{"id":88,"date":"2010-08-26T18:39:17","date_gmt":"2010-08-26T18:39:17","guid":{"rendered":"http:\/\/matematika.okamzite.eu\/?p=88"},"modified":"2010-08-26T18:39:17","modified_gmt":"2010-08-26T18:39:17","slug":"kategorie-z4-z9","status":"publish","type":"post","link":"https:\/\/matematika.besaba.com\/?p=88","title":{"rendered":"Kateg\u00f3rie Z4-Z9"},"content":{"rendered":"<p class=\"hh2\"><a name=\"top\"><\/a>59. ro\u010dn\u00edk matematickej olympi\u00e1dy 2009-2010<\/p>\n<h3 style=\"text-align: center;\">S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z4-Z9<\/h3>\n<p><!--more--><\/p>\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: 100%;\" 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\">1. kolo<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z51\">1. kolo<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z61\">1. kolo<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z71\">1. kolo<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z81\">1. kolo<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z91\">1. kolo<\/a><\/span><\/td>\n<\/tr>\n<tr align=\"center\">\n<td><span class=\"a\"><a href=\"#z42\">2. kolo<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z52\">2. kolo<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z62\">2. kolo<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z72\">2. kolo<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z82\">2. kolo<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z92\">2. kolo<\/a><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"5\" bgcolor=\"#c0c0c0\"><\/td>\n<td align=\"center\"><span class=\"a\"><a href=\"#z93\">3. kolo<\/a><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p class=\"k\">Kateg\u00f3ria <span style=\"color: #800000;\">Z9<\/span> je ur\u010den\u00e1 pre \u017eiakov 9. ro\u010dn\u00edka Z\u0160, 4. ro\u010dn\u00edka OG a 1. ro\u010dn\u00edka bilingv\u00e1lnych gymn\u00e1zi\u00ed.<br \/>\nKateg\u00f3ria <span style=\"color: #800000;\">Z8<\/span> je ur\u010den\u00e1 <span style=\"color: #ff3300;\">len<\/span> pre \u017eiakov 8. ro\u010dn\u00edka Z\u0160.<br \/>\nKateg\u00f3ria <span style=\"color: #800000;\">Z7<\/span> je ur\u010den\u00e1 pre \u017eiakov 7. ro\u010dn\u00edka Z\u0160 a 3. ro\u010dn\u00edka OG.<br \/>\nKateg\u00f3ria <span style=\"color: #800000;\">Z6<\/span> je ur\u010den\u00e1 pre \u017eiakov 6. ro\u010dn\u00edka Z\u0160 a 1. a 2. ro\u010dn\u00edka OG.<br \/>\nKateg\u00f3ria <span style=\"color: #800000;\">Z5<\/span> je ur\u010den\u00e1 <span style=\"color: #ff3300;\">len<\/span> pre \u017eiakov 5. ro\u010dn\u00edka Z\u0160.<br \/>\nKateg\u00f3ria <span style=\"color: #800000;\">Z4<\/span> je ur\u010den\u00e1 pre \u017eiakov 4. ro\u010dn\u00edka Z\u0160.<\/p>\n<p><a name=\"z41\"><\/a><\/p>\n<p><strong>Z4-I-1<\/strong><br \/>\nDopl\u0148te \u010d\u00edsla do s\u00fa\u010dinovej pyram\u00eddy. Ka\u017ed\u00e9 \u010d\u00edslo v tehli\u010dk\u00e1ch (od druh\u00e9ho  riadku)  je  rovn\u00e9 s\u00fa\u010dinu \u010d\u00edsel v tehli\u010dk\u00e1ch bezprostredne nad n\u00edm. V prvom riadku s\u00fa iba jednocifern\u00e9 \u010d\u00edsla.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/59\/59z4a1.gif\" alt=\"\" width=\"196\" height=\"106\" \/><\/p>\n<div class=\"pri\">(M. Koll\u00e1r)<\/div>\n<p><strong>Z4-I-2<\/strong><br \/>\nJanka delila lentilky sebe a sestre Danke. Sebe dala 11\u00a0lentiliek, Danke 2. Potom sebe 10\u00a0a Danke 4. Potom sebe\u00a09 a Danke 6&#8230;  t\u00fdmto sp\u00f4sobom sebe dala v\u017edy o\u00a01\u00a0menej a Danke o\u00a02\u00a0viac ako predt\u00fdm. V delen\u00ed pokra\u010dovala, a\u017e sa jej lentilky minuli. Na svoje i Dankino po\u010dudovanie diev\u010dat\u00e1 zistili, \u017ee maj\u00fa lentiliek rovnako. Ko\u013eko bolo v\u0161etk\u00fdch lentiliek?<\/p>\n<div class=\"pri\">(M. Dillingerov\u00e1)<\/div>\n<p><strong>Z4-I-3<\/strong><br \/>\nDopl\u0148 \u010d\u00edsla od 1 do 12 (ka\u017ed\u00e9 pr\u00e1ve raz) do tabu\u013eky. V jednotliv\u00fdch riadkoch s\u00fa pr\u00edklady na s\u010d\u00edtanie, od\u010d\u00edtanie, n\u00e1sobenie a delenie a tie musia by\u0165 v\u0161etky splnen\u00e9. Navy\u0161e v ka\u017edom riadku s\u00fa dop\u013a\u0148an\u00e9 \u010d\u00edsla usporiadan\u00e9 z\u013eava doprava od najv\u00e4\u010d\u0161ieho po najmen\u0161ie.<\/p>\n<table class=\"pc\" style=\"height: 120px;\" border=\"2\" cellspacing=\"0\" cellpadding=\"0\" align=\"center\" bordercolor=\"#000000\">\n<tbody>\n<tr>\n<td width=\"55\"><\/td>\n<td width=\"25\">+<\/td>\n<td width=\"55\"><\/td>\n<td width=\"25\">+<\/td>\n<td width=\"55\"><\/td>\n<td width=\"25\">=<\/td>\n<td width=\"40\">33<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>&#8211;<\/td>\n<td><\/td>\n<td>&#8211;<\/td>\n<td><\/td>\n<td>=<\/td>\n<td>0<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>.<\/td>\n<td><\/td>\n<td>.<\/td>\n<td><\/td>\n<td>=<\/td>\n<td>35<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>:<\/td>\n<td><\/td>\n<td>:<\/td>\n<td><\/td>\n<td>=<\/td>\n<td>1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"pri\">(M. Koll\u00e1r)<\/div>\n<p><strong>Z4-I-4<\/strong><br \/>\nJurko chod\u00ed domov zo \u0161koly okolo \u0161kolsk\u00e9ho latkov\u00e9ho plota. V pondelok sa rozhodol, \u017ee cestou zo \u0161koly bude na ka\u017ed\u00fa druh\u00fa latku robi\u0165 zna\u010dku bielou kriedou. V utorok, stredu i \u0161tvrtok to zopakoval s ka\u017edou druhou e\u0161te neozna\u010denou latkou. V piatok r\u00e1no zistil, \u017ee mu ostalo u\u017e iba 7\u00a0latiek neozna\u010den\u00fdch. Ko\u013eko najmenej a ko\u013eko najviac latiek mohol ma\u0165 plot \u0161koly?