{"id":377,"date":"2010-04-24T20:33:43","date_gmt":"2010-04-24T20:33:43","guid":{"rendered":"http:\/\/matematika.okamzite.eu\/?p=377"},"modified":"2010-04-24T20:33:43","modified_gmt":"2010-04-24T20:33:43","slug":"kategorie-z4-z9-6","status":"publish","type":"post","link":"https:\/\/matematika.besaba.com\/?p=377","title":{"rendered":"Kateg\u00f3rie Z4-Z9"},"content":{"rendered":"<p class=\"hh2\"><a name=\"top\"><\/a>55. ro\u010dn\u00edk matematickej olympi\u00e1dy 2005-2006<\/p>\n<h3 style=\"text-align: center;\">S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z4-Z9<\/h3>\n<p style=\"text-align: center;\"><script type=\"text\/javascript\"><!--\r\ngoogle_ad_client = \"pub-0508472359151885\";\r\n\/* 468x60, bola vytvoren\u00e1 12.11.2010 *\/\r\ngoogle_ad_slot = \"3127737040\";\r\ngoogle_ad_width = 468;\r\ngoogle_ad_height = 60;\r\n\/\/-->\r\n<\/script>\r\n<script type=\"text\/javascript\"\r\nsrc=\"http:\/\/pagead2.googlesyndication.com\/pagead\/show_ads.js\">\r\n<\/script><\/p>\n<table style=\"width: 98%;\" border=\"1\" cellspacing=\"2\" cellpadding=\"2\" align=\"center\">\n<tbody>\n<tr align=\"center\" bgcolor=\"#c0c0c0\">\n<td><span class=\"a\">Z4<\/span><\/td>\n<td><span class=\"a\">Z5<\/span><\/td>\n<td><span class=\"a\">Z6<\/span><\/td>\n<td><span class=\"a\">Z7<\/span><\/td>\n<td><span class=\"a\">Z8<\/span><\/td>\n<td><span class=\"a\">Z9<\/span><\/td>\n<\/tr>\n<tr align=\"center\">\n<td><span class=\"a\"><a href=\"#z41\">dom\u00e1ce<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z51\">dom\u00e1ce<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z61\">dom\u00e1ce<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z71\">dom\u00e1ce<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z81\">dom\u00e1ce<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z91\">dom\u00e1ce<\/a><\/span><\/td>\n<\/tr>\n<tr align=\"center\">\n<td><span class=\"a\"><a href=\"#z42\">\u0161kolsk\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z52\">obvodn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z62\">obvodn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z72\">obvodn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z82\">obvodn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z92\">obvodn\u00e9<\/a><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"5\" bgcolor=\"#c0c0c0\"><\/td>\n<td align=\"center\"><span class=\"a\"><a href=\"#z93\">krajsk\u00e9<\/a><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><!--more--><\/p>\n<p><a name=\"z41\"><\/a><\/p>\n<p><b>Z4-I-1<\/b><br \/>\nJurko r\u00e1d kresl\u00ed aut\u00ed\u010dka. V pondelok nama\u013eoval nieko\u013eko aut\u00ed\u010dok, v utorok ich nama\u013eoval trikr\u00e1t to\u013eko ako v pondelok, ale potom 12 pondelkov\u00fdch vygumoval. V stredu mal zl\u00fa n\u00e1ladu, ni\u010d nenama\u013eoval a e\u0161te roztrhal polovicu utorkov\u00fdch, \u010di\u017ee 24. Ko\u013eko aut\u00ed\u010dok nama\u013eoval v utorok? Ko\u013eko aut\u00ed\u010dok nama\u013eoval v pondelok, v utorok a v stredu spolu?<\/p>\n<div class=pri>(M. Dillingerov\u00e1)<\/div>\n<\/p>\n<p><b>Z4-I-2<\/b><br \/>\nZ \u010d\u00edsla <b>9 635 347<\/b> vy\u0161krtni nieko\u013eko \u010d\u00edslic tak, aby vzniklo \u010do najv\u00e4\u010d\u0161ie \u010d\u00edslo a s\u00fa\u010det v\u0161etk\u00fdch jeho \u010d\u00edslic bol men\u0161\u00ed ako 20.<\/p>\n<div class=pri>(M. Dillingerov\u00e1)<\/div>\n<\/p>\n<p><b>Z4-I-3<\/b><br \/>\nRozpr\u00e1vkov\u00fd nafukovac\u00ed \u0161tvorec, ktor\u00fd vie rozpr\u00e1va\u0165, mal pred 5 min\u00fatami d\u013a\u017eku strany 8 cm. Pri ka\u017edom klamstve zv\u00e4\u010d\u0161\u00ed svoj obvod dvojn\u00e1sobne. Pri ka\u017edej vyslovenej pravde sa zmen\u0161\u00ed d\u013a\u017eka ka\u017edej jeho strany o 2&nbsp;cm. Za posledn\u00fdch 5&nbsp;min\u00fat 2-kr\u00e1t klamal a 2-kr\u00e1t hovoril pravdu.<\/p>\n<p>&nbsp;&nbsp;&nbsp;<b>a)<\/b>&nbsp;&nbsp;Ak\u00fd najv\u00e4\u010d\u0161\u00ed obvod m\u00f4\u017ee ma\u0165 teraz?<br \/>\n&nbsp;&nbsp;&nbsp;<b>b)<\/b>&nbsp;&nbsp;Ak\u00fd najmen\u0161\u00ed obvod m\u00f4\u017ee ma\u0165 teraz?<\/p>\n<div class=pri>(S. Bodl\u00e1kov\u00e1)<\/div>\n<\/p>\n<p><b>Z4-I-4<\/b><br \/>\nPeter m\u00e1 na papier nap\u00edsa\u0165 z \u010d\u00edsel od 1 do 200 v\u0161etky tak\u00e9, ktor\u00e9 sa daj\u00fa deli\u0165 piatimi bezo zvy\u0161ku, ale nedaj\u00fa sa deli\u0165 bezo zvy\u0161ku siedmimi. Pavol m\u00e1 vyp\u00edsa\u0165 z \u010d\u00edsel od 1 do 200 v\u0161etky tak\u00e9, ktor\u00e9 po delen\u00ed siedmimi d\u00e1vaj\u00fa zvy\u0161ok 5. Ko\u013eko \u010d\u00edsel m\u00e1 nap\u00edsa\u0165 Peter a ko\u013eko Pavol?<\/p>\n<div class=pri>(M. Smitkov\u00e1)<\/div>\n<\/p>\n<p><b>Z4-I-5<\/b><br \/>\n\u017dabka Rosni\u010dka st\u00e1la na rebr\u00edku, ktor\u00fd mal 5&nbsp;prie\u010dok, na tretej prie\u010dke. Urobila \u0161es\u0165 skokov a zostala st\u00e1\u0165 na piatej prie\u010dke. Vyp\u00ed\u0161 v\u0161etky mo\u017enosti, ako mohla Rosni\u010dka sk\u00e1ka\u0165, ak v\u017edy sko\u010dila len o jednu prie\u010dku hore alebo o jednu prie\u010dku dole.