab\u00edk:<\/p>\n<div class=\"r2s\">ZU ZA NA NE LA LU CI SA MU EL<\/div>\n<p>P\u00edsmena chcel nahradi\u0165 \u010d\u00edslicami od 0 do 9 tak, aby r\u00f4znym p\u00edsmen\u00e1m zodpovedali r\u00f4zne \u010d\u00edslice a aby (v danom porad\u00ed) vznikla rast\u00faca postupnos\u0165 dvojcifern\u00fdch \u010d\u00edsel. Zistite, \u010di sa to d\u00e1 a ako, alebo vysvetlite, pre\u010do to mo\u017en\u00e9 nie je.<\/p>\n<div class=\"pri\">(Jaroslav Zhouf)<\/div>\n<p><strong>Z7-I-5<\/strong><br \/>\nNa obr\u00e1zku s\u00fa zn\u00e1zornen\u00e9 \u0161tvorce <em>ABCD<\/em>, <em>EFCA<\/em>, <em>GHCE<\/em> a <em>IJHE<\/em>. Body <em>S<\/em>, <em>B<\/em>, <em>F<\/em> a <em>G<\/em> s\u00fa postupne stredy t\u00fdchto \u0161tvorcov. \u00dase\u010dka <em>AC<\/em> je dlh\u00e1 1 cm. Ur\u010dte obsah trojuholn\u00edka <em>IJS<\/em>.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"72\/72z7a2.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z7-I-6<\/strong><br \/>\nEva si myslela dve prirodzen\u00e9 \u010d\u00edsla. Tieto \u010d\u00edsla najprv spr\u00e1vne s\u010d\u00edtala, potom od seba spr\u00e1vne od\u010d\u00edtala. V obidvoch pr\u00edpadoch dostala dvojcifern\u00fd v\u00fdsledok. S\u00fa\u010din takto vzniknut\u00fdch dvojcifern\u00fdch \u010d\u00edsel bol 645. Ktor\u00e9 \u010d\u00edsla si Eva myslela?<\/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 name=\"z72\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><input name=\"butt72z\" type=\"button\" value=\"Zadanie\" \/><input name=\"butt72r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z7-II-1<\/strong><\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z7-II-2<\/strong><\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z7-II-3<\/strong><\/p>\n<div class=\"pri\">()<\/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<form><a href=\"https:\/\/skmo.sk\/dokument.php?id=4169\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt81z\" type=\"button\" value=\"Zadanie\" \/><\/a><input name=\"butt81r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<div class=\"at\"><strong>Term\u00edny odovzdania ries\u030cen\u00ed:<\/strong><\/p>\n<ul>\n<li>\u00falohy 1, 2, 3: <strong><em>16. 1. 2023<\/em><\/strong><\/li>\n<li>\u00falohy 4, 5, 6: <strong><em>3. 3. 2023<\/em><\/strong><\/li>\n<\/ul>\n<\/div>\n<p><strong>Z8-I-1<\/strong><br \/>\nSu\u0301 dane\u0301 tri navza\u0301jom ro\u0302zne \u010d\u00edsla. Priemer priemeru dvoch men\u0161\u00edch \u010d\u00edsel a priemeru dvoch va\u0308c\u030cs\u030c\u00edch \u010d\u00edsel je rovny\u0301 priemeru vs\u030cetky\u0301ch troch \u010d\u00edsel. Priemer najmens\u030cieho a najva\u0308c\u030cs\u030cieho \u010d\u00edsla je 2022. Urc\u030cte su\u0301c\u030cet ty\u0301chto troch \u010d\u00edsel.<\/p>\n<div class=\"pri\">(Karel Pazourek)<\/div>\n<p><strong>Z8-I-2<\/strong><br \/>\nKosos\u030ctvorec <i>ABCD<\/i>\u00a0ma\u0301 stranu d\u013az\u030cky 6 cm a vy\u0301s\u030cku 4 cm. Bod <em>E<\/em> je stred strany <em>AD<\/em>, bod <em>G<\/em> je stred strany <em>BC<\/em>, bod <em>F<\/em> je priesec\u030cn\u00edk u\u0301sec\u030ciek <em>AG<\/em> a <em>BE<\/em>, bod <em>H<\/em> je priese\u010dn\u00edk u\u0301sec\u030ciek <em>CE<\/em> a <em>DG<\/em>. Urc\u030cte obsah s\u030ctvoruholn\u00edka <em>EFGH<\/em>.<\/p>\n<div class=\"pri\">(Karel Pazourek)<\/div>\n<p><strong>Z8-I-3<\/strong><br \/>\nPre postupnos\u0165 c\u030c\u00edsel zac\u030c\u00ednaju\u0301cu sa<\/p>\n<div class=\"r\">1, 3, 4, 7, 11, 18, \u2026<\/div>\n<p>plat\u00ed,\u00a0 z\u030ce kaz\u030cde\u0301 \u010d\u00edslo poc\u030cnu\u0301c tret\u00edm je su\u0301c\u030ctom predcha\u0301dzaju\u0301cich dvoch. Akou \u010d\u00edslicou sa konc\u030c\u00ed \u010d\u00edslo na 2023. mieste tejto postupnosti?