{"id":854,"date":"2013-09-03T19:17:42","date_gmt":"2013-09-03T19:17:42","guid":{"rendered":"http:\/\/matematika.okamzite.eu\/?p=854"},"modified":"2013-09-03T19:17:42","modified_gmt":"2013-09-03T19:17:42","slug":"sutazne-ulohy-kategorii-z5-z9-2","status":"publish","type":"post","link":"https:\/\/matematika.besaba.com\/?p=854","title":{"rendered":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9"},"content":{"rendered":"<p class=\"hh2\"><a name=\"top\"><\/a>63. ro\u010dn\u00edk matematickej olympi\u00e1dy 2013-2014<\/p>\n<p><span class=\"povinne\">aktualizovan\u00e9 20.8.2014 14:40<\/span><\/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<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\">obvodn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z62\">obvodn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z72\">obvodn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z82\">obvodn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z92\">obvodn\u00e9<\/a><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"4\" bgcolor=\"#c0c0c0\">\u00a0<\/td>\n<td align=\"center\"><span class=\"a\"><a href=\"#z92\">krajsk\u00e9<\/a><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"background-color: #ffff99;\" colspan=\"5\" align=\"center\"><span class=\"rs\">Zadania \u00faloh dom\u00e1ceho kola na stiahnutie vo form\u00e1te PDF: <a href=\"\/63\/63_z56789.pdf\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" src=\"\/img\/down.png\" width=\"48\" height=\"48\" align=\"absmiddle\" border=\"0\" \/><\/a> (ulo\u017ei\u0165 ako &#8230;)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"z51\"><\/a><\/p>\n<form><input onclick=\"window.location='http:\/\/skmo.sk\/dokument.php?id=998'\" type=\"button\" name=\"butt51\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z5-I-1<\/strong><br \/> Medzi dvoma ty\u010dami je napnut\u00e1 \u0161n\u00fara dlh\u00e1 3,8m, na ktor\u00fa chce mami\u010dka zavesi\u0165 vyprat\u00e9 vreckovky. V\u0161etky vreckovky maj\u00fa tvar \u0161tvorca so stranou 40 cm. Na \u0161n\u00fare v\u0161ak u\u017e visia dve vreckovky rovnak\u00e9ho tvaru od susedky a tie chce mami\u010dka necha\u0165 na svojich miestach. Pritom \u013eav\u00fd roh jednej z t\u00fdchto vreckoviek je 60 cm od \u013eavej ty\u010de a \u013eav\u00fd roh tej druhej je 1,3m od pravej ty\u010de. Ko\u013eko najviac vreckoviek m\u00f4\u017ee mami\u010dka na \u0161n\u00faru zavesi\u0165? Vreckovky sa ve\u0161aj\u00fa natiahnut\u00e9 za oba rohy tak, aby sa \u017eiadne dve neprekr\u00fdvali.<\/p>\n<div class=\"pri\">(M. Mach)<\/div>\n<p><strong>Z5-I-2<\/strong><br \/> Vojto m\u00e1 dve rovnak\u00e9 skl\u00ed\u010dka tvaru rovnostrann\u00e9ho trojuholn\u00edka, ktor\u00e9 sa l\u00ed\u0161ia iba svojou farbou &#8211; jedno je \u010derven\u00e9, druh\u00e9 modr\u00e9. Ke\u010f sa skl\u00ed\u010dka polo\u017eia cez seba, vznikne \u00fatvar fialovej farby. Uve\u010fte pr\u00edklad prekr\u00fdvania skl\u00ed\u010dok, pri ktorom mohol Vojto dosta\u0165:<\/p>\n<p>1. fialov\u00fd trojuholn\u00edk,<\/p>\n<p>2. fialov\u00fd \u0161tvoruholn\u00edk,<\/p>\n<p>3. fialov\u00fd p\u00e4\u0165uholn\u00edk,<\/p>\n<p>4. fialov\u00fd \u0161es\u0165uholn\u00edk.<\/p>\n<div class=\"pri\">(E. Novotn\u00e1)<\/div>\n<p><strong>Z5-I-3<\/strong><br \/> Palindr\u00f3m je tak\u00e9 \u010d\u00edslo, ktor\u00e9 je rovnak\u00e9, \u010di u\u017e ho \u010d\u00edtame spredu alebo zozadu. (Napr. \u010d\u00edslo 1881 je palindr\u00f3m.) N\u00e1jdite tak\u00fd dvojcifern\u00fd a trojcifern\u00fd palindr\u00f3m, aby ich s\u00fa\u010det bol \u0161tvorcifern\u00fd palindr\u00f3m.<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z5-I-4<\/strong><br \/> Eve sa p\u00e1\u010dia \u010d\u00edsla delite\u013en\u00e9 \u0161iestimi, Zdenke \u010d\u00edsla obsahuj\u00face aspo\u0148 jednu \u0161estku a Jane \u010d\u00edsla, ktor\u00fdch cifern\u00fd s\u00fa\u010det je 6.<\/p>\n<p>1. Ktor\u00e9 dvojcifern\u00e9 \u010d\u00edsla sa p\u00e1\u010dia v\u0161etk\u00fdm trom diev\u010dat\u00e1m?<\/p>\n<p>2. Ktor\u00e9 dvojcifern\u00e9 \u010d\u00edsla sa dvom diev\u010dat\u00e1m p\u00e1\u010dia, ale jednej sa nep\u00e1\u010dia?<\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z5-I-5<\/strong><br \/> Dopl\u0148te do pr\u00e1zdnych kr\u00fa\u017ekov na obr\u00e1zku prirodzen\u00e9 \u010d\u00edsla tak, aby s\u00fa\u010det \u010d\u00edsel na ka\u017edej strane trojuholn\u00edka bol rovnak\u00fd a aby s\u00fa\u010det v\u0161etk\u00fdch \u0161iestich \u010d\u00edsel bol 100.