<\/p>\n<div class=pri>(S. Bodl\u00e1kov\u00e1)<\/div>\n<\/p>\n<p><b>Z4-I-6<\/b><br \/>\nMajka m\u00e1 v stavebnici len rovnako ve\u013ek\u00e9 kocky s hranou d\u013a\u017eky 3&nbsp;cm. Ke\u010f z nich postav\u00ed ve\u017eu, ktor\u00e1 m\u00e1 na ka\u017edom podla\u017e\u00ed 4&nbsp;kocky, bude ma\u0165 ve\u017ea v\u00fd\u0161ku 54&nbsp;cm. Ak\u00e1 vysok\u00e1 by bola in\u00e1 ve\u017ea z tak\u00e9ho ist\u00e9ho po\u010dtu rovnak\u00fdch kociek, ktor\u00e1 by mala v ka\u017edom podla\u017e\u00ed dev\u00e4\u0165 kociek?<\/p>\n<div class=pri>(M. Dillingerov\u00e1)<\/div>\n<\/p>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tr>\n<td width=\"85%\" align=\"left\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z42\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/table>\n<p><b>Z4-II-1<\/b><br \/>\nZ \u010d\u00edsla <strong><em>9&nbsp;635&nbsp;347<\/em><\/strong> vy\u0161krtni nieko\u013eko \u010d\u00edslic tak, aby vzniklo \u010do najmen\u0161ie \u010d\u00edslo a aby s\u00fa\u010det v\u0161etk\u00fdch jeho \u010d\u00edslic bol v\u00e4\u010d\u0161\u00ed ako 23.<\/p>\n<div class=pri>(M. Dillingerov\u00e1)<\/div>\n<\/p>\n<p><b>Z4-II-2<\/b><br \/>\nRozpr\u00e1vkov\u00fd nafukovac\u00ed obd\u013a\u017enik, ktor\u00fd vie rozpr\u00e1va\u0165, mal pred 5&nbsp;min\u00fatami d\u013a\u017eky str\u00e1n 8&nbsp;cm a 10&nbsp;cm. Pri ka\u017edom klamstve zv\u00e4\u010d\u0161\u00ed jednu dvojicu svojich rovnako dlh\u00fdch str\u00e1n dvojn\u00e1sobne. Pri ka\u017edej vyslovenej pravde sa zmen\u0161\u00ed d\u013a\u017eka ka\u017edej jeho strany o&nbsp;3&nbsp;cm. Za posledn\u00fdch 5&nbsp;min\u00fat 1-kr\u00e1t klamal a 2-kr\u00e1t hovoril pravdu.<\/p>\n<p>&nbsp;&nbsp;&nbsp;<b>a)<\/b>&nbsp;&nbsp;Ak\u00fd najv\u00e4\u010d\u0161\u00ed obvod m\u00f4\u017ee ma\u0165 teraz?<br \/>\n&nbsp;&nbsp;&nbsp;<b>b)<\/b>&nbsp;&nbsp;Ak\u00fd najmen\u0161\u00ed obvod m\u00f4\u017ee ma\u0165 teraz?<\/p>\n<div class=pri>(S. Bodl\u00e1kov\u00e1, M. Dillingerov\u00e1)<\/div>\n<\/p>\n<p><b>Z4-II-3<\/b><br \/>\nKubkovi sa p\u00e1\u010dia \u010d\u00edsla, ktor\u00e9 d\u00e1vaj\u00fa po delen\u00ed piatimi zvy\u0161ok&nbsp;1. Luk\u00e1\u0161kovi sa p\u00e1\u010dia \u010d\u00edsla, ktor\u00e9 nemaj\u00fa \u017eiadne dve \u010d\u00edslice rovnak\u00e9. Nap\u00ed\u0161 najmen\u0161ie trojcifern\u00e9 \u010d\u00edslo, ktor\u00e9 sa bude p\u00e1\u010di\u0165 aj Kubkovi aj Luk\u00e1\u0161kovi.<\/p>\n<div class=pri>(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n<\/p>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tr>\n<td width=\"85%\" align=\"left\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z51\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/table>\n<p><b>Z5-I-1<\/b><br \/>\nDopl\u0148 do pr\u00e1zdnych pol\u00ed\u010dok prirodzen\u00e9 \u010d\u00edsla od 1 do 20 (ka\u017ed\u00e9 \u010d\u00edslo m\u00f4\u017ee\u0161 pou\u017ei\u0165 len raz) tak, aby platili matematick\u00e9 vz\u0165ahy.<\/p>\n<p class=pc><img decoding=\"async\" height=158 alt=\"\" src=\"\/55\/55z5a1.gif\" width=280 border=0><\/p>\n<div class=pri>(M. Smitkov\u00e1)<\/div>\n<\/p>\n<p><b>Z5-I-2<\/b><br \/>\nBlcha Sk\u00e1kalka sk\u00e1\u010de po vodorovnej \u010d\u00edselnej osi. Vie robi\u0165 iba skoky dvoch d\u013a\u017eok. Jedn\u00fdm presko\u010d\u00ed o 14 doprava alebo do\u013eava, druh\u00fdm presko\u010d\u00ed o 18 doprava alebo do\u013eava. Pr\u00e1ve stoj\u00ed na \u010d\u00edsle 2.<\/p>\n<p>&nbsp;&nbsp;&nbsp;<strong>a)<\/strong>&nbsp;&nbsp;N\u00e1jdi sp\u00f4sob, ako m\u00e1 sk\u00e1ka\u0165, aby sa \u0161tyrmi skokmi dostala na \u010d\u00edslo 10.<br \/>\n&nbsp;&nbsp;&nbsp;<strong>b)<\/strong>&nbsp;&nbsp;Tvrd\u00ed, \u017ee v\u010dera bola na \u010d\u00edsle 13. Hovor\u00ed pravdu alebo klame? Zd\u00f4vodni.<\/p>\n<div class=pri>(M. Dillingerov\u00e1)<\/div>\n<\/p>\n<p><b>Z5-I-3<\/b><br \/>\nRozpr\u00e1vkov\u00fd nafukovac\u00ed \u0161tvorec, ktor\u00fd vie rozpr\u00e1va\u0165, mal pred 5 min\u00fatami d\u013a\u017eku strany 8&nbsp;cm. Pri ka\u017edom klamstve zv\u00e4\u010d\u0161\u00ed svoj obvod dvojn\u00e1sobne. Pri ka\u017edej vyslovenej pravde sa zmen\u0161\u00ed d\u013a\u017eka ka\u017edej jeho strany o 2&nbsp;cm. Za posledn\u00fdch 5&nbsp;min\u00fat 2-kr\u00e1t klamal a 2-kr\u00e1t hovoril pravdu.<\/p>\n<p>&nbsp;&nbsp;&nbsp;<strong>a)<\/strong>&nbsp;&nbsp;Ak\u00fd najv\u00e4\u010d\u0161\u00ed obvod m\u00f4\u017ee ma\u0165 teraz?<br \/>\n&nbsp;&nbsp;&nbsp;<strong>b)<\/strong>&nbsp;&nbsp;Ak\u00fd najmen\u0161\u00ed obvod m\u00f4\u017ee ma\u0165 teraz?<\/p>\n<div class=pri>(S. Bodl\u00e1kov\u00e1)<\/div>\n<\/p>\n<p><b>Z5-I-4<\/b><br \/>\nPe\u0165o si k\u00fapil na jarmoku 4 aut\u00ed\u010dka, biele, zelen\u00e9, \u010derven\u00e9 a modr\u00e9. Biele aut\u00ed\u010dko st\u00e1lo dvakr\u00e1t to\u013eko ako \u010derven\u00e9, zelen\u00e9 trikr\u00e1t to\u013eko ako biele a za modr\u00e9 zaplatil to\u013eko ako za \u010derven\u00e9 a biele spolu. Pritom \u010derven\u00e9 st\u00e1lo o 70&nbsp;Sk menej ako zelen\u00e9. Ko\u013eko Sk st\u00e1li jednotliv\u00e9 aut\u00ed\u010dka?