<\/p>\n<div class=\"pri\">(Ja\u0301n Maza\u0301k)<\/div>\n<p><strong>Z8-I-4<\/strong><br \/>\nCyril na mape s mierkou 1 \u2236 50 000 vyznac\u030cil s\u030ctvorcovy\u0301 pozemok a vypoc\u030c\u00edtal si, z\u030ce jeho strana v skutoc\u030cnosti zodpoveda\u0301 1 km. Mapu zmens\u030cil na kop\u00edrke tak, z\u030ce vyznac\u030ceny\u0301 s\u030ctvorec mal obsah o 1,44 cm<sup>2<\/sup> mens\u030c\u00ed ako na po\u0302vodnej mape. Aka\u0301 bola mierka mapy po zmens\u030cen\u00ed?<\/p>\n<div class=\"pri\">(Michaela Petrov\u00e1)<\/div>\n<p><strong>Z8-I-5<\/strong><br \/>\nPetra mala na tabuli nap\u00edsane\u0301 vs\u030cetky prirodzene\u0301 c\u030c\u00edsla od 1 do 9, kaz\u030cde\u0301 pra\u0301ve raz. Dve z ty\u0301chto c\u030c\u00edsel sc\u030c\u00edtala, zmazala a vy\u0301sledny\u0301 su\u0301c\u030cet nap\u00edsala namiesto zmazany\u0301ch sc\u030c\u00edtancov. Mala tak teraz nap\u00edsany\u0301ch osem c\u030c\u00edsel, ktore\u0301 sa jej podarilo rozdeli\u0165 do dvoch skup\u00edn s rovnaky\u0301m su\u0301c\u030cinom. Urc\u030cte, aky\u0301 najva\u0308c\u030cs\u030c\u00ed mohol by\u0165 tento su\u0301c\u030cin.<\/p>\n<div class=\"pri\">(Erika Novotn\u00e1)<\/div>\n<p><strong>Z8-I-6<\/strong><br \/>\nJe dany\u0301 obd\u013az\u030cnik <em>ABCD<\/em> a body <em>E, F<\/em> tak, z\u030ce trojuholn\u00edky <i>BEC <\/i>a <em>CFD<\/em> su\u0301 rovnostranne\u0301 a kaz\u030cdy\u0301 z nich ma\u0301 s obd\u013az\u030cnikom <em>ABCD<\/em> spoloc\u030cnu\u0301 iba stranu. Zdo\u0302vodnite, z\u030ce aj trojuholn\u00edk <em>AEF<\/em> je rovnostrann\u00fd.<\/p>\n<div class=\"pri\">(Jaroslav \u0160vrc\u030cek)<\/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><input name=\"butt82z\" type=\"button\" value=\"Zadanie\" \/><input name=\"butt82r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z8-II-1<\/strong><\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z8-II-2<\/strong><\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z8-II-3<\/strong><\/p>\n<div class=\"pri\">()<\/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<form><a href=\"https:\/\/skmo.sk\/dokument.php?id=4170\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt91z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=4370\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt91r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<div class=\"at\"><strong>Term\u00edny odovzdania ries\u030cen\u00ed:<\/strong><\/p>\n<ul>\n<li>\u00falohy 1, 2, 3: <strong><em>16. 11. 2022<\/em><\/strong><\/li>\n<li>\u00falohy 4, 5, 6: <strong><em>16. 12. 2022<\/em><\/strong><\/li>\n<\/ul>\n<\/div>\n<p><strong>Z9-I-1<\/strong><br \/>\nAritmetick\u00e1 postupnos\u0165 je tak\u00e1 postupnos\u0165 \u010d\u00edsel, v ktorej je rozdiel ka\u017ed\u00e9ho \u010d\u00edsla a \u010d\u00edsla jemu predch\u00e1dzaj\u00faceho st\u00e1le rovnak\u00fd; tomuto rozdielu sa hovor\u00ed <em>diferencia<\/em>. (Napr\u00edklad (2, 8, 14, 20, 26, 32) je aritmetick\u00e1 postupnos\u0165 s diferenciou 6.)<br \/>\nBolek a Lolek mali ka\u017ed\u00fd svoju aritmetick\u00fa postupnos\u0165. Aj Bolkova aj Lolkova postupnos\u0165 sa za\u010d\u00ednala \u010d\u00edslom 2023 a kon\u010dila sa \u010d\u00edslom 3023. Tieto dve postupnosti mali 26 spolo\u010dn\u00fdch \u010d\u00edsel. Pomer Bolkovej a Lolkovej diferencie bol 5 : 2. Ur\u010dte rozdiel Bolkovej a Lolkovej diferencie.