<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z5-I-6<\/strong><br \/> Recep\u010dn\u00e1 v hoteli si vykladala karty a dostala nasleduj\u00facu postupnos\u0165:<\/p>\n<p class=\"r\">5, 9, 2, 7, 3, 6, 8, 4.<\/p>\n<p>Presunula dve susedn\u00e9 karty na in\u00e9 miesto tak, \u017ee t\u00e1to dvojica op\u00e4\u0165 susedila, a to v rovnakom porad\u00ed. Tento krok urobila celkom trikr\u00e1t, k\u00fdm neboli karty usporiadan\u00e9 vzostupne pod\u013ea svojej hodnoty. Zistite, ako recep\u010dn\u00e1 postupovala.<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z52\"><\/a><a href=\"#top\"><img decoding=\"async\" alt=\"\" src=\"\/img\/top.gif\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><input onclick=\"window.location='http:\/\/skmo.sk\/dokument.php?id=1000'\" type=\"button\" name=\"butt52\" value=\"Zadanie\" \/><\/form>\n<form><input onclick=\"window.location='http:\/\/skmo.sk\/dokument.php?id=1001'\" type=\"button\" name=\"butt52\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z5-II-1<\/strong><br \/> Na kraji cesty je za sebou nieko\u013eko rovnako dlh\u00fdch parkovac\u00edch miest. Jeden autobus st\u00e1l na piatom a\u017e siedmom mieste z\u013eava, druh\u00fd autobus zaberal \u00f4sme a\u017e desiate miesto sprava, inak bolo parkovisko pr\u00e1zdne. Nesk\u00f4r tu zaparkovali \u010fal\u0161ie \u0161tyri autobusy a \u017eiadny \u010fal\u0161\u00ed autobus u\u017e sa na parkovisko nezmestil. Ur\u010dte, ko\u013eko najviac parkovac\u00edch miest mohlo by\u0165 na parkovisku, ak ka\u017ed\u00fd autobus zaber\u00e1 presne tri parkovacie miesta.<\/p>\n<div class=\"pri\">(E. Pat\u00e1kov\u00e1)<\/div>\n<p><strong>Z5-II-2<\/strong><br \/> \u013dubo\u0161 rozdelil obd\u013a\u017enik jednou \u010diarou na dva men\u0161ie obd\u013a\u017eniky. Obvod ve\u013ek\u00e9ho obd\u013a\u017enika je 76 cm, obvody men\u0161\u00edch obd\u013a\u017enikov s\u00fa 40 cm a 52 cm. Ur\u010dte rozmery ve\u013ek\u00e9ho obd\u013a\u017enika.<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z5-II-3<\/strong><br \/> Juraj a Peter spustili stopky a niekedy v priebehu prv\u00fdch 15 sek\u00fand od spustenia ka\u017ed\u00fd z nich za\u010dal tlieska\u0165. Juraj po svojom prvom tlesknut\u00ed tlieskal pravidelne ka\u017ed\u00fdch 7 sek\u00fand, Peter po svojom prvom tlesknut\u00ed tlieskal pravidelne ka\u017ed\u00fdch 13 sek\u00fand. V 90. sekunde tleskli obaja naraz. Ur\u010dte v\u0161etky mo\u017enosti, kedy mohol za\u010da\u0165 tlieska\u0165 Juraj a kedy Peter.<\/p>\n<div class=\"pri\">(L. Dedkov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z61\"><\/a><a href=\"#top\"><img decoding=\"async\" alt=\"\" src=\"\/img\/top.gif\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><input onclick=\"window.location='http:\/\/skmo.sk\/dokument.php?id=1075'\" type=\"button\" name=\"butt51\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z6-I-1<\/strong><br \/> V tov\u00e1rni na v\u00fdrobu ply\u0161ov\u00fdch hra\u010diek maj\u00fa dva stroje. Prv\u00fd vyrob\u00ed \u0161tyroch zajacov za rovnak\u00fd \u010das, za ktor\u00fd vyrob\u00ed druh\u00fd p\u00e4\u0165 medve\u010fov. Aby bolo ich ovl\u00e1danie jednoduch\u0161ie, oba stroje sa sp\u00fa\u0161\u0165aj\u00fa a vyp\u00ednaj\u00fa naraz spolo\u010dn\u00fdm vyp\u00edna\u010dom. Navy\u0161e s\u00fa stroje nastaven\u00e9 tak, \u017ee prv\u00fd po spusten\u00ed najsk\u00f4r vyrob\u00ed troch ru\u017eov\u00fdch zajacov, potom jedn\u00e9ho modr\u00e9ho, potom zasa troch ru\u017eov\u00fdch at\u010f. Druh\u00fd po spusten\u00ed najsk\u00f4r vyrob\u00ed \u0161tyroch modr\u00fdch medve\u010fov, potom jedn\u00e9ho ru\u017eov\u00e9ho, potom op\u00e4\u0165 \u0161tyroch modr\u00fdch at\u010f. Po istom \u010dase bolo na t\u00fdchto dvoch strojoch vyroben\u00fdch celkom 220 modr\u00fdch hra\u010diek. Ko\u013eko bolo vtedy vyroben\u00fdch ru\u017eov\u00fdch zajacov?<\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z6-I-2<\/strong><br \/> Juro, Mi\u0161o, Peter, Filip a Samo sk\u00e1kali do dia\u013eky. Samo sko\u010dil 135 cm, Peter sko\u010dil o 4 cm viac ako Juro, Juro o 6 cm menej ako Mi\u0161o a Mi\u0161o o 7 cm menej ako Filip. Navy\u0161e Filipov skok bol presne v polovici medzi Petrov\u00fdm a Samov\u00fdm. Zistite, ko\u013eko cm sko\u010dili jednotliv\u00ed chlapci.