<\/p>\n<div class=pri>(\u0160. Pt\u00e1\u010dkov\u00e1)<\/div>\n<\/p>\n<p><b>Z5-I-5<\/b><br \/>\nMama stono\u017eka m\u00e1 dve deti a man\u017eela. Ka\u017ed\u00fd z nich m\u00e1 sto n\u00f4h a ka\u017ed\u00fd \u010dlen rodiny si berie ka\u017ed\u00fd de\u0148 \u010dist\u00e9 pono\u017eky. V sobotu r\u00e1no o 6:00 za\u010dala mami\u010dka stono\u017eka pra\u0165 \u0161pinav\u00e9 pono\u017eky. Naraz sa jej ich do pr\u00e1\u010dky zmest\u00ed 357 a jedna v\u00e1rka sa operie za dve a pol hodiny. Zisti, kedy skon\u010d\u00ed s pran\u00edm, ak vie\u0161, \u017ee pono\u017eky perie raz za t\u00fd\u017ede\u0148, ulo\u017eenie pono\u017eiek do pr\u00e1\u010dky jej trv\u00e1 2 min\u00faty a ich vybratie z pr\u00e1\u010dky 3 min\u00faty.<\/p>\n<div class=pri>(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n<\/p>\n<p><b>Z5-I-6<\/b><br \/>\nMami\u010dka m\u00e1 v chladni\u010dke tehlu syra (vi\u010f obr\u00e1zok). Postupne z nej odrez\u00e1va 1&nbsp;cm hrub\u00e9 pl\u00e1tky na vypr\u00e1\u017eanie. Najprv odrezala spredu pl\u00e1tok s rozmermi pre otecka, potom odrezala zboku pre Jurka, zozadu pre seba a nakoniec z druh\u00e9ho boku pre Ani\u010dku. Nap\u00ed\u0161, ak\u00e9 rozmery maj\u00fa jednotliv\u00e9 pl\u00e1tky. Ur\u010di rozmery zvy\u0161ku syra.<\/p>\n<p class=pc><img decoding=\"async\" height=166 alt=\"\" src=\"\/55\/55z5a2.gif\" width=363 border=0><\/p>\n<div class=pri>(M. Dillingerov\u00e1)<\/div>\n<\/p>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tr>\n<td width=\"85%\" align=\"left\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z52\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/table>\n<p><b>Z5-II-1<\/b><br \/>\nDopl\u0148 do pr\u00e1zdnych pol\u00ed\u010dok prirodzen\u00e9 \u010d\u00edsla od 1 do 16 (ka\u017ed\u00e9 \u010d\u00edslo m\u00f4\u017ee\u0161 pou\u017ei\u0165 len raz) tak, aby platili matematick\u00e9 vz\u0165ahy.<\/p>\n<div align=\"center\"><img decoding=\"async\" height=158 alt=\"\" src=\"\/55\/55z5b1.gif\" width=217 border=0><\/div>\n<div class=pri>(M. Smitkov\u00e1)<\/div>\n<\/p>\n<p><b>Z5-II-2<\/b><br \/>\nPeter zlepil 17 hrac\u00edch kociek do hada (ako na obr\u00e1zku). Kocky zliepal v\u017edy tak, \u017ee zlepil steny s rovnak\u00fdm po\u010dtom bodiek. Potom si hada obzrel zo v\u0161etk\u00fdch str\u00e1n a spr\u00e1vne spo\u010d\u00edtal v\u0161etky bodky na jeho povrchu. Ko\u013eko mu vy\u0161lo? (S\u00fa\u010det bodiek na proti\u013eahl\u00fdch sten\u00e1ch hracej kocky je 7.)<\/p>\n<p class=\"pc\"><img decoding=\"async\" height=74 alt=\"\" src=\"\/55\/55z5b2.gif\" width=286 border=0><\/div>\n<div class=pri>(S. Bodl\u00e1kov\u00e1, M. Dillingerov\u00e1)<\/div>\n<\/p>\n<p><b>Z5-II-3<\/b><br \/>\nMysl\u00edm si trojcifern\u00e9 prirodzen\u00e9 \u010d\u00edslo men\u0161ie ako 200. Ak jeho trojn\u00e1sobok zaokr\u00fahlim na stovky, zv\u00e4\u010d\u0161\u00ed sa o 36. Ktor\u00e9 \u010d\u00edslo si mysl\u00edm?<\/p>\n<div class=pri>(M. Dillingerov\u00e1)<\/div>\n<\/p>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tr>\n<td width=\"85%\" align=\"left\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z61\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/table>\n<p><b>Z6-I-1<\/b><br \/>\nDopl\u0148 do pr\u00e1zdnych pol\u00ed\u010dok prirodzen\u00e9 \u010d\u00edsla od 1 do 20 (ka\u017ed\u00e9 \u010d\u00edslo m\u00f4\u017ee\u0161 pou\u017ei\u0165 len raz) tak, aby platili matematick\u00e9 vz\u0165ahy.<\/p>\n<p class=pc><img decoding=\"async\" height=158 alt=\"\" src=\"\/55\/55z5a1.gif\" width=280 border=0><\/p>\n<div class=pri>(M. Smitkov\u00e1)<\/div>\n<\/p>\n<p><b>Z6-I-2<\/b><br \/>\nSnehulienka so siedmimi trpasl\u00edkmi zbierali lieskov\u00e9 orie\u0161ky. Snehulienka ich mala to\u013eko ako v\u0161etci trpasl\u00edci spolu. Ke\u010f sa vracali domov, stretli veveri\u010dku Finku. Snehulienka aj ka\u017ed\u00fd trpasl\u00edk jej dali rovnak\u00fd po\u010det orie\u0161kov. Doma trpasl\u00edci a Snehulienka vysypali orie\u0161ky na k\u00f4pky na stole, ka\u017ed\u00fd na in\u00fa k\u00f4pku a Vedko zap\u00edsal po\u010dty orie\u0161kov v k\u00f4pkach: 120, 316, 202, 185, 333, 297, 111 a 1672. Ko\u013eko orie\u0161kov dostala veveri\u010dka Finka?<\/p>\n<div class=pri>(L. Hozov\u00e1)<\/div>\n<\/p>\n<p><b>Z6-I-3<\/b><br \/>\nKe\u010f sme \u010d\u00edsla 80 a 139 vydelili t\u00fdm ist\u00fdm prirodzen\u00fdm \u010d\u00edslom, z\u00edskali sme zvy\u0161ky 8 a 13. Ktor\u00fdm \u010d\u00edslom sme delili?<\/p>\n<div class=pri>(M. Volfov\u00e1)<\/div>\n<\/p>\n<p><b>Z6-I-4<\/b><br \/>\nObvod trojuholn\u00edka je 16&nbsp;cm. Ak\u00e9 m\u00f4\u017eu by\u0165 d\u013a\u017eky jeho str\u00e1n, ke\u010f s\u00fa to prirodzen\u00e9 \u010d\u00edsla a s\u00fa\u010det d\u013a\u017eok dvoch str\u00e1n je o&nbsp;6&nbsp;cm v\u00e4\u010d\u0161\u00ed ako d\u013a\u017eka tretej strany?