<\/p>\n<div class=\"pri\">(Erika Novotn\u00e1)<\/div>\n<p><strong>Z9-I-2<\/strong><br \/>\nS\u00fa dan\u00e9 dva zhodn\u00e9 rovnostrann\u00e9 trojuholn\u00edky <em>ABC<\/em> a <em>BDE<\/em> tak, \u017ee ve\u013ekos\u0165 uhla <em>ABD<\/em> je v\u00e4\u010d\u0161ia ako 120\u00b0 a men\u0161ia ako 180\u00b0 a body <em>C, E<\/em> le\u017eia v rovnakej polrovine vymedzenej priamkou <em>AD<\/em>. Priese\u010dn\u00edk <em>CD<\/em> a <em>AE<\/em> je ozna\u010den\u00fd <em>F<\/em>. Ur\u010dte ve\u013ekos\u0165 uhla <em>AFD<\/em>.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"72\/72z9a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Iveta Janc\u030cigova\u0301)<\/div>\n<p><strong>Z9-I-3<\/strong><br \/>\nTraja k\u00fazeln\u00edci k\u00fazlia s \u010d\u00edslami, ka\u017ed\u00fd v\u0161ak vie len jedno k\u00fazlo:<\/p>\n<ul>\n<li>Prv\u00fd k\u00fazeln\u00edk vie od \u013eubovo\u013en\u00e9ho \u010d\u00edsla od\u010d\u00edta\u0165 \u010d\u00edslo jedna.<\/li>\n<li>Druh\u00fd k\u00fazeln\u00edk vie \u013eubovo\u013en\u00e9 \u010d\u00edslo vydeli\u0165 \u010d\u00edslom dva.<\/li>\n<li>Tret\u00ed k\u00fazeln\u00edk vie \u013eubovo\u013ene \u010d\u00edslo vyn\u00e1sobi\u0165 \u010d\u00edslom tri.<\/li>\n<\/ul>\n<p>K\u00fazeln\u00edci sa pri \u010darovan\u00ed m\u00f4\u017eu \u013eubovo\u013ene strieda\u0165, ka\u017ed\u00fd v\u0161ak m\u00f4\u017ee svoje k\u00fazlo po\u010das jedn\u00e9ho vyst\u00fapenia pou\u017ei\u0165 najviac p\u00e4\u0165kr\u00e1t a \u017eiadny medziv\u00fdsledok nesmie by\u0165 v\u00e4\u010d\u0161\u00ed ako 10. Pri jednom vyst\u00fapen\u00ed mali z p\u00e4tice \u010d\u00edsel (3, 8, 9, 2, 4) vyk\u00fazli\u0165 p\u00e4ticu trojok, pri inom vyst\u00fapen\u00ed mali z tej istej p\u00e4tice \u010d\u00edsel vyk\u00fazli\u0165 p\u00e4ticu p\u00e4tiek Ako si mohli s probl\u00e9mom poradi\u0165? N\u00e1jdite mo\u017en\u00e9 rie\u0161enie alebo vysvetlite, pre\u010do to mo\u017en\u00e9 nie je.<\/p>\n<div class=\"pri\">(Erika Novotn\u00e1)<\/div>\n<p><strong>Z9-I-4<\/strong><br \/>\nNa\u0301jdite najmens\u030cie kladne\u0301 cele\u0301 \u010d\u00edsla <em>a<\/em> a <em>b<\/em>, pre ktore\u0301 plat\u00ed<\/p>\n<div class=\"r\">7<em>a<\/em><sup>3<\/sup> = 11<em>b<\/em><sup>5<\/sup><\/div>\n<div class=\"pri\">(Alz\u030cbeta Bohin\u00edkov\u00e1)<\/div>\n<p><strong>Z9-I-5<\/strong><br \/>\nNa snovom trhovisku ponu\u0301kla Sfinga cestovate\u013eovi za s\u030ctyri sny sedem ilu\u0301zi\u0131\u0301, dva s\u030clof\u0131\u0301ky a jednu noc\u030cnu\u0301 moru. Ine\u0301mu cestovate\u013eovi ponu\u0301kla za sedem snov s\u030ctyri ilu\u0301zie, s\u030ctyri s\u030clof\u0131\u0301ky a dve noc\u030cne\u0301 mory. Sfinga je pri svojich ponuka\u0301ch spravodliva\u0301 a vymeriava vz\u030cdy rovnako. Ko\u013eko ilu\u0301zi\u0131\u0301 stoj\u0131\u0301 jeden sen?<\/p>\n<div class=\"pri\">(Karel Pazourek)<\/div>\n<p><strong>Z9-I-6<\/strong><br \/>\nVrcholy s\u030ctvorca <em>ABCD<\/em> spa\u0301ja lomena\u0301 c\u030ciara <em>DEFGHB<\/em>. Uhly <em>DEF<\/em>, <em>EFG<\/em>, <em>FGH<\/em>, <em>GHB<\/em> su\u0301 prave\u0301 a u\u0301sec\u030cky <em>DE<\/em>, <em>EF<\/em>, <em>FG<\/em>, <em>GH<\/em>, <em>HB<\/em> meraju\u0301 postupne 6 cm, 4 cm, 4 cm, 1 cm, 2 cm. Urc\u030cte obsah s\u030ctvorca <em>ABCD<\/em>.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"72\/72z9a2.