<\/p>\n<div class=\"pri\">(M. Dillingerov\u00e1)<\/div>\n<p><strong>Z4-I-5<\/strong><br \/>\nSamo m\u00e1 tri star\u0161ie sestry. Ka\u017ed\u00e1 rob\u00ed nie\u010do in\u00e9 a ka\u017ed\u00e1 chov\u00e1 jedno zvieratko. \u013dudka hr\u00e1 tenis. Najmlad\u0161ia je klaviristka. Ivana ne\u0161tuduje jazyky. M\u00e1ria nie je prostredn\u00e1. Na klav\u00edri sed\u00e1va ve\u013ek\u00e1 \u010dierna ma\u010dka. Prostredn\u00e1 m\u00e1 papag\u00e1ja. Tret\u00edm zviera\u0165om je pes. Vyp\u00ed\u0161, ako sa vol\u00e1 najstar\u0161ia sestra, \u010do rob\u00ed a ak\u00e9 zviera chov\u00e1.<\/p>\n<div class=\"pri\">(M. Dillingerov\u00e1)<\/div>\n<p><strong>Z4-I-6<\/strong><br \/>\nMartin m\u00e1 vystrihnut\u00fd z papiera jeden obd\u013a\u017enik s rozmermi 2\u00a0cm a 6\u00a0cm. Okrem toho m\u00e1 e\u0161te jeden obd\u013a\u017enik a jeden \u0161tvorec. Zo v\u0161etk\u00fdch troch \u00fatvarov vie bez prekr\u00fdvania a dier zlo\u017ei\u0165 jeden ve\u013ek\u00fd \u0161tvorec. Ak\u00e9 rozmery m\u00f4\u017eu ma\u0165 jeho \u00fatvary? N\u00e1jdi dve rie\u0161enia.<\/p>\n<div class=\"pri\">(M. Dillingerov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\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<\/tbody>\n<\/table>\n<p><strong>Z4-II-1<\/strong><br \/>\nJana si na poli\u010dku ulo\u017eila svoje tri ob\u013e\u00faben\u00e9 b\u00e1biky. Ka\u017ed\u00e1 b\u00e1bika m\u00e1 samozrejme meno, in\u00fa farbu vlasov, \u0161at\u00f4\u010diek a ma\u0161le na nich. B\u00e1bika Ariel m\u00e1 modr\u00e9 \u0161at\u00f4\u010dky. Blond\u00ednka m\u00e1 ru\u017eov\u00e9 \u0161at\u00f4\u010dky. Jedna b\u00e1bika m\u00e1 strieborn\u00fa ma\u0161\u013eu. Hanna nem\u00e1 hned\u00e9 vlasy. Na ru\u017eov\u00fdch \u0161at\u00f4\u010dkach je ve\u013ek\u00e1 zlat\u00e1 ma\u0161\u013ea. Hnedovl\u00e1ska m\u00e1 modr\u00fa ma\u0161\u013eu. Zuzka nem\u00e1 biele \u0161at\u00f4\u010dky. Ktor\u00e1 m\u00e1 biele \u0161at\u00f4\u010dky? Vyp\u00ed\u0161, ako sa vol\u00e1 \u010diernovl\u00e1ska, ak\u00e9 m\u00e1 \u0161at\u00f4\u010dky a ma\u0161\u013eu.<\/p>\n<div class=\"pri\">(M. Dillingerov\u00e1)<\/div>\n<p><strong>Z4-II-2<\/strong><br \/>\nDopl\u0148te miesto hviezdi\u010diek \u010d\u00edsla tak, aby s\u00fa\u010det v\u00fdsledkov oboch pr\u00edkladov bol 9 345.<\/p>\n<table class=\"r\" style=\"width: 350px;\" border=\"0\" cellspacing=\"0\" cellpadding=\"0\" align=\"center\">\n<tbody>\n<tr height=\"15\">\n<td>*<\/td>\n<td>*<\/td>\n<td>*<\/td>\n<td>6<\/td>\n<td width=\"80\"><\/td>\n<td><\/td>\n<td>2<\/td>\n<td>*<\/td>\n<td>3<\/td>\n<td>*<\/td>\n<\/tr>\n<tr height=\"15\">\n<td>1<\/td>\n<td>5<\/td>\n<td>7<\/td>\n<td>*<\/td>\n<td><\/td>\n<td><\/td>\n<td>2<\/td>\n<td>8<\/td>\n<td>*<\/td>\n<td>4<\/td>\n<\/tr>\n<tr height=\"1\">\n<td colspan=\"4\">\n<hr size=\"1\" \/><\/td>\n<td><\/td>\n<td colspan=\"5\">\n<hr size=\"1\" \/><\/td>\n<\/tr>\n<tr height=\"15\">\n<td>*<\/td>\n<td>*<\/td>\n<td>0<\/td>\n<td>0<\/td>\n<td><\/td>\n<td><\/td>\n<td>*<\/td>\n<td>5<\/td>\n<td>4<\/td>\n<td>*<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"pri\">(M. Dillingerov\u00e1)<\/div>\n<p><strong>Z4-II-3<\/strong><br \/>\nIvan mal z papiera so \u0161tvor\u010dekovou sie\u0165ou vystrihnut\u00fd obd\u013a\u017enik. Strany obd\u013a\u017enika le\u017eali na \u010diarach siete. Ke\u010f zo svojho obd\u013a\u017enika z ka\u017edej strany odstrihol p\u00e1s o \u0161\u00edrke jedn\u00e9ho \u0161tvor\u010deka siete, dok\u00e1zal z t\u00fdchto p\u00e1sikov (bez \u010fal\u0161ieho strihania a bez prekr\u00fdvania \u010di dier) zlo\u017ei\u0165 \u0161tvorec obsahuj\u00faci 16 \u0161tvor\u010dekov. Ur\u010di ak\u00e9 rozmery (v \u0161tvor\u010dekoch siete) mal Ivanov p\u00f4vodn\u00fd obd\u013a\u017enik.<\/p>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\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<\/tbody>\n<\/table>\n<p><strong>Z5-I-1<\/strong><br \/>\nH\u00fasenica Leona spadla doprostred \u0161tvorcovej siete. Rozhodla sa, \u017ee polezie &#8222;<em>do \u0161pir\u00e1ly<\/em>&#8220; tak, ako je nazna\u010den\u00e9 na obr\u00e1zku; na \u017eiadnom \u0161tvor\u010deku nebude dvakr\u00e1t a \u017eiaden \u0161tvor\u010dek nevynech\u00e1.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/59\/59z5a1.gif\" alt=\"\" width=\"229\" height=\"229\" \/><\/p>\n<p>Z prv\u00e9ho \u0161tvor\u010deka na druh\u00fd liezla smerom na v\u00fdchod, z druh\u00e9ho na tret\u00ed smerom na sever, z tretieho na \u0161tvrt\u00fd smerom na z\u00e1pad, zo \u0161tvrt\u00e9ho na piaty tie\u017e na z\u00e1pad, z piateho na \u0161iesty na juh&#8230; Ktor\u00fdm smerom liezla z 81. na 82. \u0161tvor\u010dek?<\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z5-I-2<\/strong><br \/>\nMi\u0161a si z papiera vystrihla dva rovnak\u00e9 \u0161tvorce, jeden obd\u013a\u017enik s rozmermi 10\u00a0cm\u00d724\u00a0cm a e\u0161te jeden obd\u013a\u017enik. Ak\u00e9 rozmery mohol ma\u0165 tento obd\u013a\u017enik, ak sa zo v\u0161etk\u00fdch \u0161tyroch \u00fatvarov dal zlo\u017ei\u0165 \u0161tvorec bez toho, aby sa jednotliv\u00e9 diely prekr\u00fdvali? Tak\u00fdch obd\u013a\u017enikov sa d\u00e1 n\u00e1js\u0165 nieko\u013eko, uve\u010f aspo\u0148 \u0161tyri.<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z5-I-3<\/strong><br \/>\nVyrie\u0161 nasleduj\u00faci algebrogram a n\u00e1jdi v\u0161etky rie\u0161enia. Rovnak\u00e9 p\u00edsmen\u00e1 nahra\u010f rovnak\u00fdmi \u010d\u00edslicami, r\u00f4zne r\u00f4znymi.