<\/p>\n<div class=pri>(L. Hozov\u00e1)<\/div>\n<\/p>\n<p><b>Z6-I-5<\/b><br \/>\nMaru\u0161ka dostala p\u00e4\u0165 r\u00f4zne \u0165a\u017ek\u00fdch kol\u00e1\u010dov. Priemern\u00e1 hmotnos\u0165 kol\u00e1\u010da bola 200&nbsp;gramov. Maru\u0161ka jeden kol\u00e1\u010d zjedla a priemern\u00e1 hmotnos\u0165 zvy\u0161n\u00fdch kol\u00e1\u010dov potom bola 160&nbsp;gramov. Ko\u013eko gramov v\u00e1\u017eil kol\u00e1\u010d, ktor\u00fd Maru\u0161ka zjedla?<\/p>\n<div class=pri>(B. \u0160\u0165astn\u00e1)<\/div>\n<\/p>\n<p><b>Z6-I-6<\/b><br \/>\nUr\u010dte ve\u013ekos\u0165 plochy \u0161ed\u00e9ho \u0161tvoruholn\u00edka na obr\u00e1zku, kde v\u0161etky rozmery s\u00fa ud\u00e1van\u00e9 v centimetroch.<\/p>\n<p class=pc><img decoding=\"async\" height=159 alt=\"\" src=\"\/55\/55z6a1.gif\" width=171 border=0><\/p>\n<div class=pri>(P. Tlust\u00fd)<\/div>\n<\/p>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tr>\n<td width=\"85%\" align=\"left\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z62\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/table>\n<p><b>Z6-II-1<\/b><br \/>\nP\u00e1n \u0160ikovn\u00fd chcel v k\u00fape\u013eni nafarbi\u0165 na 20 dla\u017ed\u00edc hviezdicov\u00e9 ornamenty (pozri jednu dla\u017edicu na obr\u00e1zku). Na plechovke s farbou bolo nap\u00edsan\u00e9, \u017ee farba vysta\u010d\u00ed na 750&nbsp;cm<sup>2<\/sup>. Ko\u013eko najmenej plechoviek musel p\u00e1n \u0160ikovn\u00fd k\u00fapi\u0165, ak jeden \u0161tvor\u010dek siete m\u00e1 obsah 1&nbsp;cm<sup>2<\/sup>.<\/p>\n<p class=pc><img loading=\"lazy\" decoding=\"async\" height=\"177\" alt=\"\" src=\"\/55\/55z6b1.gif\" width=\"177\" border=\"0\"><\/p>\n<div class=pri>(L. Hozov\u00e1)<\/div>\n<\/p>\n<p><b>Z6-II-2<\/b><br \/>\nV jaskyni \u017eije p\u00e4\u0165 obrov. Ich priemern\u00e1 v\u00fd\u0161ka je 15&nbsp;504,63&nbsp;cm. Obor Drob\u010dek meria 174,53&nbsp;m, obor Lomikame\u0148 173&nbsp;530,5&nbsp;mm, obor Zlobor 1&nbsp;745,23&nbsp;dm, obor Hromp\u00e1c 0,017&nbsp;34&nbsp;km.<\/p>\n<p>&nbsp;&nbsp;&nbsp;<strong>a)<\/strong>&nbsp;&nbsp;Zisti ko\u013eko meria piaty obor &#8211; Kolodej.<br \/>\n&nbsp;&nbsp;&nbsp;<strong>b)<\/strong>&nbsp;&nbsp;Zora\u010f obrov pod\u013ea ve\u013ekosti od najmen\u0161ieho po najv\u00e4\u010d\u0161ieho.<\/p>\n<div class=pri>(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n<\/p>\n<p><b>Z6-II-3<\/b><br \/>\nSnehulienka dala ka\u017ed\u00e9mu trpasl\u00edkovi rovnako ve\u013ea gu\u013e\u00f4\u010diek. Trpasl\u00edci potom h\u00e1dzali kockou. Ko\u013eko im padlo bodiek, to\u013eko gu\u013e\u00f4\u010diek museli vr\u00e1ti\u0165 Snehulienke. Ka\u017ed\u00e9mu z prv\u00fdch \u0161iestich trpasl\u00edkov padol in\u00fd po\u010det bodiek, tak\u017ee ka\u017ed\u00fd z nich vr\u00e1til Snehulienke in\u00fd po\u010det gu\u013e\u00f4\u010diek. Ko\u013eko bodiek padlo posledn\u00e9mu trpasl\u00edkovi, ak v\u0161etk\u00fdm siedmim trpasl\u00edkom po tomto hode ostalo spolu 46 gu\u013e\u00f4\u010diek? Ko\u013eko gu\u013e\u00f4\u010diek dala Snehulienka ka\u017ed\u00e9mu trpasl\u00edkovi na za\u010diatku?<\/p>\n<div class=pri>(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n<\/p>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tr>\n<td width=\"85%\" align=\"left\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z71\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/table>\n<p><b>Z7-I-1<\/b><br \/>\nPat a Mat upravovali nov\u00fd asfalt na ceste. Najprv i\u0161li s valcom 10&nbsp;m dopredu, potom 7&nbsp;m c\u00favli, op\u00e4\u0165 pre\u0161li 10&nbsp;m dopredu a 7&nbsp;m c\u00favli&#8230; Takto pokra\u010dovali, a\u017e k\u00fdm po prv\u00fd raz nezi\u0161li z nov\u00e9ho asfaltu.<\/p>\n<p>&nbsp;&nbsp;&nbsp;<strong>a)<\/strong>&nbsp;&nbsp;Ko\u013eko metrov najazdil valec na novom 540 metrovom \u00faseku cesty?<br \/>\n&nbsp;&nbsp;&nbsp;<strong>b)<\/strong>&nbsp;&nbsp;Ko\u013ekokr\u00e1t pre\u0161li po dev\u00e4tn\u00e1stom metri nov\u00e9ho asfaltu?<\/p>\n<div class=pri>(M. Dillingerov\u00e1)<\/div>\n<\/p>\n<p><b>Z7-I-2<\/b><br \/>\nZistite obsah a ve\u013ekosti v\u0161etk\u00fdch vn\u00fatorn\u00fdch uhlov mnohouholn\u00edka zn\u00e1zornen\u00e9ho v koso\u0161tvorcovej sieti na obr\u00e1zku, ak viete, \u017ee priamky siete zvieraj\u00fa uhol 80&nbsp;stup\u0148ov a jeden mal\u00fd koso\u0161tvor\u010dek m\u00e1 obsah 1&nbsp;cm<sup>2<\/sup>.<\/p>\n<p>(Pozor, obr\u00e1zok je nepresn\u00fd, neoplat\u00ed sa  mera\u0165.)<img loading=\"lazy\" decoding=\"async\" height=\"146\" src=\"\/55\/55z7a1.gif\" align=\"center\" width=\"219\" border=\"0\"><\/p>\n<div class=pri>(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n<\/p>\n<p><b>Z7-I-3<\/b><br \/>\nNa obr\u00e1zku vid\u00edte \u0161peci\u00e1lny druh rol\u00e1dy, tzv. <em>KVADRIL\u00c1DU<\/em>. Je vyroben\u00e1 z bielej a hnedej marcip\u00e1novej hmoty, pri\u010dom obe hmoty maj\u00fa rovnak\u00fa hr\u00fabku 1&nbsp;cm. Cel\u00e1 kvadril\u00e1da m\u00e1 d\u013a\u017eku 15&nbsp;cm. Pred\u00e1va sa rozkr\u00e1jan\u00e1 na 10 rovnak\u00fdch pl\u00e1tkov, pri\u010dom sa kr\u00e1ja rovnobe\u017ene s prednou stenou. Zistite <\/p>\n<p>&nbsp;&nbsp;&nbsp;<strong>a)<\/strong>&nbsp;&nbsp;rozmery jedn\u00e9ho pl\u00e1tku,<br \/>\n&nbsp;&nbsp;&nbsp;<strong>b)<\/strong>&nbsp;&nbsp;ko\u013eko gramov hnedej hmoty a ko\u013eko gramov bielej hmoty treba na jej pr\u00edpravu, ak  viete, \u017ee jeden cm<sup>3<\/sup> marcip\u00e1novej hmoty v\u00e1\u017ei 2&nbsp;gramy.