<\/p>\n<div class=\"pri\">(M. Dillingerov\u00e1)<\/div>\n<p><strong>Z6-I-3<\/strong><br \/> Ko\u013eko mus\u00edme nap\u00edsa\u0165 cifier, ak chceme vyp\u00edsa\u0165 v\u0161etky prirodzen\u00e9 \u010d\u00edsla od 1 do 2013?<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z6-I-4<\/strong><br \/> Spr\u00e1vne vyplnen\u00e1 tabu\u013eka na obr\u00e1zku m\u00e1 obsahova\u0165 \u0161es\u0165 prirodzen\u00fdch \u010d\u00edsel, pri\u010dom v ka\u017edom sivom pol\u00ed\u010dku m\u00e1 by\u0165 s\u00fa\u010det \u010d\u00edsel z dvoch bielych pol\u00ed\u010dok, ktor\u00e9 s n\u00edm susedia.<\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td width=\"50\">\u00a0<\/td>\n<td bgcolor=\"#999999\" width=\"50\">\u00a0<\/td>\n<td width=\"50\">\u00a0<\/td>\n<td width=\"50\">\u00a0<\/td>\n<td bgcolor=\"#999999\" width=\"50\">\u00a0<\/td>\n<td width=\"50\">\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Ur\u010dte \u010d\u00edsla spr\u00e1vne vyplnenej tabu\u013eky, ak viete, \u017ee s\u00fa\u010det prv\u00fdch dvoch \u010d\u00edsel z\u013eava je 33, s\u00fa\u010det prv\u00fdch dvoch \u010d\u00edsel sprava je 28 a s\u00fa\u010det v\u0161etk\u00fdch \u0161iestich \u010d\u00edsel je 64.<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z6-I-5<\/strong><br \/> Adam dostal od deda dreven\u00e9 kocky. V\u0161etky boli rovnak\u00e9 a mali hranu dlh\u00fa 4 cm. Rozhodol sa, \u017ee z nich bude stava\u0165 kom\u00edny, a to tak\u00e9:<\/p>\n<ul type=\"disc\">\n<li>aby boli pou\u017eit\u00e9 v\u0161etky kocky,<\/li>\n<li>aby kom\u00edn pri poh\u013eade zhora vyzeral ako &#8222;dut\u00fd obd\u013a\u017enik&#8220; alebo &#8222;dut\u00fd \u0161tvorec&#8220; ohrani\u010den\u00fd jedn\u00fdm radom kociek (podobne ako na obr\u00e1zku),<\/li>\n<li>aby ani v najvy\u0161\u0161ej vrstve \u017eiadna kocka nech\u00fdbala.<\/li>\n<\/ul>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td width=\"25\" height=\"25\">\u00a0<\/td>\n<td width=\"25\">\u00a0<\/td>\n<td width=\"25\">\u00a0<\/td>\n<td width=\"25\">\u00a0<\/td>\n<td width=\"25\">\u00a0<\/td>\n<\/tr>\n<tr>\n<td height=\"25\">\u00a0<\/td>\n<td rowspan=\"2\" colspan=\"3\">\u00a0<\/td>\n<td>\u00a0<\/td>\n<\/tr>\n<tr>\n<td height=\"25\">\u00a0<\/td>\n<td>\u00a0<\/td>\n<\/tr>\n<tr>\n<td height=\"25\">\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Adam zistil, \u017ee kom\u00edn vysok\u00fd 16 cm, 20 cm aj 24 cm sa pod\u013ea t\u00fdchto pravidiel ur\u010dite d\u00e1 z jeho kociek postavi\u0165.<\/p>\n<ul style=\"list-style-type: decimal;\">\n<li>Ak\u00fd najmen\u0161\u00ed po\u010det kociek mohol Adam dosta\u0165 od deda?<\/li>\n<li>Ak\u00fd vysok\u00fd je najvy\u0161\u0161\u00ed kom\u00edn, ktor\u00fd m\u00f4\u017ee Adam s t\u00fdmto najmen\u0161\u00edm po\u010dtom kociek postavi\u0165 pod\u013ea uveden\u00fdch pravidiel?<\/li>\n<\/ul>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z6-I-6<\/strong><br \/> Na obr\u00e1zku je sie\u0165 zlo\u017een\u00e1 z 20 zhodn\u00fdch obd\u013a\u017enikov, do ktorej sme zakreslili tri \u00fatvary a vyfarbili ich. Obd\u013a\u017enik ozna\u010den\u00fd p\u00edsmenom A a \u0161es\u0165uholn\u00edk ozna\u010den\u00fd p\u00edsmenom B maj\u00fa zhodn\u00e9 obvody, a to 56 cm. Vypo\u010d\u00edtajte obvod tretieho \u00fatvaru ozna\u010den\u00e9ho p\u00edsmenom C.<\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td bgcolor=\"#999999\" width=\"40\" height=\"25\">\u00a0<\/td>\n<td align=\"center\" bgcolor=\"#999999\" width=\"40\">A<\/td>\n<td bgcolor=\"#999999\" width=\"40\">\u00a0<\/td>\n<td width=\"40\">\u00a0<\/td>\n<\/tr>\n<tr>\n<td height=\"25\">\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<\/tr>\n<tr>\n<td bgcolor=\"#999999\" height=\"25\">\u00a0<\/td>\n<td>\u00a0<\/td>\n<td>\u00a0<\/td>\n<td bgcolor=\"#999999\">\u00a0<\/td>\n<\/tr>\n<tr>\n<td align=\"center\" bgcolor=\"#999999\" height=\"25\">B<\/td>\n<td bgcolor=\"#999999\">\u00a0<\/td>\n<td>\u00a0<\/td>\n<td align=\"center\" bgcolor=\"#999999\">C<\/td>\n<\/tr>\n<tr>\n<td bgcolor=\"#999999\" height=\"25\">\u00a0<\/td>\n<td bgcolor=\"#999999\">\u00a0<\/td>\n<td>\u00a0<\/td>\n<td bgcolor=\"#999999\">\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/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\" alt=\"\" src=\"\/img\/top.gif\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><input