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Monika Dillingerova\u0301)<\/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=\"https:\/\/skmo.sk\/dokument.php?id=4374\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt92z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=4376\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt92r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-II-1<\/strong><br \/>\nNa\u0301jdite vs\u030cetky s\u030ctvorciferne\u0301 \u010d\u00edsla, ktore\u0301 maju\u0301 presne pa\u0308\u0165 s\u030ctvorciferny\u0301ch a presne deva\u0308\u0165 jednociferny\u0301ch delite\u013eov.<\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z9-II-2<\/strong><br \/>\nTrojuholn\u00edk <em>ABC<\/em> ma\u0301 stranu <em>AC<\/em> dlhu\u0301 24 cm a vy\u0301s\u030cku z vrcholu <em>B<\/em> dlhu\u0301 25 cm. Strana <em>AB<\/em> je rozdelena\u0301 na pa\u0308\u0165 zhodny\u0301ch c\u030cast\u0131\u0301, deliace body su\u0301 postupne od <em>A<\/em> k <em>B<\/em> oznac\u030cene\u0301 <em>K, L, M, N<\/em>. Kaz\u030cdy\u0301m z ty\u0301chto bodov precha\u0301dza rovnobez\u030cka so stranou <em>AC<\/em>. Priese\u010dn\u00edky rovnobez\u030ciek so stranou <em>BC<\/em><br \/>\nsu\u0301 postupne od <em>B<\/em> k <em>C<\/em> oznac\u030cene\u0301 <em>O, P, Q, R<\/em>.<\/p>\n<p>Vypo\u010d\u00edtajte su\u0301c\u030cet obsahov lichobe\u017en\u00edkov <em>KLQR<\/em> a <em>MNOP<\/em>.<\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z9-II-3<\/strong><br \/>\nPomocou premennej <em>n<\/em> boli zap\u00edsane\u0301 nasleduju\u0301ce s\u030ctyri vy\u0301razy:<\/p>\n<div class=\"r\">2023n,&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;n<sup>2<\/sup> + n + 23,&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3n<sup>3<\/sup>,&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;10n<sup>2<\/sup> + 2023.<\/div>\n<p>Vy\u0301raz nazveme <em>nep\u00e1rnotvorn\u00fd<\/em>, ak pre kaz\u030cde\u0301 prirodzene\u0301 \u010d\u00edslo <em>n<\/em> plat\u00ed, z\u030ce hodnota vy\u0301razu je nepa\u0301rna. Rozhodnite, ktore\u0301 z uvedeny\u0301ch s\u030ctyroch vy\u0301razov su\u0301 nepa\u0301rnotvorne\u0301, a zdo\u0302vodnite prec\u030co.<\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z9-II-4<\/strong><br \/>\nV istom mnohouholn\u00edku plat\u00ed, z\u030ce pomer su\u0301c\u030ctu ve\u013ekost\u0131\u0301 jeho vnu\u0301torny\u0301ch uhlov a su\u0301c\u030ctu ve\u013ekost\u0131\u0301 k nim doplnkovy\u0301ch uhlov je 3 \u2236 5.?<br \/>Ko\u013eko vrcholov ma\u0301 tento mnohouholn\u00edk?<\/p>\n<p><em>(Doplnkov\u00fd uhol dop\u013a\u0148a dany\u0301 uhol do uhla plne\u0301ho.)<\/em><\/p>\n<div class=\"pri\">()<\/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><input name=\"butt93z\" type=\"button\" value=\"Zadanie\" \/><input name=\"butt93r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z9-III-1<\/strong><\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z9-III-2<\/strong><\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z9-III-3<\/strong><br \/>\n(Karel Pazourek)<\/p>\n<p><strong>Z9-III-4<\/strong><\/p>\n<div class=\"pri\">()<\/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><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>72. ro\u010dn\u00edk matematickej olympi\u00e1dy 2022-2023 S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9 aktualizovan\u00e9 12.11.2022 18:30<\/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":[38],"tags":[],"class_list":["post-1540","post","type-post","status-publish","format-standard","hentry","category-72-rocnik-2022-23"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.6 