<\/p>\n<p>(Neozna\u010den\u00e1 \u0161\u00edpka v obr\u00e1zku m\u00e1 d\u013a\u017eku 2 cm.)<\/p>\n<p class=pc><img decoding=\"async\" height=286 alt=\"\" src=\"\/55\/55z7a2.gif\" width=298 border=0><\/p>\n<div class=pri>(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n<\/p>\n<p><b>Z7-I-4<\/b><br \/>\nN\u00e1jdite v\u0161etky p\u00e4\u0165cifern\u00e9 \u010d\u00edsla, ktor\u00e9 sa \u0161krtnut\u00edm prvej a poslednej cifry zmen\u0161ia 250-kr\u00e1t.<\/p>\n<div class=pri>(L. \u0160im\u016fnek)<\/div>\n<\/p>\n<p><b>Z7-I-5<\/b><br \/>\nPavol dostal na dom\u00e1cu \u00falohu vyjadri\u0165 desatinn\u00fdm \u010d\u00edslom zlomky <img decoding=\"async\" height=21 alt=\"\" src=\"\/55\/55z7a3.gif\" width=11 align=absMiddle border=0> a <img decoding=\"async\" height=21 alt=\"\" src=\"\/55\/55z7a4.gif\" width=15 align=absMiddle border=0>. Aby urobil pani u\u010dite\u013eke rados\u0165, p\u00edsal \u00falohu miesto do zo\u0161ita na latky \u0161kolsk\u00e9ho plota. Najprv vyjadroval <img decoding=\"async\" height=21 alt=\"\" src=\"\/55\/55z7a3.gif\" width=11 align=absMiddle border=0>, teda na prv\u00fa latku navrch nap\u00edsal 0, na druh\u00fa desatinn\u00fa \u010diarku, na tretiu 4 at\u010f. Ke\u010f skon\u010dil, nap\u00edsal pod tieto \u010d\u00edsla vyjadrenie zlomku <img decoding=\"async\" height=21 alt=\"\" src=\"\/55\/55z7a4.gif\" width=15 align=absMiddle border=0>. Na prv\u00fa latku dole nap\u00edsal 0, na druh\u00fa desatinn\u00fa \u010diarku, na tretiu 5 at\u010f. Ko\u013eko bolo latiek na plote, ak viete, \u017ee \u010d\u00edslicu 5 nap\u00edsal presne 667 kr\u00e1t a \u017ee na 668 latk\u00e1ch bola dvojica rovnak\u00fdch \u010d\u00edsel.<\/p>\n<div class=pri>(M. Dillingerov\u00e1, P. Tlust\u00fd)<\/div>\n<\/p>\n<p><b>Z7-I-6<\/b><br \/>\nV Koc\u00farkove maj\u00fa dve zmen\u00e1rne. V s\u00fa\u010dasnosti maj\u00fa tak\u00e9to kurzy:<\/p>\n<table class=pc width=\"450\" align=\"center\" cellspacing=\"1\" cellpadding=\"1\" border=\"1\">\n<tr>\n<td colspan=\"3\" bgcolor=\"#dcdcc0\"><strong>1. zmen\u00e1re\u0148<\/strong><\/td>\n<td colspan=\"3\" bgcolor=\"#c0dcdc\"><strong>2. zmen\u00e1re\u0148<\/strong><\/td>\n<\/tr>\n<tr>\n<td bgcolor=\"#dcdcc0\">MENA<\/td>\n<td bgcolor=\"#dcdcc0\">N\u00c1KUP<\/td>\n<td bgcolor=\"#dcdcc0\">PREDAJ<\/td>\n<td bgcolor=\"#c0dcdc\">MENA<\/td>\n<td bgcolor=\"#c0dcdc\">N\u00c1KUP<\/td>\n<td bgcolor=\"#c0dcdc\">PREDAJ<\/td>\n<\/tr>\n<tr>\n<td bgcolor=\"#dcdcc0\">1 euro<\/td>\n<td bgcolor=\"#dcdcc0\">123 Kk<\/td>\n<td bgcolor=\"#dcdcc0\">132 Kk<\/td>\n<td bgcolor=\"#c0dcdc\">1 euro<\/td>\n<td bgcolor=\"#c0dcdc\">143 Kk<\/td>\n<td bgcolor=\"#c0dcdc\">154 Kk<\/td>\n<\/tr>\n<\/table>\n<p>\nSlavo \u0160pakulat\u00fd mal nieko\u013eko euro. V 2.&nbsp;zmen\u00e1rni ich zamenil za Koc\u00farkovsk\u00e9 koruny (Kk) a tie potom zamenil v 1.&nbsp;zmen\u00e1rni sp\u00e4\u0165 na euro. Takto zarobil 1&nbsp;euro. Ko\u013eko euro mal na za\u010diatku?<\/p>\n<div class=pri>(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n<\/p>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tr>\n<td width=\"85%\" align=\"left\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z72\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/table>\n<p><b>Z7-II-1<\/b><br \/>\nSie\u0165 na obr\u00e1zku je zlo\u017een\u00e1 zo zhodn\u00fdch rovnoramenn\u00fdch trojuholn\u00edkov. Ka\u017ed\u00fd z nich m\u00e1 obsah 2&nbsp;cm<sup>2<\/sup> a jeden z vn\u00fatorn\u00fdch uhlov trojuholn\u00edkov meria 100\u00b0. Zisti obsah a ve\u013ekos\u0165 vn\u00fatorn\u00fdch uhlov mnohouholn\u00edka zn\u00e1zornen\u00e9ho v tejto trojuholn\u00edkovej sieti. (Pozor, obr\u00e1zok m\u00f4\u017ee by\u0165 ve\u013emi nepresn\u00fd.)<\/p>\n<div class=pc><img loading=\"lazy\" decoding=\"async\" height=\"186\" alt=\"\" src=\"\/55\/55z7b1.gif\" width=\"146\" border=\"0\" vspace=\"15\"><\/div>\n<\/p>\n<p><b>Z7-II-2<\/b><br \/>\nN\u00e1jdi v\u0161etky \u0161tvorcifern\u00e9 \u010d\u00edsla, ktor\u00e9 sa \u0161krtnut\u00edm prostredn\u00fdch dvoch cifier zmen\u0161ia 120-kr\u00e1t.<\/p>\n<p><b>Z7-II-3<\/b><br \/>\nKr\u00e1\u013e Origin\u00e1l&nbsp;I. si s\u00e1m navrhol a dal vybudova\u0165 baz\u00e9n. Na obr\u00e1zku vid\u00ed\u0161, ako vyzer\u00e1 jeho baz\u00e9n pri poh\u013eade zhora. Dno je pokryt\u00e9 \u0161tvorcov\u00fdmi dla\u017edicami so stranou d\u013a\u017eky 2&nbsp;m, bet\u00f3nov\u00e9 steny s\u00fa 3&nbsp;m vysok\u00e9 a 0,5&nbsp;m hrub\u00e9. Zisti, ko\u013eko kg bet\u00f3nu spotrebovali na steny baz\u00e9nu, ak 1&nbsp;m<sup>3<\/sup> bet\u00f3nu m\u00e1 hmotnos\u0165 2000&nbsp;kg.<\/p>\n<div class=pc><img loading=\"lazy\" decoding=\"async\" height=\"180\" alt=\"\" src=\"\/55\/55z7b2.gif\" width=\"409\" border=\"0\" vspace=\"15\"><\/div>\n<\/p>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tr>\n<td width=\"85%\" align=\"left\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z81\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/table>\n<p><b>Z8-I-1<\/b><br \/>\nUr\u010dte v\u0161etky dvojcifern\u00e9ho \u010d\u00edsla, pre ktor\u00e9 s\u00fa\u010din cifern\u00e9ho s\u00fa\u010dtu a cifern\u00e9ho s\u00fa\u010dinu je 126.