<\/p>\n<p class=r>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;O&nbsp;S&nbsp;E&nbsp;L<br \/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;S&nbsp;E&nbsp;L<br \/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;E&nbsp;L<br \/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;L<br \/>&#8212;&#8212;&#8212;&#8212;&#8212;&#8211;<br \/>1&nbsp;0&nbsp;0&nbsp;3&nbsp;4<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z5-I-4<\/strong><br \/>\nNina dostala od pani u\u010dite\u013eky nasleduj\u00face karti\u010dky:<\/p>\n<div class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/59\/59z5a2.gif\" alt=\"\" width=\"632\" height=\"48\" \/><\/div>\n<p>M\u00e1 z nich zostavi\u0165 pr\u00edklad pre svojich spolu\u017eiakov, pri\u010dom ka\u017ed\u00fa karti\u010dku pou\u017eije pr\u00e1ve raz. Pom\u00f4\u017e Nine a zostav jeden tak\u00fd pr\u00edklad tak, aby ka\u017ed\u00e9 delenie vy\u0161lo bezo zvy\u0161ku. Ak\u00fd bude v\u00fdsledok?<\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z5-I-5<\/strong><br \/>\nNa\u0161ich 84\u00a0\u017eiakov i\u0161lo do kina. L\u00edstok s\u00edce st\u00e1l 2\u20ac, ale ka\u017ed\u00fd 12.\u00a0\u017eiak mal polovi\u010dn\u00fa z\u013eavu a ka\u017ed\u00fd 35.\u00a0vstup zdarma. Ko\u013eko st\u00e1lo vstupn\u00e9 pre v\u0161etk\u00fdch \u017eiakov?<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z5-I-6<\/strong><br \/>\nChlapci na\u0161li star\u00fd pl\u00e1n m\u00ednov\u00e9ho po\u013ea (na obr\u00e1zku). \u010c\u00edsla s\u00fa na pol\u00ed\u010dkach, kde \u017eiadne m\u00edny nie s\u00fa, a ud\u00e1vaj\u00fa po\u010det zam\u00ednovan\u00fdch susedn\u00fdch pol\u00ed\u010dok. Ur\u010di, ko\u013eko je v poli m\u00edn spolu a kde s\u00fa. (Pol\u00ed\u010dka susedia pr\u00e1ve vtedy, ke\u010f maj\u00fa spolo\u010dn\u00fd vrchol alebo stranu.)<\/p>\n<table class=\"pc\" style=\"width: 150px; height: 150px;\" border=\"2\" cellspacing=\"0\" cellpadding=\"0\" align=\"center\" bordercolor=\"#000000\">\n<tbody>\n<tr>\n<td>1<\/td>\n<td><\/td>\n<td>2<\/td>\n<td><\/td>\n<td>2<\/td>\n<\/tr>\n<tr align=\"center\">\n<td><\/td>\n<td>3<\/td>\n<td><\/td>\n<td>3<\/td>\n<td><\/td>\n<\/tr>\n<tr align=\"center\">\n<td>3<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td>3<\/td>\n<\/tr>\n<tr align=\"center\">\n<td><\/td>\n<td>2<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr align=\"center\">\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td>2<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td 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<\/tbody>\n<\/table>\n<p><strong>Z5-II-1<\/strong><br \/>\nMat\u00fa\u0161 a jeho kamar\u00e1ti i\u0161li na \u0160tefana koledova\u0165. Okrem jab\u013a\u010dok, orie\u0161kov a pern\u00ed\u010dkov dostal ka\u017ed\u00fd z chlapcov aj pomaran\u010de. Jaro dostal 1 pomaran\u010d, Milan tie\u017e. Po dvoch pomaran\u010doch dostali Rado, Patrik, Michal a Du\u0161an. Mat\u00fa\u0161 dostal dokonca \u0161tyri pomaran\u010de, \u010do bolo najviac zo v\u0161etk\u00fdch chlapcov. Ostatn\u00ed chlapci dostali po troch pomaran\u010doch. Ko\u013eko chlapcov i\u0161lo koledova\u0165, ke\u010f v\u0161etci spolu dostali 23 pomaran\u010dov?<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z5-II-2<\/strong><br \/>\nBzdocha Jozef\u00edna dopadla na st\u00f4l doprostred \u0161tvorcovej siete tvorenej 81 \u0161tvorcami (vi\u010f obr\u00e1zok). Rozhodla sa, \u017ee zo siete nezlezie na st\u00f4l priamo, ale nasleduj\u00facim sp\u00f4sobom:<br \/>\nnajprv jeden \u0161tvoreek na juh, potom jeden na v\u00fdchod, potom dva na sever, dva na z\u00e1pad a op\u00e4\u0165 jeden na juh, jeden na v\u00fdchod, dva na sever, dva na z\u00e1pad&#8230; Na ktorom \u0161tvor\u010deku bola tesne pred t\u00fdm, ne\u017e zliezla z tejto siete na st\u00f4l? Po ko\u013ek\u00fdch \u0161tvor\u010dekoch tejto siete liezla?<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/59\/59z5b1.gif\" alt=\"\" width=\"237\" height=\"237\" \/><\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z5-II-3<\/strong><br \/>\nJurko m\u00e1 pali\u010dky d\u013a\u017eok 2\u00a0cm, 3\u00a0cm, 3\u00a0cm, 3\u00a0cm, 4\u00a0cm, 5\u00a0cm, 5\u00a0cm, 5\u00a0cm, 6\u00a0cm, 6\u00a0cm a 9\u00a0cm. Sklad\u00e1 z nich strany trojuholn\u00edkov. \u017diadna pali\u010dka nie je s\u00fa\u010das\u0165ou strany dvoch alebo viac trojuholn\u00edkov. Jurko m\u00f4\u017ee pou\u017ei\u0165 to\u013eko pali\u010diek, ko\u013eko chce, ale nesmie ich l\u00e1ma\u0165 a ka\u017ed\u00e1 pou\u017eit\u00e1 pali\u010dka mus\u00ed le\u017ea\u0165 cel\u00e1 na obvode trojuholn\u00edka. Jurko tvrd\u00ed, \u017ee sa jeho pali\u010dky daj\u00fa pou\u017ei\u0165 na poskladanie str\u00e1n troch trojuholn\u00edkov s rovnak\u00fdmi obvodmi. M\u00e1 pravdu? Ak\u00fd najv\u00e4\u010d\u0161\u00ed obvod by tak\u00e9 trojuholn\u00edky mali?<\/p>\n<div class=\"pri\">(M. Dillingerov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\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<\/tbody>\n<\/table>\n<p><strong>Z6-I-1<\/strong><br \/>\nJanko s Marienkou chodia k babi\u010dke, ktor\u00e1 m\u00e1 cukr\u00e1re\u0148 a pred\u00e1va pern\u00edky. Obe deti jej samozrejme pom\u00e1haj\u00fa, hlavne so zdoben\u00edm. K\u00fdm  babi\u010dka ozdob\u00ed p\u00e4\u0165 pern\u00edkov, ozdob\u00ed Marienka tri a Janko dva. Pri poslednej n\u00e1v\u0161teve ozdobili v\u0161etci traja spolu p\u00e4\u0165 pln\u00fdch t\u00e1cok. Marienka s babi\u010dkou zdobili po cel\u00fd \u010das, Janko najprv zdobil a potom zrovn\u00e1val pern\u00edky po dvan\u00e1s\u0165 na jednu t\u00e1cku a odn\u00e1\u0161al ich do komory. V\u0161etci traja v ten ist\u00fd okamih za\u010dali i skon\u010dili.