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9 - Matematick\u00e1 olympi\u00e1da<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/matematika.besaba.com\/?p=1540\" \/>\n<meta property=\"og:locale\" content=\"sk_SK\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9 - Matematick\u00e1 olympi\u00e1da\" \/>\n<meta property=\"og:description\" content=\"72. ro\u010dn\u00edk matematickej olympi\u00e1dy 2022-2023 S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9 aktualizovan\u00e9 12.11.2022 18:30\" \/>\n<meta property=\"og:url\" content=\"https:\/\/matematika.besaba.com\/?p=1540\" \/>\n<meta property=\"og:site_name\" content=\"Matematick\u00e1 olympi\u00e1da\" \/>\n<meta property=\"article:published_time\" content=\"2022-09-20T20:00:13+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2023-01-25T21:05:02+00:00\" \/>\n<meta name=\"author\" content=\"admin\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Autor\" \/>\n\t<meta name=\"twitter:data1\" content=\"admin\" \/>\n\t<meta name=\"twitter:label2\" content=\"Predpokladan\u00fd \u010das \u010d\u00edtania\" \/>\n\t<meta name=\"twitter:data2\" content=\"13 min\u00fat\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/matematika.besaba.com\/?p=1540\",\"url\":\"https:\/\/matematika.besaba.com\/?p=1540\",\"name\":\"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9 - Matematick\u00e1 olympi\u00e1da\",\"isPartOf\":{\"@id\":\"https:\/\/matematika.besaba.com\/#website\"},\"datePublished\":\"2022-09-20T20:00:13+00:00\",\"dateModified\":\"2023-01-25T21:05:02+00:00\",\"author\":{\"@id\":\"https:\/\/matematika.besaba.com\/#\/schema\/person\/0b550d607535616720c775cf1d895c32\"},\"breadcrumb\":{\"@id\":\"https:\/\/matematika.besaba.com\/?p=1540#breadcrumb\"},\"inLanguage\":\"sk-SK\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/matematika.besaba.com\/?p=1540\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/matematika.besaba.com\/?p=1540#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Domovsk\u00e1 str\u00e1nka\",\"item\":\"https:\/\/matematika.besaba.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/matematika.besaba.com\/#website\",\"url\":\"https:\/\/matematika.besaba.com\/\",\"name\":\"Matematick\u00e1 olympi\u00e1da\",\"description\":\"arch\u00edv zadan\u00ed a rie\u0161en\u00ed matematickej olympi\u00e1dy\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/matematika.besaba.com\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"sk-SK\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/matematika.besaba.com\/#\/schema\/person\/0b550d607535616720c775cf1d895c32\",\"name\":\"admin\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"sk-SK\",\"@id\":\"https:\/\/matematika.besaba.com\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/eb87de157eb8bac30b30a5d51e9e53a81806a5b7015a13f1242444d33c18731e?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/eb87de157eb8bac30b30a5d51e9e53a81806a5b7015a13f1242444d33c18731e?s=96&d=mm&r=g\",\"caption\":\"admin\"},\"url\":\"https:\/\/matematika.besaba.com\/?author=1\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9 - Matematick\u00e1 olympi\u00e1da","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/matematika.besaba.com\/?p=1540","og_locale":"sk_SK","og_type":"article","og_title":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9 - Matematick\u00e1 olympi\u00e1da","og_description":"72. ro\u010dn\u00edk matematickej olympi\u00e1dy 2022-2023 