onclick=\"window.location='http:\/\/skmo.sk\/dokument.php?id=1082'\" type=\"button\" name=\"butt51\" value=\"Zadanie\" \/><input onclick=\"window.location='http:\/\/skmo.sk\/dokument.php?id=1085'\" type=\"button\" name=\"butt51\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z6-II-1<\/strong><br \/> Ke\u010f kr\u00e1\u010dame pozd\u013a\u017e plota zo severu na juh, s\u00fa rozstupy medzi jeho st\u013apikmi zoza\u010diatku zhodn\u00e9. Od ur\u010dit\u00e9ho st\u013apika sa rozstup zmen\u0161\u00ed na 2,9 metra a tak\u00fd zost\u00e1va a\u017e po ju\u017en\u00fd koniec plota. Medzi 1. a 16. st\u013apikom (po\u010d\u00edtan\u00e9 od severu) sa rozstupy nemenia a vzdialenos\u0165 medzi t\u00fdmito dvoma st\u013apikmi je 48 metrov. Vzdialenos\u0165 medzi 16. a 28. st\u013apikom je 36 metrov. Ko\u013ek\u00fd st\u013apik m\u00e1 od svojich susedn\u00fdch st\u013apikov r\u00f4zne rozstupy<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z6-II-2<\/strong><br \/> Ivana, Majka, Lucka, Sa\u0161a a Zuzka pretekali v \u010d\u00edtan\u00ed tej istej knihy. Za jednu hodinu stihla Lucka pre\u010d\u00edta\u0165 32 str\u00e1n, \u010do bolo presne v strede medzi po\u010dtami str\u00e1n, ktor\u00e9 stihli pre\u010d\u00edta\u0165 Sa\u0161a a Zuzka. Ivana pre\u010d\u00edtala o 5 str\u00e1n viac ako Zuzka a Majka pre\u010d\u00edtala o 8 str\u00e1n menej ako Sa\u0161a. Ivanin v\u00fdsledok bol presne v strede medzi Majkin\u00fdm a Zuzkin\u00fdm. Ur\u010dte, ko\u013eko str\u00e1n pre\u010d\u00edtali jednotliv\u00e9 diev\u010dat\u00e1.<\/p>\n<div class=\"pri\">(M. Dillingerov\u00e1)<\/div>\n<p><strong>Z6-II-3<\/strong><br \/> Na obr\u00e1zku je zn\u00e1zornen\u00fdch nieko\u013eko pravouholn\u00edkov s nieko\u013ek\u00fdmi spolo\u010dn\u00fdmi vrcholmi. Ich d\u013a\u017eky str\u00e1n zap\u00edsan\u00e9 v centimetroch s\u00fa cel\u00e9 \u010d\u00edsla. Obsah pravouholn\u00edka DRAK je rovn\u00fd 44 cm<sup>2<\/sup>, obsah pravouholn\u00edka DUPE je rovn\u00fd 64 cm<sup>2<\/sup> a obsah mnohouholn\u00edka DUPLAK je rovn\u00fd 92 cm<sup>2<\/sup>. Ur\u010dte d\u013a\u017eky str\u00e1n mnohouholn\u00edka DUPLAK.<\/p>\n<p class=\"pc\"><img decoding=\"async\" alt=\"\" src=\"..\/63\/63z6b1.png\" \/><\/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 align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z71\"><\/a><a href=\"#top\"><img decoding=\"async\" alt=\"\" src=\"\/img\/top.gif\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><input onclick=\"window.location='http:\/\/skmo.sk\/dokument.php?id=1074'\" type=\"button\" name=\"butt51\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z7-I-1<\/strong><br \/> Na lavi\u010dke v parku sedia ved\u013ea seba Ani\u010dka, Barborka, Cilka, Dominik a Edo. Ani\u010dka m\u00e1 4 roky, Edo m\u00e1 10 rokov, s\u00fa\u010din vekov Ani\u010dky, Barborky a Cilky je 140, s\u00fa\u010din vekov Barborky, Cilky a Dominika je 280 a s\u00fa\u010din vekov Cilky, Dominika a Eda je 560. Ko\u013eko rokov m\u00e1 Cilka?<\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z7-I-2<\/strong><br \/> K starej mame pri\u0161li na pr\u00e1zdniny vnuci &#8211; p\u00e4\u0165 r\u00f4zne star\u00fdch bratov. Star\u00e1 mama im povedala, \u017ee pre nich m\u00e1 celkom 60\u20ac ako vreckov\u00e9, ktor\u00e9 si maj\u00fa rozdeli\u0165 tak, aby:<\/p>\n<ul>\n<li>najstar\u0161\u00ed dostal najviac,<\/li>\n<li>ka\u017ed\u00fd mlad\u0161\u00ed dostal o ur\u010dit\u00fa \u010diastku menej ako jeho star\u0161\u00ed vekom najbli\u017e\u0161\u00ed s\u00farodenec,<\/li>\n<li>t\u00e1to \u010diastka bola st\u00e1le rovnak\u00e1,<\/li>\n<li>najmlad\u0161\u00ed dostal sumu, ktor\u00e1 sa d\u00e1 vyplati\u0165 v jednoeurovk\u00e1ch a ktor\u00e1 nie je men\u0161ia ako 5\u20ac, ale nie je v\u00e4\u010d\u0161ia ako 8\u20ac.<\/li>\n<\/ul>\n<p>Ur\u010dte v\u0161etky mo\u017enosti, ako si mohli vnuci vreckov\u00e9 rozdeli\u0165.<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z7-I-3<\/strong><br \/> Juro, Mi\u0161o, Peter, Filip a Samo sk\u00e1kali do dia\u013eky. Samo sko\u010dil 135 cm, Peter sko\u010dil o 4 cm viac ako Juro a Mi\u0161o o 7 cm menej ako Filip. Navy\u0161e Filipov skok bol presne v polovici medzi t\u00fdm Petrov\u00fdm a Samov\u00fdm a najkrat\u0161\u00ed skok meral 127 cm. Zistite, ko\u013eko cm sko\u010dili jednotliv\u00ed chlapci.