<\/p>\n<p><strong>1.<\/strong> Ko\u013eko pern\u00edkov ozdobil Janko?<br \/>\n<strong>2.<\/strong> Ako dlho im trvala cel\u00e1 pr\u00e1ca, ak babi\u010dka ozdob\u00ed jeden pern\u00edk za 4 min\u00faty?<br \/>\n<strong>3.<\/strong> Ako dlho pom\u00e1hal Janko zdobi\u0165?<\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z6-I-2<\/strong><br \/>\n\u0160tvormiestny PIN k\u00f3d Ras\u0165ovho mobilu je zauj\u00edmav\u00fd:<br \/>\n\u2022\u00a0\u00a0jednotliv\u00e9 \u010d\u00edslice s\u00fa prvo\u010d\u00edsla,<br \/>\n\u2022\u00a0\u00a01. a 2. \u010d\u00edslica v tomto porad\u00ed vytvor\u00ed prvo\u010d\u00edslo,<br \/>\n\u2022\u00a0\u00a02. a 3. \u010d\u00edslica v tomto porad\u00ed vytvor\u00ed prvo\u010d\u00edslo,<br \/>\n\u2022\u00a0\u00a03. a 4. \u010d\u00edslica v tomto porad\u00ed vytvor\u00ed prvo\u010d\u00edslo.<\/p>\n<p>Ras\u0165o zabudol svoj PIN k\u00f3d, ale pam\u00e4t\u00e1 si v\u0161etky vy\u0161\u0161ie uveden\u00e9 vlastnosti. Sna\u017e\u00ed sa zaktivova\u0165 vypnut\u00fd mobil. Ktor\u00e9 \u010d\u00edsla by mal vysk\u00fa\u0161a\u0165?<\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z6-I-3<\/strong><br \/>\nNa nasleduj\u00facom obr\u00e1zku je \u00fatvar zlo\u017een\u00fd zo siedmich rovnak\u00fdch \u0161tvoruholn\u00edkov\u00fdch dielikov stavebnice. Ak\u00fd je obvod tohto \u00fatvaru, ak obvod jedn\u00e9ho \u0161tvoruholn\u00edkov\u00e9ho dielika je 17\u00a0cm?<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/59\/59z6a1.gif\" alt=\"\" width=\"154\" height=\"183\" \/><\/p>\n<div class=\"pri\">(K. Pazourek)<\/div>\n<p><strong>Z6-I-4<\/strong><br \/>\nOcko sa rozhodol, \u017ee bude d\u00e1va\u0165 svojmu synovi Mojm\u00edrovi raz za mesiac vreckov\u00e9. Prv\u00e9 vreckov\u00e9 dostal Mojm\u00edr v janu\u00e1ri. Ocko ka\u017ed\u00fd mesiac vreckov\u00e9 zvy\u0161oval v\u017edy o 4\u00a0\u20ac. Keby Mojm\u00edr neutr\u00e1cal, mal by po dvan\u00e1stom vreckovom pred Vianocami 900\u00a0\u20ac. Ko\u013eko \u20ac dostal Mojm\u00edr v janu\u00e1ri ako prv\u00e9 vreckov\u00e9?<\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z6-I-5<\/strong><br \/>\nDopl\u0148 miesto hviezdi\u010diek \u010d\u00edsla tak, aby s\u00fa\u010det v\u00fdsledkov nasleduj\u00facich dvoch pr\u00edkladov bol 5842.<\/p>\n<table class=\"r\" style=\"width: 350px;\" border=\"0\" cellspacing=\"0\" cellpadding=\"0\" align=\"center\">\n<tbody>\n<tr height=\"15\">\n<td>*<\/td>\n<td>2<\/td>\n<td>*<\/td>\n<td>7<\/td>\n<td width=\"80\"><\/td>\n<td><\/td>\n<td>2<\/td>\n<td>*<\/td>\n<td>9<\/td>\n<td>*<\/td>\n<\/tr>\n<tr height=\"15\">\n<td>3<\/td>\n<td>*<\/td>\n<td>4<\/td>\n<td>*<\/td>\n<td><\/td>\n<td>&#8211;<\/td>\n<td>*<\/td>\n<td>2<\/td>\n<td>*<\/td>\n<td>4<\/td>\n<\/tr>\n<tr height=\"1\">\n<td colspan=\"4\">\n<hr size=\"1\" \/><\/td>\n<td><\/td>\n<td colspan=\"5\">\n<hr size=\"1\" \/><\/td>\n<\/tr>\n<tr height=\"15\">\n<td>4<\/td>\n<td>*<\/td>\n<td>0<\/td>\n<td>0<\/td>\n<td><\/td>\n<td><\/td>\n<td>2<\/td>\n<td>*<\/td>\n<td>9<\/td>\n<td>*<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"pri\">(M. Dillingerov\u00e1)<\/div>\n<p><strong>Z6-I-6<\/strong><br \/>\nNa \u0161kolsk\u00fa olympi\u00e1du vytvorili \u017eiaci stupne v\u00ed\u0165azov z dreven\u00fdch kociek (pozri obr\u00e1zok). Ko\u013eko kociek pou\u017eili celkom?<br \/>\nZostaven\u00e9 stupne natreli po celom povrchu (okrem podstavy) na bielo a po vyhl\u00e1sen\u00ed v\u00fdsledkov svoj v\u00fdtvor rozobrali. Ko\u013eko kociek malo 6, ko\u013eko 5, 4, 3, 2, 1 \u010di \u017eiadnu stenu bielu?<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/59\/59z6a2.gif\" alt=\"\" width=\"194\" height=\"80\" \/><\/p>\n<div class=\"pri\">(M. Dillingerov\u00e1, M. Volfov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\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<\/tbody>\n<\/table>\n<p><b>Z6-II-1<\/b><\/p>\n<p>Vp\u00ed\u0161te do kr\u00fa\u017ekov \u010d\u00edsla 1, 2, 3, 4, 5, 6, 7 a 8 tak, aby \u017eiadne dve po sebe id\u00face \u010d\u00edsla neboli spojen\u00e9 \u010diarou a v&nbsp;siv\u00fdch kr\u00fa\u017ekoch boli nep\u00e1rne \u010d\u00edsla.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/59\/59z6b1.gif\" width=\"235\" height=\"172\" alt=\"\" \/><\/p>\n<div class=pri>(S.Bedn\u00e1\u0159ov\u00e1)<\/div>\n<\/p>\n<p><b>Z6-II-2<\/b><\/p>\n<p>Mysl\u00edm si \u0161tvorcifern\u00e9 \u010d\u00edslo, ktor\u00e9ho ka\u017ed\u00e1 \u010d\u00edslica je in\u00e1. Ke\u010f \u0161krtnem posledn\u00e9 dve \u010d\u00edslice v tomto \u010d\u00edsle, dostanem prvo\u010d\u00edslo. Rovnako dostanem prvo\u010d\u00edslo aj v pr\u00edpade, ke\u010f vy\u0161krtnem z myslen\u00e9ho \u010d\u00edsla druh\u00fa a \u0161tvrt\u00fa \u010d\u00edslicu, a dokonca aj v pr\u00edpade, ke\u010f z neho vy\u0161krtnem prostredn\u00e9 dve \u010d\u00edslice. Moje myslen\u00e9 \u010d\u00edslo ale prvo\u010d\u00edslo nie je &#8211; d\u00e1 sa bezozvy\u0161ku deli\u0165 tromi. \u010c\u00edsel, ktor\u00e9 maj\u00fa tieto vlastnosti, je viac. To moje je najv\u00e4\u010d\u0161ie z nich. Ktor\u00e9 \u010d\u00edslo si mysl\u00edm?<\/p>\n<div class=pri>(M.Petrov\u00e1)<\/div>\n<\/p>\n<p><b>Z6-II-3<\/b><\/p>\n<p>Krabi\u010dka tvaru kocky s hranou d\u0139\u017eky 4&nbsp;cm je \u00faplne naplnen\u00e1 ulo\u017een\u00fdmi mal\u00fdmi kockami s&nbsp;hranou d\u013a\u017eky 1&nbsp;cm. Vymysli v\u0161etky r\u00f4zne krabi\u010dky tak\u00e9, ktor\u00e9 maj\u00fa \u0161tvorcov\u00e9 dno a v\u0161etky kocky sa do ka\u017edej z nich presne vojd\u00fa. Nap\u00ed\u0161 ich rozmery.