<\/p>\n<div class=pri>(M. Raabov\u00e1)<\/div>\n<\/p>\n<p><b>Z8-I-2<\/b><br \/>\nPani Zru\u010dn\u00e1 sa uch\u00e1dzala o miesto vo v\u00fdrobe viano\u010dn\u00fdch pern\u00edkov. Pri pohovore s ved\u00facim chcela poveda\u0165, ko\u013eko pern\u00edkov ozdob\u00ed za ko\u013eko min\u00fat. Bola nerv\u00f3zna, a preto omylom prehodila po\u010det min\u00fat s po\u010dtom pern\u00edkov. Ved\u00faci pod\u013ea jej \u00fadajov vypo\u010d\u00edtal, ko\u013eko pern\u00edkov by mala pani Zru\u010dn\u00e1 stihn\u00fa\u0165 ozdobi\u0165 za p\u00e4\u0165hodinov\u00fa pracovn\u00fa dobu, a presne tento po\u010det jej dal za \u00falohu ozdobi\u0165. Pani Zru\u010dnej trvala pr\u00e1ca o&nbsp;2&nbsp;hodiny a 12&nbsp;min\u00fat dlh\u0161ie. Ko\u013eko pern\u00edkov ozdobila?<\/p>\n<div class=pri>(L. \u0160im\u016fnek)<\/div>\n<\/p>\n<p><b>Z8-I-3<\/b><br \/>\nNa obr\u00e1zku vid\u00edte \u0161peci\u00e1lny druh rol\u00e1dy, tzv. <em>KVADRIL\u00c1DU<\/em>. Je vyroben\u00e1 z bielej a hnedej marcip\u00e1novej hmoty, pri\u010dom obe hmoty maj\u00fa rovnak\u00fa hr\u00fabku 1&nbsp;cm. Cel\u00e1 kvadril\u00e1da m\u00e1 d\u013a\u017eku 15&nbsp;cm. Pred\u00e1va sa rozkr\u00e1jan\u00e1 na 10 rovnak\u00fdch pl\u00e1tkov, pri\u010dom sa kr\u00e1ja rovnobe\u017ene s prednou stenou. Zistite <\/p>\n<p>&nbsp;&nbsp;&nbsp;<strong>a)<\/strong>&nbsp;&nbsp;rozmery jedn\u00e9ho pl\u00e1tku,<br \/>\n&nbsp;&nbsp;&nbsp;<strong>b)<\/strong>&nbsp;&nbsp;ko\u013eko gramov hnedej hmoty a ko\u013eko gramov bielej hmoty treba na jej pr\u00edpravu, ak  viete, \u017ee jeden cm<sup>3<\/sup> marcip\u00e1novej hmoty v\u00e1\u017ei 2&nbsp;gramy.<\/p>\n<p>(Neozna\u010den\u00e1 \u0161\u00edpka v obr\u00e1zku m\u00e1 d\u013a\u017eku 2 cm.)<\/p>\n<div class=pc><img decoding=\"async\" height=286 alt=\"\" src=\"\/55\/55z7a2.gif\" width=298 border=0><\/div>\n<div class=pri>(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n<\/p>\n<p><b>Z8-I-4<\/b><br \/>\nRoman p\u00edsal na papier za sebou cel\u00e9 \u010d\u00edsla tak, \u017ee nasleduj\u00face z\u00edskal z predch\u00e1dzaj\u00faceho striedavo n\u00e1soben\u00edm dvoma a od\u010d\u00edtan\u00edm troch. Napr. postupnos\u0165 \u010d\u00edsel 1; 2; -1; -2; -5; -10 vyhovuje jeho pravidlu, ale postupnos\u0165 10; 7; 4; 8; 16; 32 jeho pravidlo nespl\u0148uje. Po chv\u00edli s\u010d\u00edtal posledn\u00fdch 5 \u010d\u00edsel, ktor\u00e9 nap\u00edsal a vy\u0161lo mu 114. Ktor\u00fdch 5 \u010d\u00edsel s\u010ditoval?<\/p>\n<div class=pri>(M. Raabov\u00e1)<\/div>\n<\/p>\n<p><b>Z8-I-5<\/b><br \/>\nUr\u010dte polomer v\u00e4\u010d\u0161ej kru\u017enice na obr\u00e1zku, ak ka\u017ed\u00e1 mal\u00e1 kru\u017enica m\u00e1 polomer 1 cm.<\/p>\n<div class=pc><img decoding=\"async\" height=125 alt=\"\" src=\"\/55\/55z8a1.gif\" width=125 vspace=8 border=0><\/div>\n<div class=pri>(P. Tlust\u00fd)<\/div>\n<\/p>\n<p><b>Z8-I-6<\/b><br \/>\nJakub m\u00e1 tento \u0161kolsk\u00fd rok priemer v\u0161etk\u00fdch svojich zn\u00e1mok 1,85. Za cel\u00fd \u0161kolsk\u00fd rok dostal iba \u0161tyri p\u00e4torky a pr\u00e1ve tretina jeho zn\u00e1mok boli jednotky. Najmenej ko\u013eko zn\u00e1mok musel tento \u0161kolsk\u00fd rok dosta\u0165?<\/p>\n<div class=pri>(L. \u0160im\u016fnek)<\/div>\n<\/p>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tr>\n<td width=\"85%\" align=\"left\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z82\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/table>\n<p><b>Z8-II-1<\/b><br \/>\nKa\u017ed\u00e9 p\u00edsmeno tabu\u013eky nahr\u00e1dza ur\u010dit\u00e9 prirodzen\u00e9 \u010d\u00edslo (rovnak\u00e9 p\u00edsmen\u00e1 nahr\u00e1dzaj\u00fa rovnak\u00e9 \u010d\u00edsla). S\u00fa\u010det \u010d\u00edsel v tre\u0165om riadku je 50. S\u00fa\u010det \u010d\u00edsel vo \u0161tvrtom riadku je 45. S\u00fa\u010det \u010d\u00edsel v prvom st\u013apci je 85. Ur\u010dte aspo\u0148 jednu vyhovuj\u00facu \u0161tvoricu \u010d\u00edsel, ktorou mo\u017eno nahradi\u0165 p\u00edsmen\u00e1 A, B, C, D. Ur\u010dte s\u00fa\u010det v\u0161etk\u00fdch \u010d\u00edsel v tabu\u013eke.<\/p>\n<table bordercolor=\"#511566\" align=\"center\" width=\"120\" height=\"120\" cellspacing=\"2\" cellpadding=\"2\" border=\"1\" class=\"a\">\n<tr>\n<td width=\"30\" align=\"center\">D<\/td>\n<td width=\"30\" align=\"center\">A<\/td>\n<td width=\"30\" align=\"center\">C<\/td>\n<td width=\"30\" align=\"center\">B<\/td>\n<\/tr>\n<tr>\n<td>C<\/td>\n<td>B<\/td>\n<td>D<\/td>\n<td>A<\/td>\n<\/tr>\n<tr>\n<td>B<\/td>\n<td>C<\/td>\n<td>B<\/td>\n<td>C<\/td>\n<\/tr>\n<tr>\n<td>D<\/td>\n<td>A<\/td>\n<td>A<\/td>\n<td>A<\/td>\n<\/tr>\n<\/table>\n<p><b>Z8-II-2<\/b><br \/>\nM\u00e1me obrazec tvaru <em>T<\/em> zlo\u017een\u00fd z 22&nbsp;\u0161tvor\u010dekov so stranou d\u013a\u017eky 1&nbsp;cm. Bodom V sme viedli priamku <em>p<\/em>, ktor\u00e1 rozdelila obrazec na dve \u010dasti rovnak\u00e9ho obsahu. V akom pomere rozde\u013euje priamka <em>p<\/em> \u00fase\u010dku <em>AB<\/em>?