<\/p>\n<div class=\"pri\">(M. Dillingerov\u00e1)<\/div>\n<p><strong>Z7-I-4<\/strong><br \/> V hostinci U troch prasiatok obsluhuj\u00fa Pa\u0161\u00edk, Ra\u0161\u00edk a Sa\u0161\u00edk. Pa\u0161\u00edk je ne\u010destn\u00fd, tak\u017ee ka\u017ed\u00e9mu hos\u0165ovi pripo\u010d\u00edta k celkovej cene 6 grajciarov. Ra\u0161\u00edk je poctivec, ka\u017ed\u00e9mu vy\u00fa\u010dtuje presne to, \u010do zjedol a vypil. Sa\u0161\u00edk je dobr\u00e1k, tak\u017ee ka\u017ed\u00e9mu hos\u0165ovi d\u00e1 z\u013eavu z celkovej ceny vo v\u00fd\u0161ke 20%. Prasiatka sa na seba tak podobaj\u00fa, \u017ee \u017eiadny hos\u0165 nepozn\u00e1, ktor\u00e9 pr\u00e1ve obsluhuje. Koza Lujza za\u0161la v pondelok, v utorok aj v stredu do tohto hostinca na \u010du\u010doriedkov\u00fa buchtu. Napriek tomu, \u017ee vedela, \u017ee v pondelok bol Ra\u0161\u00edk chor\u00fd a neobsluhoval, utratila za svoju pondelkov\u00fa, utorkov\u00fa aj stredaj\u0161iu buchtu dokopy rovnako, ako keby ju v\u017edy obsluhoval Ra\u0161\u00edk. Ko\u013eko grajciarov \u00fa\u010dtuje Ra\u0161\u00edk za jednu \u010du\u010doriedkov\u00fa buchtu? N\u00e1jdite v\u0161etky mo\u017enosti. (Ceny uv\u00e1dzan\u00e9 v jed\u00e1lnom l\u00edstku sa v tieto dni nemenili.)<\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z7-I-5<\/strong><br \/> Mami\u010dka del\u00ed \u010dokol\u00e1du, ktor\u00e1 m\u00e1 6 \u00d7 4 rovnak\u00fdch dielikov, svojim trom de\u0165om. Ako m\u00f4\u017ee mami\u010dka \u010dokol\u00e1du rozdeli\u0165 na pr\u00e1ve tri \u010dasti s rovnak\u00fdm obsahom tak, aby jeden \u00fatvar bol trojuholn\u00edk, jeden \u0161tvoruholn\u00edk a jeden p\u00e4\u0165uholn\u00edk?<\/p>\n<div class=\"pri\">(E. Novotn\u00e1)<\/div>\n<p><strong>Z7-I-6<\/strong><br \/> Ke\u010f C\u00e9zar stoj\u00ed na psej b\u00fade a Dun\u010do na zemi, je C\u00e9zar o 70 cm vy\u0161\u0161\u00ed ako Dun\u010do. Ke\u010f Dun\u010do stoj\u00ed na psej b\u00fade a C\u00e9zar na zemi, je Dun\u010do o 90 cm vy\u0161\u0161\u00ed ako C\u00e9zar. Ak\u00e1 vysok\u00e1 je psia b\u00fada?<\/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 align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z72\"><\/a><a href=\"#top\"><img decoding=\"async\" alt=\"\" src=\"\/img\/top.gif\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><input onclick=\"window.location='http:\/\/skmo.sk\/dokument.php?id=1081'\" type=\"button\" name=\"butt71\" value=\"Zadanie\" \/><input onclick=\"window.location='http:\/\/skmo.sk\/dokument.php?id=1084'\" type=\"button\" name=\"butt72\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z7-II-1<\/strong><br \/> Tabu\u013eka na obr\u00e1zku m\u00e1 obsahova\u0165 sedem prirodzen\u00fdch \u010d\u00edsel, pri\u010dom v ka\u017edom sivom pol\u00ed\u010dku m\u00e1 by\u0165 s\u00fa\u010din \u010d\u00edsel z dvoch bielych pol\u00ed\u010dok, ktor\u00e9 s n\u00edm susedia. \u010c\u00edsla v bielych pol\u00ed\u010dkach s\u00fa navz\u00e1jom r\u00f4zne a s\u00fa\u010din \u010d\u00edsel v siv\u00fdch pol\u00ed\u010dkach je rovn\u00fd 525. Ak\u00fd je s\u00fa\u010det \u010d\u00edsel v siv\u00fdch pol\u00ed\u010dkach? N\u00e1jdite v\u0161etky mo\u017enosti.<\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td width=\"40\" height=\"20\">\u00a0<\/td>\n<td bgcolor=\"#999999\" width=\"40\">\u00a0<\/td>\n<td width=\"40\">\u00a0<\/td>\n<td bgcolor=\"#999999\" width=\"40\">\u00a0<\/td>\n<td width=\"40\">\u00a0<\/td>\n<td bgcolor=\"#999999\" width=\"40\">\u00a0<\/td>\n<td width=\"40\">\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"pri\">(E. Pat\u00e1kov\u00e1)<\/div>\n<p><strong>Z7-II-2<\/strong><br \/> Ivana, Majka, Lucka, Sa\u010da a Zuzka pretekali v \u010d\u00edtan\u00ed tej istej knihy. Za jednu hodinu stihla Lucka pre\u010d\u00edta\u0165 32 str\u00e1n, \u010do bolo presne v strede medzi po\u010dtami str\u00e1n, ktor\u00e9 stihli pre\u010d\u00edta\u0165 Sa\u0161a a Zuzka. Ivana pre\u010d\u00edtala o 5 str\u00e1n viac ako Zuzka a Majka pre\u010d\u00edtala o 8 str\u00e1n menej ako Sa\u0161a. \u017diadne dve diev\u010dat\u00e1 nepre\u010d\u00edtali rovnak\u00fd po\u010det str\u00e1n a najhor\u0161\u00ed v\u00fdsledok bol 27 str\u00e1n. Ur\u010dte, ko\u013eko str\u00e1n pre\u010d\u00edtali jednotliv\u00e9 diev\u010dat\u00e1.