<\/p>\n<div class=pri>(M.Krej\u010dov\u00e1)<\/div>\n<\/p>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\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<\/tbody>\n<\/table>\n<p><strong>Z7-I-1<\/strong><br \/>\nDo  predajne  v\u00edna  sa  v  noci  dostal  koc\u00far.  Vysko\u010dil  na  policu,  na  ktorej  boli  v dlhom  rade vyrovnan\u00e9 f\u013ea\u0161e  s  v\u00ednom \u2014 prv\u00e1  tretina  flia\u0161  skraja  st\u00e1la  po  8\u00a0\u20ac, nasleduj\u00faca  tretina  flia\u0161 st\u00e1la po 6,5\u00a0\u20ac a posledn\u00e1 tretina po 5\u00a0\u20ac. Najprv koc\u00far zhodil na zem f\u013ea\u0161u za 8\u00a0\u20ac, ktor\u00e1 st\u00e1la \u00faplne na za\u010diatku radu, a potom postupoval \u010falej a zhadzoval bez vynechania jednu f\u013ea\u0161u za druhou. Ne\u017e ho  to prestalo bavi\u0165, zhodil 25\u00a0flia\u0161 a  tie sa v\u0161etky rozbili. R\u00e1no majite\u013e \u013eutoval, \u017ee koc\u00far neza\u010dal so zhadzovan\u00edm na druhom okraji police. Aj keby  toti\u017e rozbil rovnako ve\u013ea flia\u0161, bola by \u0161koda o\u00a033\u00a0\u20ac ni\u017e\u0161ia.<br \/>\nKo\u013eko flia\u0161 bolo p\u00f4vodne na polici?<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z7-I-2<\/strong><br \/>\nNa tabuli s\u00fa nap\u00edsan\u00e9 tri prirodzen\u00e9 \u010d\u00edsla <em>a, b, c<\/em>, pre ktor\u00e9 plat\u00ed:<\/p>\n<p>\u2022  najv\u00e4\u010d\u0161\u00ed spolo\u010dn\u00fd delite\u013e \u010d\u00edsel <em>a, b<\/em> je 15,<\/p>\n<p>\u2022  najv\u00e4\u010d\u0161\u00ed spolo\u010dn\u00fd delite\u013e \u010d\u00edsel <em>b, c<\/em> je 6,<\/p>\n<p>\u2022  s\u00fa\u010din \u010d\u00edsel <em>b, c<\/em> je 1800,<\/p>\n<p>\u2022  najmen\u0161\u00ed spolo\u010dn\u00fd n\u00e1sobok \u010d\u00edsel <em>a, b<\/em> je 3150.<\/p>\n<p>Ktor\u00e9 s\u00fa to \u010d\u00edsla?<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<\/p>\n<p><strong>Z7-I-3<\/strong><br \/>\nV \u0161tvoruholn\u00edku <em>KLMN<\/em> pozn\u00e1me vyzna\u010den\u00e9 uhly a vieme, \u017ee plat\u00ed |<em>KN<\/em>| = |<em>LM<\/em>|. Zisti ve\u013ekos\u0165 uhla <em>KNM<\/em>.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/59\/59z7a2.gif\" alt=\"\" width=\"203\" height=\"161\" \/><\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z7-I-4<\/strong><br \/>\nKocka bola zlo\u017een\u00e1 z 64\u00a0koc\u00f4\u010dok s hranou d\u013a\u017eky 2\u00a0cm. Potom bolo nieko\u013eko koc\u00f4\u010dok z vidite\u013enej strany odobran\u00fdch (pozri obr\u00e1zok).<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/59\/59z7a3.gif\" alt=\"\" width=\"115\" height=\"109\" \/><\/p>\n<p><strong>1.<\/strong> Ur\u010di objem a povrch z\u00edskan\u00e9ho telesa.<br \/>\n<strong>2.<\/strong> Teleso bolo na celom povrchu natret\u00e9 \u010dervenou farbou, potom rozobran\u00e9 na p\u00f4vodn\u00e9 koc\u00f4\u010dky. Ko\u013eko z nich malo 6, ko\u013eko 5, 4, 3, 2, 1 \u010di \u017eiadnu stenu \u010derven\u00fa?<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z7-I-5<\/strong><br \/>\nNa \u010d\u00edselnej osi s\u00fa zn\u00e1zornen\u00e9 \u010d\u00edsla <em>12x<\/em> a <em>\u20134x<\/em>. Zn\u00e1zorni na tejto osi nulu a \u010d\u00edslo <em>x<\/em>.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/59\/59z7a1.gif\" alt=\"\" width=\"325\" height=\"16\" \/><\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z7-I-6<\/strong> &#8211; <span class=\"povinne\">opraven\u00e9<\/span><br \/>\nDopl\u0148 miesto hviezdi\u010diek \u010d\u00edsla tak, aby s\u00fa\u010det v\u00fdsledkov nasleduj\u00facich dvoch  pr\u00edkladov bol 5842.<\/p>\n<table class=\"r\" style=\"width: 350px;\" border=\"0\" cellspacing=\"0\" cellpadding=\"0\" align=\"center\">\n<tbody>\n<tr height=\"15\">\n<td>*<\/td>\n<td>2<\/td>\n<td>*<\/td>\n<td>7<\/td>\n<td width=\"80\"><\/td>\n<td><\/td>\n<td>2<\/td>\n<td>*<\/td>\n<td>9<\/td>\n<td>*<\/td>\n<\/tr>\n<tr height=\"15\">\n<td>3<\/td>\n<td>*<\/td>\n<td>4<\/td>\n<td>*<\/td>\n<td><\/td>\n<td>&#8211;<\/td>\n<td>*<\/td>\n<td>2<\/td>\n<td>5<\/td>\n<td>4<\/td>\n<\/tr>\n<tr height=\"1\">\n<td colspan=\"4\">\n<hr size=\"1\" \/><\/td>\n<td><\/td>\n<td colspan=\"5\">\n<hr size=\"1\" \/><\/td>\n<\/tr>\n<tr height=\"15\">\n<td>4<\/td>\n<td>*<\/td>\n<td>0<\/td>\n<td>*<\/td>\n<td><\/td>\n<td><\/td>\n<td>*<\/td>\n<td>5<\/td>\n<td>*<\/td>\n<td>*<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u00daloha m\u00e1 viac rie\u0161en\u00ed, ur\u010di aspo\u0148 dve.<\/p>\n<div class=\"pri\">(M. Dillingerov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\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<\/tbody>\n<\/table>\n<p><b>Z7-II-1<\/b><\/p>\n<p>Kremienok a Chochol\u00fa\u0161ik na\u0161li debni\u010dku s pokladom. Ka\u017ed\u00fd z nich si nabral do jedn\u00e9ho vrecka strieborn\u00e9 mince a do druh\u00e9ho vrecka zlat\u00e9 mince. Kremienok mal na pravom vrecku dieru a cestou polovicu zlatiek stratil. Chochol\u00fa\u0161ik mal dieru na \u013eavom vrecku a cestou domov stratil polovicu strieborniakov. Doma venoval Chochol\u00fa\u0161ik tretinu svojich zlatiek Kremienkovi a Kremienok \u0161tvrtinu svojich strieborniakov Chochol\u00fa\u0161ikovi. Ka\u017ed\u00fd potom mal presne 12&nbsp;zlatiek a 18&nbsp;strieborniakov. Ko\u013eko zlatiek a ko\u013eko strieborniakov si vzal ka\u017ed\u00fd z nich z n\u00e1jden\u00e9ho pokladu?