<\/p>\n<div class=pc><img loading=\"lazy\" decoding=\"async\" src=\"\/55\/55z8b1.gif\" alt=\"\" width=\"201\" height=\"206\" vspace=\"15\" border=\"0\"><\/div>\n<\/p>\n<p><b>Z8-II-3<\/b><br \/>\nMajka vytvorila postupnos\u0165 \u010d\u00edsel, v ktorej je ka\u017ed\u00e9 \u010d\u00edslo (s v\u00fdnimkou prv\u00e9ho) s\u00fa\u010dtom druh\u00fdch mocn\u00edn \u010d\u00edslic predch\u00e1dzaj\u00faceho \u010d\u00edsla. Vyp\u00ed\u0161 prv\u00fdch 10 \u010dlenov tejto postupnosti, ak jej prv\u00fdm \u010dlenom je \u010d\u00edslo 29. Ktor\u00e9 \u010d\u00edslo je v postupnosti na 2006. mieste?<\/p>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tr>\n<td width=\"85%\" align=\"left\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z91\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/table>\n<p><b>Z9-I-1<\/b><br \/>\nUr\u010dte po\u010det trojcifern\u00fdch prirodzen\u00fdch \u010d\u00edsel, ktor\u00e9 maj\u00fa pr\u00e1ve dve rovnak\u00e9 cifry.<\/p>\n<div class=pri>(P. Tlust\u00fd)<\/div>\n<\/p>\n<p><b>Z9-I-2<\/b><br \/>\nNa obr\u00e1zku s\u00fa tri rovnostrann\u00e9 trojuholn\u00edky, jedna ve\u013ek\u00e1 polkru\u017enica s polomerom 1&nbsp;dm a tri mal\u00e9 polkru\u017enice. Ur\u010dte d\u013a\u017eku \u00fase\u010dky <em>AB<\/em>.<\/p>\n<div class=pc><img decoding=\"async\" height=80 alt=\"\" src=\"\/55\/55z9a1.gif\" width=159 border=0><\/div>\n<div class=pri>(P. Tlust\u00fd)<\/div>\n<\/p>\n<p><b>Z9-I-3<\/b><br \/>\nV s\u00faradnicovej s\u00fastave sme zn\u00e1zornili body <em>A<\/em>[3; 2], <em>B<\/em>[-1; 1], <em>C<\/em>[-2; 4] a ich obrazy <em>A\u00b4, B\u00b4, C\u00b4<\/em> v stredovej s\u00famernosti so stredom v za\u010diatku s\u00faradnicovej s\u00fastavy. Vypo\u010d\u00edtajte obsah \u0161es\u0165uholn\u00edka <em>ABCA\u00b4B\u00b4C\u00b4<\/em>.<\/p>\n<div class=pri>(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n<\/p>\n<p><b>Z9-I-4<\/b><br \/>\nStar\u00fd podnikate\u013e zomrel a zanechal po sebe dva \u00fa\u010dty, jeden dlh a testament. V testamente sa p\u00edsalo, \u017ee peniaze z prv\u00e9ho \u00fa\u010dtu si maj\u00fa rozdeli\u0165 1. a 2. syn v pomere 1:2, peniaze z druh\u00e9ho \u00fa\u010dtu 1. a 3. syn v pomere 1:3 a dlh maj\u00fa zaplati\u0165 2. a 3. syn v pomere 2:3. Zistite, ko\u013eko Sk bolo na jednotliv\u00fdch \u00fa\u010dtoch a ak\u00fd dlh museli synovia splati\u0165, ak viete, \u017ee v kone\u010dnom d\u00f4sledku ka\u017ed\u00fd z nich z\u00edskal 123&nbsp;456 Sk.<\/p>\n<div class=pri>(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n<\/p>\n<p><b>Z9-I-5<\/b><br \/>br \/><br \/>\nDva rovnostrann\u00e9 papierov\u00e9 trojuholn\u00edky, z ktor\u00fdch men\u0161\u00ed m\u00e1 obsah 60&nbsp;cm<sup>2<\/sup>, sme polo\u017eili cez seba tak, \u017ee ich prienikom bol pravouhl\u00fd trojuholn\u00edk s obsahom 30&nbsp;cm<sup>2<\/sup>. Ak\u00fd najmen\u0161\u00ed obsah mohol ma\u0165 v\u00e4\u010d\u0161\u00ed z rovnostrann\u00fdch trojuholn\u00edkov?<\/p>\n<div class=pri>(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n<\/p>\n<p><b>Z9-I-6<\/b><br \/>\nZadanie p\u00edsomnej pr\u00e1ce obsahovalo 26 ot\u00e1zok, ktor\u00e9 boli rozdelen\u00e9 pod\u013ea obtia\u017enosti do troch skup\u00edn. V prvej skupine bola spr\u00e1vna odpove\u010f hodnoten\u00e1 tromi bodmi, v druhej piatimi bodmi a v tretej \u00f4smimi bodmi. Maxim\u00e1lny mo\u017en\u00fd po\u010det z\u00edskan\u00fdch bodov bol 111. Ko\u013eko ot\u00e1zok mohlo by\u0165 v ka\u017edej skupine?<\/p>\n<div class=pri>(L. \u0160im\u016fnek)<\/div>\n<\/p>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tr>\n<td width=\"85%\" align=\"left\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z92\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/table>\n<p><b>Z9-II-1<\/b><br \/>\nPeter a Jano i\u0161li pl\u00e1va\u0165. Vzdialenosti, ktor\u00e9 odpl\u00e1vali, boli v pomere 4:5. (Jano odpl\u00e1val viac.) \u010eal\u0161\u00ed de\u0148 \u0161li op\u00e4\u0165 pl\u00e1va\u0165, teraz prepl\u00e1val Peter o&nbsp;200 metrov menej a Jano o&nbsp;100 metrov viac ako predch\u00e1dzaj\u00faci de\u0148. Pomer vzdialenost\u00ed za druh\u00fd de\u0148 bol 5:8. Ko\u013eko metrov prepl\u00e1val prv\u00fd de\u0148 Peter a ko\u013eko Jano?<\/p>\n<div class=pri>(B. \u0160\u0165astn\u00e1)<\/div>\n<\/p>\n<p><b>Z9-II-2<\/b><br \/>\nUr\u010dte obsah \u0161edej plochy na obr\u00e1zku, ak ka\u017ed\u00fd z troch bielych kruhov m\u00e1 polomer 1&nbsp;cm a \u00fase\u010dky, ktor\u00e9 plochu ohrani\u010duj\u00fa, s\u00fa ich doty\u010dnicami.