<\/p>\n<div class=\"pri\">(M. Dillingerov\u00e1)<\/div>\n<p><strong>Z7-II-3<\/strong><br \/> Po n\u00e1raze kamienka praskla sklenen\u00e1 tabu\u013ea tak, \u017ee vznikli \u0161tyri men\u0161ie trojuholn\u00edky so spolo\u010dn\u00fdm vrcholom v mieste n\u00e1razu. Pritom plat\u00ed, \u017ee:<\/p>\n<ul>\n<li>sklenen\u00e1 tabu\u013ea mala tvar obd\u013a\u017enika, ktor\u00fd bol 8 dm \u0161irok\u00fd a 6 dm vysok\u00fd,<\/li>\n<li>trojuholn\u00edk vpravo mal trikr\u00e1t v\u00e4\u010d\u0161\u00ed obsah ako trojuholn\u00edk v\u013eavo,<\/li>\n<li>trojuholn\u00edk v\u013eavo mal dvakr\u00e1t men\u0161\u00ed obsah ako trojuholn\u00edk dole.<\/li>\n<\/ul>\n<p>Ur\u010dte vzdialenosti bodu n\u00e1razu od str\u00e1n obd\u013a\u017enika.<\/p>\n<div class=\"pri\">(E. 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=\"z81\"><\/a><a href=\"#top\"><img decoding=\"async\" alt=\"\" src=\"\/img\/top.gif\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><input onclick=\"window.location='http:\/\/skmo.sk\/dokument.php?id=1073'\" type=\"button\" name=\"butt51\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z8-I-1<\/strong><\/p>\n<p>Po okru\u017enej linke v meste ide elektri\u010dka, v ktorej je 300 cestuj\u00facich. Na ka\u017edej zast\u00e1vke sa odohr\u00e1 jedna z nasleduj\u00facich situ\u00e1ci\u00ed:<\/p>\n<ul>\n<li>ak je v elektri\u010dke aspo\u0148 7 cestuj\u00facich, tak ich 7 vyst\u00fapi,<\/li>\n<li>ak je v elektri\u010dke menej ako 7 cestuj\u00facich, tak 5 nov\u00fdch cestuj\u00facich prist\u00fapi.<\/li>\n<\/ul>\n<p>Vysvetlite, pre\u010do v istom okamihu v elektri\u010dke neostane \u017eiadny cestuj\u00faci. Potom zistite, ko\u013eko by malo by\u0165 na za\u010diatku v elektri\u010dke cestuj\u00facich, aby sa elektri\u010dka nikdy nevypr\u00e1zdnila.<\/p>\n<div class=\"pri\">(J. Maz\u00e1k)<\/div>\n<p><strong>Z8-I-2<\/strong><\/p>\n<p>Mami\u010dka del\u00ed \u010dokol\u00e1du, ktor\u00e1 m\u00e1 6 \u00d7 4 rovnak\u00fdch dielikov, svojim \u0161tyrom de\u0165om. Ako m\u00f4\u017ee\u00a0mami\u010dka \u010dokol\u00e1du rozdeli\u0165 na pr\u00e1ve \u0161tyri \u010dasti s rovnak\u00fdm obsahom tak, aby jeden \u00fatvar bol\u00a0trojuholn\u00edk, jeden \u0161tvoruholn\u00edk, jeden p\u00e4\u0165uholn\u00edk a jeden \u0161es\u0165uholn\u00edk?<\/p>\n<div class=\"pri\">(E. Novotn\u00e1)<\/div>\n<p><strong>Z8-I-3<\/strong><\/p>\n<p>Zme\u0148te v ka\u017edom z troch \u010d\u00edsel jednu cifru tak, aby bol pr\u00edklad na od\u010d\u00edtanie bez chyby:<\/p>\n<p class=\"r\">\u00a0\u00a07\u00a02\u00a04<br \/><span style=\"text-decoration: underline;\">&#8211;\u00a03\u00a00\u00a07<\/span><br \/>\u00a0\u00a01\u00a08\u00a08<\/p>\n<p>N\u00e1jdite v\u0161etky rie\u0161enia.<\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z8-I-4<\/strong><\/p>\n<p>Trojuholn\u00edky <em>ABC<\/em> a <em>DEF<\/em> s\u00fa rovnostrann\u00e9 s d\u013a\u017ekou strany 5 cm. Tieto trojuholn\u00edky s\u00fa polo\u017een\u00e9 cez seba tak, aby strany jedn\u00e9ho trojuholn\u00edka boli rovnobe\u017en\u00e9 so stranami druh\u00e9ho a aby prienikom t\u00fdchto dvoch trojuholn\u00edkov bol \u0161es\u0165uholn\u00edk (na obr\u00e1zku ozna\u010den\u00fd ako <em>GHIJKL<\/em>).<\/p>\n<p class=\"pc\"><img decoding=\"async\" alt=\"\" src=\"..\/63\/63z8a1.png\" \/><\/p>\n<p>Je mo\u017en\u00e9 ur\u010di\u0165 obvod dvan\u00e1s\u0165uholn\u00edka <em>AGEHBIFJCKDL<\/em> bez toho, aby sme poznali presnej\u0161ie inform\u00e1cie o polohe trojuholn\u00edkov? Ak \u00e1no, spo\u010d\u00edtajte ho; ak nie, vysvetlite pre\u010do.<\/p>\n<div class=\"pri\">(E. Pat\u00e1kov\u00e1)<\/div>\n<p><strong>Z8-I-5<\/strong><\/p>\n<p>Z\u00e1kazn\u00edk priv\u00e1\u017eaj\u00faci odpad do zbern\u00fdch surov\u00edn je povinn\u00fd zastavi\u0165 nalo\u017een\u00fdm autom na\u00a0v\u00e1he a po vykl\u00e1dke odpadu znova. Rozdiel nameran\u00fdch hmotnost\u00ed tak zodpoved\u00e1 privezen\u00e9mu\u00a0odpadu. Pat a Mat spravili chybu. Pri v\u00e1\u017een\u00ed nalo\u017een\u00e9ho auta sa na v\u00e1hu priplietol Pat a pri\u00a0v\u00e1\u017een\u00ed vylo\u017een\u00e9ho auta sa tam namiesto Pata ocitol Mat. Ved\u00faci zbern\u00fdch surov\u00edn si tak\u00a0zaznamenal rozdiel 332 kg. N\u00e1sledne sa na pr\u00e1zdnu v\u00e1hu postavili spolu ved\u00faci a Pat, potom\u00a0samotn\u00fd Mat a v\u00e1ha uk\u00e1zala rozdiel 86 kg. \u010ealej sa spolu zv\u00e1\u017eili ved\u00faci a Mat, potom samotn\u00fd\u00a0Pat a v\u00e1ha uk\u00e1zala rozdiel 64 kg. Ko\u013eko v skuto\u010dnosti v\u00e1\u017eil privezen\u00fd odpad?