<\/p>\n<div class=pri>(M. Dillingerov\u00e1)<\/div>\n<\/p>\n<p><b>Z7-II-2<\/b><\/p>\n<p>Na tabuli s\u00fa nap\u00edsan\u00e9 tri prirodzen\u00e9 \u010d\u00edsla <i>x, y, z<\/i>. Ur\u010di ktor\u00e9, ak vie\u0161, \u017ee s\u00fa\u010dasne plat\u00ed:<\/p>\n<p>&nbsp;&nbsp;&nbsp;\u2022&nbsp;&nbsp;<i>x<\/i> je z nich najv\u00e4\u010d\u0161ie,<br \/>\n&nbsp;&nbsp;&nbsp;\u2022&nbsp;&nbsp;najmen\u0161\u00ed spolo\u010dn\u00fd n\u00e1sobok \u010d\u00edsel <i>x<\/i> a <i>y<\/i> je 200,<br \/>\n&nbsp;&nbsp;&nbsp;\u2022&nbsp;&nbsp;najmen\u0161\u00ed spolo\u010dn\u00fd n\u00e1sobok \u010d\u00edsel <i>y<\/i> a <i>z<\/i> je 300,<br \/>\n&nbsp;&nbsp;&nbsp;\u2022&nbsp;&nbsp;najmen\u0161\u00ed spolo\u010dn\u00fd n\u00e1sobok \u010d\u00edsel <i>x<\/i> a <i>z<\/i> je 120.<\/p>\n<div class=pri>(L. \u0160im\u016fnek)<\/div>\n<\/p>\n<p><b>Z7-II-3<\/b><br \/>\nPravideln\u00e1 \u0161es\u0165c\u00edpa hviezda <i>ABCDEFGHIJKL<\/i> so stredom&nbsp;S, zn\u00e1zornen\u00e1 na obr\u00e1zku, vznikla zjednoten\u00edm dvoch rovnostrann\u00fdch trojuholn\u00edkov, z&nbsp;ktor\u00fdch ka\u017ed\u00fd mal obsah&nbsp;72cm<sup>2<\/sup>. Vypo\u010d\u00edtaj obsah \u0161tvoruholn\u00edka <i>ABCF<\/i>.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/59\/59z7b1.gif\" width=\"146\" height=\"163\" alt=\"\" \/><\/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<tbody>\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<\/tbody>\n<\/table>\n<p><strong>Z8-I-1<\/strong><br \/>\nNap\u00ed\u0161te \u010d\u00edslo 75 ako s\u00fa\u010det nieko\u013eko bezprostredne po sebe id\u00facich prirodzen\u00fdch \u010d\u00edsel. N\u00e1jdite aspo\u0148 \u0161tyri rie\u0161enia.<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z8-I-2<\/strong><br \/>\nTri kamar\u00e1tky sa zi\u0161li na chalupe a vyrazili na huby. Na\u0161li spolu 55\u00a0hr\u00edbov. Po n\u00e1vrate si urobili pra\u017eenicu, rozdelili ju na \u0161tyri rovnak\u00e9 porcie a pozvali na \u0148u kamar\u00e1ta Petra. \u013duba dala na pra\u017eenicu \u0161es\u0165 zo svojich hr\u00edbov, Marienka osem a \u0160\u00e1rka p\u00e4\u0165. Ka\u017edej po tom zostal rovnak\u00fd po\u010det hr\u00edbov. Peter im daroval bonboni\u00e9ru, v ktorej bolo 38\u00a0bonb\u00f3nov, a povedal, \u017ee sa maj\u00fa spravodlivo rozdeli\u0165 pod\u013ea toho, ak\u00fdm dielom prispeli na jeho jedlo.<\/p>\n<\/p>\n<p>\u2022  Ko\u013eko hr\u00edbov na\u0161la ka\u017ed\u00e1?<br \/>\n\u2022  Ako si mali pod\u013ea Petra bonb\u00f3ny podeli\u0165? Ur\u010dte ko\u013eko bonb\u00f3nov si mali jednotliv\u00e9 kamar\u00e1tky vzia\u0165.<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<\/p>\n<p><strong>Z8-I-3<\/strong><br \/>\nSedadl\u00e1 v divadelnom s\u00e1le s\u00fa rozdelen\u00e9 do troch kateg\u00f3ri\u00ed pod\u013ea ich vzdialenosti od javiska. \u201e<em>I.\u00a0miesta<\/em>\u201c s\u00fa najbli\u017e\u0161ie k javisku, tvoria dve p\u00e4tiny kapacity s\u00e1ly a pred\u00e1vaj\u00fa sa za 11\u00a0\u20ac. \u201e<em>II.\u00a0miesta<\/em>\u201c tvoria \u010fal\u0161ie dve p\u00e4tiny kapacity s\u00e1ly a pred\u00e1vaj\u00fa sa za 10\u00a0\u20ac. Ostatn\u00e9 \u201e<em>III.\u00a0miesta<\/em>\u201c sa pred\u00e1vaj\u00fa za 9\u00a0\u20ac. Pred zah\u00e1jen\u00edm predpredaja na sl\u00e1vnostn\u00fa premi\u00e9ru bolo rozdan\u00fdch 150\u00a0vstupeniek zadarmo pozvan\u00fdm hos\u0165om. Vstupenky boli rozd\u00e1van\u00e9 postupne od predn\u00fdch miest s\u00e1ly dozadu. V\u0161etky ostatn\u00e9 vstupenky sa potom predali. Keby sa v\u0161ak vo\u013en\u00e9 vstupenky rozd\u00e1vali postupne od zadn\u00fdch miest dopredu, bola by tr\u017eba o\u00a0216\u00a0\u20ac v\u00e4\u010d\u0161ia.<\/p>\n<p>Ko\u013eko miest bolo v s\u00e1le?<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z8-I-4<\/strong><br \/>\nDostali sme kocku, ktor\u00e1 mala d\u013a\u017eku hrany vyjadren\u00fa v centimetroch cel\u00fdm \u010d\u00edslom. V\u0161etky jej steny sme nafarbili na \u010derveno a potom sme ju rozrezali bezo zvy\u0161ku na koc\u00f4\u010dky s hranou d\u013a\u017eky 1\u00a0cm.<\/p>\n<p>\u2022  Luk\u00e1\u0161 tvrd\u00ed, \u017ee koc\u00f4\u010dok s dvomi nafarben\u00fdmi stenami je desa\u0165kr\u00e1t viac ne\u017e t\u00fdch s tromi nafarben\u00fdmi stenami.<br \/>\n\u2022  Martina tvrd\u00ed, \u017ee koc\u00f4\u010dok s dvomi nafarben\u00fdmi stenami je p\u00e4tn\u00e1s\u0165kr\u00e1t viac ne\u017e t\u00fdch s tromi nafarben\u00fdmi stenami.<\/p>\n<p>Pravdu m\u00e1 v\u0161ak iba jeden \u2014 kto? Ko\u013eko cm merala hrana p\u00f4vodnej kocky?<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<\/p>\n<p><strong>Z8-I-5<\/strong><br \/>\nZo \u0161tvorca so stranou d\u013a\u017eky 6\u00a0cm odstrihneme od ka\u017ed\u00e9ho vrcholu zhodn\u00fd rovnoramenn\u00fd pravouhl\u00fd trojuholn\u00edk tak, aby sa obsah \u0161tvorca zmen\u0161il o\u00a032%. Zistite d\u013a\u017eku odvesien odstrihnut\u00fdch trojuholn\u00edkov.<\/p>\n<div class=\"pri\">(M. Krej\u010dov\u00e1)<\/div>\n<p><strong>Z8-I-6<\/strong><br \/>\nV dvoch miestnostiach vzdel\u00e1vacieho centra sa konali predn\u00e1\u0161ky. Priemern\u00fd vek \u00f4smych \u013eud\u00ed pr\u00edtomn\u00fdch v prvej miestnosti bol 20\u00a0rokov, priemern\u00fd vek dvan\u00e1stich \u013eud\u00ed pr\u00edtomn\u00fdch v druhej miestnosti bol 45\u00a0rokov. Po\u010das predn\u00e1\u0161ky odi\u0161iel jeden \u00fa\u010dastn\u00edk a t\u00fdm sa priemern\u00fd vek v\u0161etk\u00fdch os\u00f4b v oboch miestnostiach zv\u00fd\u0161il o\u00a0jeden rok. Ko\u013eko rokov mal \u00fa\u010dastn\u00edk, ktor\u00fd odi\u0161iel?