<\/p>\n<p class=pc><img decoding=\"async\" height=169 alt=\"\" src=\"\/55\/55z9b1.gif\" width=183 border=0><\/p>\n<div class=pri>(P. Tlust\u00fd)<\/div>\n<\/p>\n<p><b>Z9-II-3<\/b><br \/>\nMarek m\u00e1 kalkula\u010dku, ktor\u00e1 prevedie po\u010dtov\u00e9 \u00fakony v takom porad\u00ed, v akom ich Marek zad\u00e1va. Na papier si nap\u00edsal jedno \u010d\u00edslo. Zadal ho do kalkula\u010dky a potom postupne stl\u00e1\u010dal tla\u010didl\u00e1<\/p>\n<p class=pc><img decoding=\"async\" height=27 alt=\"\" src=\"\/55\/55z9b2.gif\" width=273 border=0><\/p>\n<p>V\u00fdsledok odp\u00edsal z displeja na papier. Potom s nap\u00edsan\u00fdm \u010d\u00edslom urobil to ist\u00e9 ako s predch\u00e1dzaj\u00facim \u010d\u00edslom, teda zadal ho do kalkula\u010dky a stl\u00e1\u010dal plus, \u0161tyri, delen\u00e9, \u0161tyri, m\u00ednus, \u0161tyri, kr\u00e1t, \u0161tyri. V\u00fdsledok si zap\u00edsal na papier. Cel\u00fd postup s t\u00fdmto novo z\u00edskan\u00fdm \u010d\u00edslom zopakoval a v\u00fdsledok op\u00e4\u0165 zap\u00edsal na papier. Potom zistil, \u017ee s\u00fa\u010det \u0161tyroch \u010d\u00edsel zap\u00edsan\u00fdch na papieri je 80. Ktor\u00e9 \u010d\u00edsla a v akom porad\u00ed zapisoval Marek na papier?<\/p>\n<div class=pri>(M. Raabov\u00e1)<\/div>\n<\/p>\n<p><b>Z9-II-4<\/b><br \/>\nV&nbsp;rovnostrannom trojuholn\u00edku vyzna\u010dte ka\u017ed\u00fd bod, ktor\u00e9ho vzdialenos\u0165 od najbli\u017e\u0161ieho vrcholu je men\u0161ia ako vzdialenos\u0165 od \u0165a\u017eiska. Ko\u013eko percent plochy rovnostrann\u00e9ho trojuholn\u00edka tvoria v\u0161etky vyzna\u010den\u00e9 body?<\/p>\n<div class=pri>(L. \u0160im\u016fnek)<\/div>\n<\/p>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tr>\n<td width=\"85%\" align=\"left\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z93\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/table>\n<p><b>Z9-III-1<\/b><br \/>\nDopl\u0148te do \u0161tvor\u010dekov prirodzen\u00e9 \u010d\u00edsla tak, aby:<\/p>\n<p>\u2022 s\u00fa\u010det v\u0161etk\u00fdch doplnen\u00fdch \u010d\u00edsel bol 44,<br \/>\n\u2022 s\u00fa\u010det \u010d\u00edsel v ka\u017edom \u0161tvorci zlo\u017eenom zo \u0161tyroch \u0161tvor\u010dekov bol rovnak\u00fd,<br \/>\n\u2022 najmen\u0161ie doplnen\u00e9 \u010d\u00edslo bolo nep\u00e1rne,<br \/>\n\u2022 uprostred \u0161tvorca bolo jednocifern\u00e9 \u010d\u00edslo.<\/p>\n<table align=\"center\" width=\"90\" cellspacing=\"2\" cellpadding=\"2\" border=\"2\" class=\"r\" bordercolor=\"#808080\">\n<tr>\n<td width=\"30\" align=\"middle\">&nbsp;<\/td>\n<td width=\"30\" align=\"middle\">7<\/td>\n<td width=\"30\" align=\"middle\">&nbsp;<\/td>\n<\/tr>\n<tr>\n<td align=\"middle\">8<\/td>\n<td>&nbsp;<\/td>\n<td align=\"middle\">4<\/td>\n<\/tr>\n<tr>\n<td>&nbsp;<\/td>\n<td align=\"middle\">2<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<\/table>\n<div class=pri>(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n<\/p>\n<p><b>Z9-III-2<\/b><br \/>\nUr\u010dte obsah \u0161ed\u00e9ho mesia\u010dika z obr\u00e1zku, ak viete, \u017ee priemer ve\u013ek\u00e9ho polkruhu m\u00e1 d\u013a\u017eku |<em>CD<\/em>|&nbsp;=&nbsp;2&nbsp;cm, priemer men\u0161ieho polkruhu m\u00e1 d\u013a\u017eku |<em>AB<\/em>|&nbsp;=&nbsp;1&nbsp;cm a plat\u00ed <em>AB<\/em>&nbsp;||&nbsp;<em>CD<\/em>.<\/p>\n<div class=pc><img loading=\"lazy\" decoding=\"async\" src=\"\/55\/55z9c1.gif\" width=\"206\" height=\"135\" border=\"0\" alt=\"\"><\/div>\n<div class=pri>(P. Tlust\u00fd)<\/div>\n<\/p>\n<p><b>Z9-III-3<\/b><br \/>\nJakub mal v pivnici tri krabice tvaru kv\u00e1dra so \u0161tvorcovou podstavou. Prv\u00e1 bola dvojn\u00e1sobne vy\u0161\u0161ia ako druh\u00e1. Druh\u00e1 bola jeden a pol n\u00e1sobne \u0161ir\u0161ia ako prv\u00e1. Tretia bola trojn\u00e1sobne vy\u0161\u0161ia ako prv\u00e1 a dvojn\u00e1sobne u\u017e\u0161ia ako prv\u00e1. V akom pomere s\u00fa objemy n\u00e1dob?<\/p>\n<div class=pri>(\u0160. Pt\u00e1\u010dkov\u00e1)<\/div>\n<\/p>\n<p><b>Z9-III-4<\/b><br \/>\nPri prij\u00edmac\u00edch sk\u00fa\u0161kach na univerzitu sa ka\u017ed\u00e9mu z\u00e1ujemcovi o \u0161t\u00fadium pride\u013euje kryc\u00ed k\u00f3d zlo\u017een\u00fd z piatich \u010d\u00edslic. Sk\u00fa\u0161ky organizoval d\u00f4kladn\u00fd, ale pover\u010div\u00fd docent, ktor\u00fd sa pred pride\u013eovan\u00edm k\u00f3dov rozhodol vyradi\u0165 zo v\u0161etk\u00fdch mo\u017en\u00fdch k\u00f3dov (t.j. 00000 a\u017e 99999) tie, ktor\u00e9 v sebe obsahovali \u010d\u00edslo 13, teda \u010d\u00edslicu 3 bezprostredne nasleduj\u00facu po \u010d\u00edslici 1. Ko\u013eko k\u00f3dov musel docent vyradi\u0165?<\/p>\n<div class=pri>(L. \u0160im\u016fnek)<\/div>\n<\/p>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tr>\n<td width=\"85%\" align=\"left\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/table>\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>55. ro\u010dn\u00edk matematickej olympi\u00e1dy 2005-2006 S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z4-Z9 Z4 Z5 Z6 Z7 Z8 Z9 dom\u00e1ce dom\u00e1ce dom\u00e1ce dom\u00e1ce dom\u00e1ce dom\u00e1ce \u0161kolsk\u00e9 obvodn\u00e9 obvodn\u00e9 obvodn\u00e9 obvodn\u00e9 obvodn\u00e9 krajsk\u00e9<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[4],"tags":[],"class_list":["post-377","post","type-post","status-publish","format-standard","hentry","category-55-rocnik"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.6 - 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