<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z8-I-6<\/strong><\/p>\n<p>V dome m\u00e1me medzi dvoma poschodiami dve r\u00f4zne schodisk\u00e1. Na ka\u017edom z t\u00fdchto schod\u00edsk\u00a0s\u00fa v\u0161etky schody rovnako vysok\u00e9. Jedno zo schod\u00edsk m\u00e1 ka\u017ed\u00fd schod vysok\u00fd 10 cm, druh\u00e9 m\u00e1\u00a0o 11 schodov menej ako to prv\u00e9. Behom d\u0148a som i\u0161iel p\u00e4\u0165kr\u00e1t nahor a p\u00e4\u0165kr\u00e1t nadol, pri\u010dom\u00a0som si medzi t\u00fdmito dvoma schodiskami vyberal n\u00e1hodne. Celkom som na ka\u017edom zo schod\u00edsk\u00a0zdolal rovnak\u00fd po\u010det schodov. Ak\u00fd je v\u00fd\u0161kov\u00fd rozdiel medzi poschodiami?<\/p>\n<div class=\"pri\">(M. Mach)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z82\"><\/a><a href=\"#top\"><img decoding=\"async\" alt=\"\" src=\"\/img\/top.gif\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><input onclick=\"window.location='http:\/\/skmo.sk\/dokument.php?id=1080'\" type=\"button\" name=\"butt51\" value=\"Zadanie\" \/><input onclick=\"window.location='http:\/\/skmo.sk\/dokument.php?id=1083'\" type=\"button\" name=\"butt51\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z8-II-1<\/strong><br \/> Angela, Barbora, Jano, Vlado a Mat\u00fa\u0161 s\u00fa\u0165a\u017eili v hode papierov\u00fdm lietadielkom. Ka\u017ed\u00fd h\u00e1dzal raz a s\u00fa\u010det d\u013a\u017eok ich hodov bol 41 metrov. Mat\u00fa\u0161 hodil najmenej, \u010do bolo o 90 cm menej ako hodila Angela, a t\u00e1 hodila o 60 cm menej ako Vlado. Jano hodil naj\u010falej a trafil lietadielkom do p\u00e1sky ozna\u010duj\u00facej cel\u00e9 metre. Ak by s\u00fa\u0165a\u017eili iba Mat\u00fa\u0161, Vlado a Angela, priemern\u00e1 d\u013a\u017eka hodu by bola o 20 cm krat\u0161ia. Ur\u010dte d\u013a\u017eky hodov v\u0161etk\u00fdch menovan\u00fdch det\u00ed.<\/p>\n<div class=\"pri\">(L. Dedkov\u00e1)<\/div>\n<p><strong>Z8-II-2<\/strong><br \/> Dan\u00fd je \u0161tvoruholn\u00edk <em>ABCD<\/em>, pozri obr\u00e1zok. Bod <em>T<\/em><sub>1<\/sub> je \u0165a\u017eiskom trojuholn\u00edka <em>BCD<\/em>, bod <em>T<\/em><sub>2<\/sub> je \u0165a\u017eiskom trojuholn\u00edka <em>ABD<\/em> a body <em>T<\/em><sub>1<\/sub> a <em>T<\/em><sub>2<\/sub> le\u017eia na \u00fase\u010dke <em>AC<\/em>. D\u013a\u017eka \u00fase\u010dky <em>T<\/em><sub>1<\/sub><em>T<\/em><sub>2<\/sub> je 3 cm a bod <em>D<\/em> m\u00e1 od \u00fase\u010dky <em>AC<\/em> vzdialenos\u0165 3 cm. Ur\u010dte obsah \u0161tvoruholn\u00edka <em>ABCD<\/em>.<\/p>\n<p class=\"pc\"><img decoding=\"async\" alt=\"\" src=\"..\/63\/63z8b1.png\" \/><\/p>\n<div class=\"pri\">(E. Pat\u00e1kov\u00e1)<\/div>\n<p><strong>Z8-II-3<\/strong><br \/> V meste rekordov a kurioz\u00edt postavili pyram\u00eddu z kociek. V hornej vrstve je jedna kocka a po\u010dty kociek v jednotliv\u00fdch vrstv\u00e1ch sa smerom nadol zv\u00e4\u010d\u0161uj\u00fa v\u017edy o dve (nieko\u013eko horn\u00fdch vrstiev stavby je zn\u00e1zornen\u00fdch na obr\u00e1zku). Prv\u00e1, teda najspodnej\u0161ia vrstva m\u00e1 \u010diernu farbu, druh\u00e1 siv\u00fa, tretia bielu, \u0161tvrt\u00e1 op\u00e4\u0165 \u010diernu, piata siv\u00fa, \u0161iesta bielu a takto sa farby pravidelne striedaj\u00fa a\u017e k hornej vrstve. Ur\u010dte, ko\u013eko m\u00e1 pyram\u00edda vrstiev, ak viete, \u017ee \u010diernych kociek je pou\u017eit\u00fdch o 55 viac ako bielych.<\/p>\n<p class=\"pc\"><img decoding=\"async\" alt=\"\" src=\"..\/63\/63z8b2.png\" \/><\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/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\" alt=\"\" src=\"\/img\/top.gif\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><input onclick=\"window.location='http:\/\/skmo.sk\/dokument.php?id=999'\" type=\"button\" name=\"butt91\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z9-I-1<\/strong><\/p>\n<p>Peter si mysl\u00ed dvojcifern\u00e9 \u010d\u00edslo. Ke\u010f toto \u010d\u00edslo nap\u00ed\u0161e dvakr\u00e1t za sebou, vznikne \u0161tvorcifern\u00e9 \u010d\u00edslo delite\u013en\u00e9 deviatimi. Ke\u010f to ist\u00e9 \u010d\u00edslo nap\u00ed\u0161e trikr\u00e1t za sebou, vznikne \u0161es\u0165cifern\u00e9 \u010d\u00edslo delite\u013en\u00e9 \u00f4smimi. Zistite, ak\u00e9 \u010d\u00edslo si m\u00f4\u017ee Peter myslie\u0165.