S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9 aktualizovan\u00e9 12.11.2022 18:30","og_url":"https:\/\/matematika.besaba.com\/?p=1540","og_site_name":"Matematick\u00e1 olympi\u00e1da","article_published_time":"2022-09-20T20:00:13+00:00","article_modified_time":"2023-01-25T21:05:02+00:00","author":"admin","twitter_card":"summary_large_image","twitter_misc":{"Autor":"admin","Predpokladan\u00fd \u010das \u010d\u00edtania":"13 min\u00fat"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/matematika.besaba.com\/?p=1540","url":"https:\/\/matematika.besaba.com\/?p=1540","name":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9 - Matematick\u00e1 olympi\u00e1da","isPartOf":{"@id":"https:\/\/matematika.besaba.com\/#website"},"datePublished":"2022-09-20T20:00:13+00:00","dateModified":"2023-01-25T21:05:02+00:00","author":{"@id":"https:\/\/matematika.besaba.com\/#\/schema\/person\/0b550d607535616720c775cf1d895c32"},"breadcrumb":{"@id":"https:\/\/matematika.besaba.com\/?p=1540#breadcrumb"},"inLanguage":"sk-SK","potentialAction":[{"@type":"ReadAction","target":["https:\/\/matematika.besaba.com\/?p=1540"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/matematika.besaba.com\/?p=1540#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Domovsk\u00e1 str\u00e1nka","item":"https:\/\/matematika.besaba.com\/"},{"@type":"ListItem","position":2,"name":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9"}]},{"@type":"WebSite","@id":"https:\/\/matematika.besaba.com\/#website","url":"https:\/\/matematika.besaba.com\/","name":"Matematick\u00e1 olympi\u00e1da","description":"arch\u00edv zadan\u00ed a rie\u0161en\u00ed matematickej olympi\u00e1dy","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/matematika.besaba.com\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"sk-SK"},{"@type":"Person","@id":"https:\/\/matematika.besaba.com\/#\/schema\/person\/0b550d607535616720c775cf1d895c32","name":"admin","image":{"@type":"ImageObject","inLanguage":"sk-SK","@id":"https:\/\/matematika.besaba.com\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/eb87de157eb8bac30b30a5d51e9e53a81806a5b7015a13f1242444d33c18731e?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/eb87de157eb8bac30b30a5d51e9e53a81806a5b7015a13f1242444d33c18731e?s=96&d=mm&r=g","caption":"admin"},"url":"https:\/\/matematika.besaba.com\/?author=1"}]}},"_links":{"self":[{"href":"https:\/\/matematika.besaba.com\/index.php?rest_route=\/wp\/v2\/posts\/1540","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/matematika.besaba.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/matematika.besaba.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/matematika.besaba.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/matematika.besaba.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1540"}],"version-history":[{"count":35,"href":"https:\/\/matematika.besaba.com\/index.php?rest_route=\/wp\/v2\/posts\/1540\/revisions"}],"predecessor-version":[{"id":1596,"href":"https:\/\/matematika.besaba.com\/index.php?rest_route=\/wp\/v2\/posts\/1540\/revisions\/1596"}],"wp:attachment":[{"href":"https:\/\/matematika.besaba.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1540"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/matematika.besaba.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1540"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/matematika.besaba.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1540"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}