<\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\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<\/tbody>\n<\/table>\n<p><b>Z8-II-1<\/b><\/p>\n<p>Priemern\u00fd vek rodiny Gebu\u013eov\u00fdch, ktor\u00fa tvor\u00ed otec, mama a nieko\u013eko det\u00ed, je 18&nbsp;rokov. Otec m\u00e1 38&nbsp;rokov a priemern\u00fd vek rodiny bez neho je&nbsp;14. Ko\u013eko det\u00ed maj\u00fa Gebu\u013eovci?<\/p>\n<div class=pri>(L.Hozov\u00e1)<\/div>\n<\/p>\n<p><b>Z8-II-2<\/b><\/p>\n<p>Ko\u013eko existuje \u0161es\u0165cifern\u00fdch prirodzen\u00fdch \u010d\u00edsel delite\u013en\u00fdch bezo zvy\u0161ku 45 tak\u00fdch, ktor\u00e9 maj\u00fa na mieste stotis\u00edcok \u010d\u00edslicu 1, na mieste tis\u00edcok \u010d\u00edslicu 2 a na mieste desiatok \u010d\u00edslicu 3?<\/p>\n<div class=pri>(L. \u0160im\u016fnek)<\/div>\n<\/p>\n<p><b>Z8-II-3<\/b><\/p>\n<p>Na obr\u00e1zku je \u0161es\u0165uholn\u00edk <i>ABEFGD<\/i>. \u0160tvoruholn\u00edky <i>ABCD<\/i> a <i>EFGC<\/i> s\u00fa zhodn\u00e9 obd\u013a\u017eniky a \u0161tvoruholn\u00edk <i>BEGD<\/i> je tie\u017e obd\u013a\u017enik. Ur\u010dte pomer obsahov bielej a sivej \u010dasti \u0161es\u0165uholn\u00edka, ak |<i>AB<\/i>|&nbsp;=&nbsp;5&nbsp;cm a trojuholn\u00edk <i>BEC<\/i> je rovnostrann\u00fd.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/59\/59z8b1.gif\" width=\"211\" height=\"130\" alt=\"\" \/><\/p>\n<div class=pri>(K.Pazourek)<\/div>\n<\/p>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\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<\/tbody>\n<\/table>\n<p><strong>Z9-I-1<\/strong><br \/>\nDostal som zadan\u00e9 dve prirodzen\u00e1 \u010d\u00edsla. Potom som ich obe zaokr\u00fahlil na desiatky. Ktor\u00e9 \u010d\u00edsla som dostal zadan\u00e9, ak viete, \u017ee s\u00fa\u010dasne plat\u00ed:<\/p>\n<p>\u2022  podiel zaokr\u00fahlen\u00fdch \u010d\u00edsel je rovnak\u00fd ako podiel p\u00f4vodn\u00fdch \u010d\u00edsel,<br \/>\n\u2022  s\u00fa\u010din zaokr\u00fahlen\u00fdch \u010d\u00edsel je o\u00a0295 v\u00e4\u010d\u0161\u00ed ne\u017e s\u00fa\u010din p\u00f4vodn\u00fdch \u010d\u00edsel,<br \/>\n\u2022  s\u00fa\u010det zaokr\u00fahlen\u00fdch \u010d\u00edsel je o\u00a06 v\u00e4\u010d\u0161\u00ed ne\u017e s\u00fa\u010det p\u00f4vodn\u00fdch \u010d\u00edsel.<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<\/p>\n<p><strong>Z9-I-2<\/strong><br \/>\nPat a Mat boli na v\u00fdlete. Vy\u0161li r\u00e1no po \u00f4smej hodine v \u010dase, ke\u010f ve\u013ek\u00e1 a mal\u00e1 ru\u010di\u010dka na Patov\u00fdch hodink\u00e1ch le\u017eali na opa\u010dn\u00fdch polpriamkach. Na opa\u010dn\u00fdch polpriamkach le\u017eali ru\u010di\u010dky Patov\u00fdch hodiniek aj v \u010dase, ke\u010f sa obaja priatelia pred poludn\u00edm vr\u00e1tili. Mat dobu trvania v\u00fdletu meral stopkami. Ur\u010dite aj vy s presnos\u0165ou na sekundy, ako dlho trval v\u00fdlet.<br \/>\n(<em>Predpokladajte, \u017ee Patove hodinky a Matove stopky i\u0161li presne.<\/em>)<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z9-I-3<\/strong><br \/>\nNa obr\u00e1zku je kocka s hranou d\u013a\u017eky 2\u00a0cm tvoren\u00e1 \u00f4smimi koc\u00f4\u010dkami s hranou d\u013a\u017eky 1\u00a0cm. Osem stien koc\u00f4\u010dok je nafarben\u00fdch na \u010dierno, ostatn\u00e9 s\u00fa biele. Pritom sa z nich d\u00e1 zlo\u017ei\u0165 kocka, ktorej povrch je biely. Ko\u013ek\u00fdmi sp\u00f4sobmi m\u00f4\u017eu by\u0165 koc\u00f4\u010dky nafarben\u00e9?<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/59\/59z9a1.gif\" alt=\"\" width=\"72\" height=\"71\" \/><\/p>\n<p>(<em>Predpokladajte, \u017ee rovnako nafarben\u00e9 koc\u00f4\u010dky nedok\u00e1\u017eeme odl\u00ed\u0161i\u0165, mo\u017eno ich teda zamie\u0148a\u0165.<\/em>)<\/p>\n<div class=\"pri\">(K. Pazourek)<\/div>\n<p><strong>Z9-I-4<\/strong><br \/>\nAdam a Eva dostali ko\u0161\u00edk, v ktorom bolo 31\u00a0jab\u013ak. Prv\u00fd de\u0148 zjedla Eva tri \u0161tvrtiny toho, \u010do zjedol Adam. Druh\u00fd de\u0148 zjedla Eva dve tretiny toho, \u010do zjedol druh\u00fd de\u0148 Adam. Druh\u00fd de\u0148 ve\u010der bol ko\u0161\u00edk pr\u00e1zdny. Ko\u013eko jab\u013ak zjedla z ko\u0161\u00edku Eva?<br \/>\n(<em>Adam i Eva jablk\u00e1 jedia cel\u00e9 a nedelia si ich.<\/em>)<\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z9-I-5<\/strong><\/p>\n<p>Vodi\u010d prev\u00e1\u017ea mlieko v cisterne tvaru valca. Priemer podstavy je 180\u00a0cm, d\u013a\u017eka cisterny je 4\u00a0m. Ko\u013eko hl mlieka je v cisterne, ak je naplnen\u00e1 do troch \u0161tvrt\u00edn priemeru?<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/59\/59z9a3.gif\" alt=\"\" width=\"334\" height=\"97\" \/><\/p>\n<div class=\"pri\">(M. Krej\u010dov\u00e1)<\/div>\n<p><strong>Z9-I-6<\/strong><br \/>\nV lichobe\u017en\u00edku <em>ABCD<\/em> so z\u00e1klad\u0148ami <em>AB<\/em> a <em>CD<\/em> d\u013a\u017eok 7\u00a0cm a 4\u00a0cm s\u00fa body <em>S<\/em> a <em>T<\/em> stredy str\u00e1n <em>AD<\/em> a <em>BC<\/em> (pozri obr\u00e1zok). Bod <em>X<\/em> je priese\u010dn\u00edk \u00fase\u010diek <em>AC<\/em> a <em>ST<\/em>, bod <em>Y<\/em> je priese\u010dn\u00edk \u00fase\u010dky <em>AB<\/em> a priamky <em>DX<\/em>. Obsah \u0161tvoruholn\u00edka <em>AYCD<\/em> je 12\u00a0cm<sup>2<\/sup>. Vypo\u010d\u00edtajte obsah lichobe\u017en\u00edka <em>ABCD<\/em>.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"59z9a2.gif\" alt=\"\" width=\"257\" height=\"131\" \/><\/p>\n<div class=\"pri\">(M. Dillingerov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\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<\/tbody>\n<\/table>\n<p><strong>Z9-II-1<\/strong><br \/>\nDopl\u0148te do pr\u00e1zdnych pol\u00ed\u010dok na obr\u00e1zku \u010d\u00edsla tak, aby v ka\u017edom pol\u00ed\u010dku bol s\u00fa\u010det \u010d\u00edsel zo v\u0161etk\u00fdch s n\u00edm priamo susediacich pol\u00ed\u010dok, ktor\u00e9 s\u00fa svetlej\u0161ie ako dop\u013a\u0148an\u00e9. To znamen\u00e1, \u017ee vo svetlosivom pol\u00ed\u010dku je s\u00fa\u010det \u010d\u00edsel zo susedn\u00fdch bielych pol\u00ed\u010dok a v tmavosivom pol\u00ed\u010dku je s\u00fa\u010det \u010d\u00edsel zo susedn\u00fdch svetlosiv\u00fdch pol\u00ed\u010dok.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/59\/59z9b1.gif\" alt=\"\" width=\"173\" height=\"187\" \/><\/p>\n<div class=\"pri\">(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z9-II-2<\/strong><\/p>\n<p>\u0160\u00e1rka naplnila poh\u00e1r a hrn\u010dek, ktor\u00fd mal dvakr\u00e1t v\u00e4\u010d\u0161\u00ed objem ako poh\u00e1r, vodou s d\u017e\u00fasom. Pomer vody a d\u017e\u00fasu bol v poh\u00e1ri <em>2\u00a0:\u00a01<\/em> a v hrn\u010deku <em>4\u00a0:\u00a01<\/em>. Potom preliala obsah poh\u00e1ra i obsah hrn\u010deka do d\u017eb\u00e1nu. Ak\u00fd bol pomer vody a d\u017e\u00fasu v d\u017eb\u00e1ne?<\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z9-II-3<\/strong><br \/>\nDostal som zadan\u00e9 dve dvojcifern\u00e9 prirodzen\u00e9 \u010d\u00edsla. Potom som ich obe zaokr\u00fahlil na desiatky. Ur\u010dite, ktor\u00e9 \u010d\u00edsla som mal zadan\u00e9, ak viete, \u017ee s\u00fa\u010dasne plat\u00ed:<\/p>\n<p>\u2022  rozdiel zaokr\u00fahlen\u00fdch \u010d\u00edsiel je rovnak\u00fd ako rozdiel p\u00f4vodn\u00fdch \u010d\u00edsiel,<br \/>\n\u2022  s\u00fa\u010din zaokr\u00fahlen\u00fdch \u010d\u00edsiel je o\u00a0184 v\u00e4\u010d\u0161\u00ed ne\u017e s\u00fa\u010din p\u00f4vodn\u00fdch \u010d\u00edsiel.<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<\/p>\n<p><strong>Z9-II-4<\/strong><br \/>\nDo rovnostrann\u00e9ho trojuholn\u00edka <em>ABC<\/em> je vp\u00edsan\u00fd pravideln\u00fd \u0161es\u0165uholn\u00edk <em>KLMNOP<\/em> tak, \u017ee body <em>K, L<\/em> le\u017eia na strane <em>AB<\/em>, body <em>M, N<\/em> le\u017eia na strane <em>BC<\/em> a body <em>O, P<\/em> le\u017eia na strane <em>AC<\/em>. Vypo\u010d\u00edtajte obsah \u0161es\u0165uholn\u00edka <em>KLMNOP<\/em>, ak obsah trojuholn\u00edka <em>ABC<\/em> je 60\u00a0cm<sup>2<\/sup>.<\/p>\n<div class=\"pri\">(K. Pazourek)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\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<\/tbody>\n<\/table>\n<p><strong>Z9-III-1<\/strong><br \/>\nPani u\u010dite\u013eka potrebovala vymyslie\u0165 pr\u00edklady na rovnice do p\u00edsomky. Preto si vyp\u00edsala v\u0161etky rovnice tvaru<\/p>\n<p class=\"r\">\u2665\u00a0.\u00a0x\u00a0+\u00a0\u2663\u00a0=\u00a013,<\/p>\n<p>kde \u2665 a \u2663 s\u00fa jednocifern\u00e9 prirodzen\u00e9 \u010d\u00edsla. Zo v\u0161etk\u00fdch rovn\u00edc vybrala tie, ktor\u00fdch kore\u0148 bol\u00a03. V p\u00edsomke dala do ka\u017edej skupiny jednu z nich. Ko\u013eko najviac skup\u00edn mohla ma\u0165?<\/p>\n<div class=\"pri\">(K. Pazourek)<\/div>\n<p><strong>Z9-III-2<\/strong><br \/>\nPo\u010det dom\u00e1cich a doch\u00e1dzaj\u00facich \u017eiakov \u0161koly je v pomere <em>3:1<\/em>. Dom\u00e1ci chodia pe\u0161o. U doch\u00e1dzaj\u00facich je pomer po\u010dtu t\u00fdch, \u010do vyu\u017e\u00edvaj\u00fa verejn\u00fa dopravu, a t\u00fdch, \u010do jazdia sami na bicykli alebo s rodi\u010dmi autom, <em>3:2<\/em>. U verejnej dopravy je pomer t\u00fdch, \u010do jazdia vlakom, a t\u00fdch, \u010do autobusom, <em>7:5<\/em>. \u010ealej vieme, \u017ee pomer t\u00fdch, \u010do doch\u00e1dzaj\u00fa na bicykli, k po\u010dtu t\u00fdch, ktor\u00fdch vozia rodi\u010dia autom, je <em>5:3<\/em>. O ko\u013eko viac \u017eiakov doch\u00e1dza vlakom ako vozia rodi\u010dia, ke\u010f verejnou dopravou ich doch\u00e1dza 24? Ko\u013eko je \u017eiakov \u0161koly?<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z9-III-3<\/strong><br \/>\nDostali sme kocku, ktor\u00e1 mala d\u013a\u017eku hrany vyjadren\u00fa v centimetroch cel\u00fdm \u010d\u00edslom v\u00e4\u010d\u0161\u00edm ako 2. V\u0161etky jej steny sme nafarbili na \u017elto a potom sme ju rozrezali bezo zvy\u0161ku na koc\u00f4\u010dky s hranou d\u013a\u017eky 1\u00a0cm. Tieto koc\u00f4\u010dky sme roztriedili na \u0161tyri k\u00f4pky. Na prvej boli koc\u00f4\u010dky s jednou \u017eltou stenou, na druhej s dvomi \u017elt\u00fdmi stenami a na tretej s tromi. Na \u0161tvrtej k\u00f4pke potom boli koc\u00f4\u010dky bez \u017eltej steny. Ur\u010dte d\u013a\u017eku hrany p\u00f4vodnej kocky, ak viete, \u017ee aspo\u0148 jedno z nasleduj\u00facich tvrden\u00ed je pravdiv\u00e9:<\/p>\n<p>\u2022  Po\u010dty koc\u00f4\u010dok v prvej a \u0161tvrtej k\u00f4pke boli v pomere 4:9.<br \/>\n\u2022  Na prvej k\u00f4pke bolo trikr\u00e1t viac koc\u00f4\u010dok ako na druhej.<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<\/p>\n<p><strong>Z9-III-4<\/strong><br \/>\nV rovnostrannom trojuholn\u00edku <em>ABC<\/em> le\u017e\u00ed pravideln\u00fd \u0161es\u0165uholn\u00edk <em>KLMNOP<\/em> tak, \u017ee body <em>K, M, O<\/em> s\u00fa postupne stredmi str\u00e1n <em>AB<\/em>, <em>BC<\/em> a <em>AC<\/em>. Vypo\u010d\u00edtajte obsah \u0161es\u0165uholn\u00edka <em>KLMNOP<\/em>, ak obsah trojuholn\u00edka <em>ABC<\/em> je 60\u00a0cm<sup>2<\/sup>.<\/p>\n<div class=\"pri\">(K. 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