<\/p>\n<div class=\"pri\">(E. Novotn\u00e1)<\/div>\n<p><strong>Z9-I-2<\/strong><\/p>\n<p>Dan\u00fd je rovnoramenn\u00fd lichobe\u017en\u00edk s d\u013a\u017ekami str\u00e1n |<em>AB<\/em>| = 31 cm, |<em>BC<\/em>| = 26 cm a |<em>CD<\/em>| = 11 cm. Na strane <em>AB<\/em> je bod <em>E<\/em> ur\u010den\u00fd pomerom vzdialenost\u00ed |<em>AE<\/em>|:|<em>EB<\/em>| = = 3:28. Vypo\u010d\u00edtajte obvod trojuholn\u00edka <em>CDE<\/em>.<\/p>\n<div class=\"pri\">(L. Dedkov\u00e1)<\/div>\n<p><strong>Z9-I-3<\/strong><\/p>\n<p>Podlahu tvaru obd\u013a\u017enika so stranami 360 cm a 540 cm m\u00e1me pokry\u0165 (bez medzier) zhodn\u00fdmi \u0161tvorcov\u00fdmi dla\u017edicami. M\u00f4\u017eeme si vybra\u0165 z dvoch typov \u0161tvorcov\u00fdch dla\u017ed\u00edc, ktor\u00fdch strany s\u00fa v pomere 2 : 3. V oboch pr\u00edpadoch sa d\u00e1 pokry\u0165 cel\u00e1 plocha jedn\u00fdm typom dla\u017ed\u00edc bez p\u00edlenia. Men\u0161\u00edch dla\u017ed\u00edc by sme potrebovali o 30 viac ako v\u00e4\u010d\u0161\u00edch. Ur\u010dte, ako dlh\u00e9 s\u00fa strany dla\u017ed\u00edc.<\/p>\n<div class=\"pri\">(K. Pazourek)<\/div>\n<p><strong>Z9-I-4<\/strong><\/p>\n<p>V pravouholn\u00edku <em>ACKI<\/em> s\u00fa vyzna\u010den\u00e9 dve rovnobe\u017eky so susedn\u00fdmi stranami a jedna uhloprie\u010dka (obr\u00e1zok). Pritom trojuholn\u00edky <em>ABD<\/em> a <em>GHK<\/em> s\u00fa zhodn\u00e9. Ur\u010dte pomer obsahov pravouholn\u00edkov <em>ABFE<\/em> a <em>FHKJ<\/em>.<\/p>\n<p class=\"pc\"><img decoding=\"async\" alt=\"\" src=\"..\/63\/63z9a1.png\" \/><\/p>\n<div class=\"pri\">(V. \u017d\u00e1dn\u00edk)<\/div>\n<p><strong>Z9-I-5<\/strong><\/p>\n<p>Eva rie\u0161ila experiment\u00e1lnu \u00falohu Fyzik\u00e1lnej olympi\u00e1dy. Dopoludnia od 9:15 robila v trojmin\u00fatov\u00fdch odstupoch 4 merania. Z\u00edskan\u00e9 hodnoty zapisovala do tabu\u013eky, ktor\u00fa si pripravila v po\u010d\u00edta\u010di:<\/p>\n<table style=\"border-color: #000000; border-width: 1px; ; width: 250px;\" border=\"1\" cellspacing=\"0\" cellpadding=\"1\" align=\"center\">\n<tbody>\n<tr align=\"center\">\n<td>hod\u00edn<\/td>\n<td>min\u00fat<\/td>\n<td>hodnota<\/td>\n<\/tr>\n<tr align=\"center\">\n<td>9<\/td>\n<td>15<\/td>\n<td>\u00a0<\/td>\n<\/tr>\n<tr align=\"center\">\n<td>9<\/td>\n<td>18<\/td>\n<td>\u00a0<\/td>\n<\/tr>\n<tr align=\"center\">\n<td>9<\/td>\n<td>21<\/td>\n<td>\u00a0<\/td>\n<\/tr>\n<tr align=\"center\">\n<td>9<\/td>\n<td>24<\/td>\n<td>\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Popoludn\u00ed v experimente pokra\u010dovala. Tentoraz urobila v trojmin\u00fatov\u00fdch odstupoch 9 meran\u00ed a hodnoty zapisovala do podobnej tabu\u013eky. Omylom do po\u010d\u00edta\u010da zadala, aby sa zobrazil s\u00fa\u010det deviatich \u010d\u00edsel z prostredn\u00e9ho st\u013apca. Tento zbyto\u010dn\u00fd v\u00fdpo\u010det vy\u0161iel 258. Ktor\u00e9 \u010d\u00edsla boli v danom st\u013apci?<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z9-I-6<\/strong><\/p>\n<p>V hostinci U troch prasiatok obsluhuj\u00fa Pa\u0161\u00edk, Ra\u0161\u00edk a Sa\u0161\u00edk. Pa\u0161\u00edk je ne\u010destn\u00fd, tak\u017ee ka\u017ed\u00e9mu hos\u0165ovi pripo\u010d\u00edta k celkovej cene 10 grajciarov. Ra\u0161\u00edk je poctivec, ka\u017ed\u00e9mu vy\u00fa\u010dtuje presne to, \u010do zjedol a vypil. Sa\u0161\u00edk je dobr\u00e1k, tak\u017ee ka\u017ed\u00e9mu hos\u0165ovi d\u00e1 z\u013eavu z celkovej ceny vo v\u00fd\u0161ke 20%. Prasiatka sa na seba tak podobaj\u00fa, \u017ee \u017eiadny hos\u0165 nepozn\u00e1, ktor\u00e9 pr\u00e1ve obsluhuje. Bar\u00e1nok Vendel\u00edn si v pondelok objednal tri kol\u00e1\u010diky a d\u017eb\u00e1nok d\u017e\u00fasu a zaplatil za to 56 grajciarov. Bol spokojn\u00fd, tak\u017ee hne\u010f v utorok zjedol p\u00e4\u0165 kol\u00e1\u010dikov, vypil k nim tri d\u017eb\u00e1nky d\u017e\u00fasu a platil 104 grajciarov. V stredu zjedol osem kol\u00e1\u010dikov, vypil \u0161tyri d\u017eb\u00e1nky d\u017e\u00fasu a zaplatil 112 grajciarov.<\/p>\n<ul style=\"list-style-type: decimal;\">\n<li>Kto obsluhoval Vendel\u00edna v pondelok, kto v utorok a kto v stredu?<\/li>\n<li>Ko\u013eko grajciarov \u00fa\u010dtuje Ra\u0161\u00edk za jeden kol\u00e1\u010dik a ko\u013eko za jeden d\u017eb\u00e1nok d\u017e\u00fasu?<\/li>\n<\/ul>\n<p><em>(V\u0161etky kol\u00e1\u010diky s\u00fa rovnak\u00e9, rovnako tak v\u0161etky d\u017eb\u00e1nky d\u017e\u00fasu. Ceny uv\u00e1dzan\u00e9 v jed\u00e1lnom l\u00edstku sa v uveden\u00fdch d\u0148och nemenili.)<\/em><\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z92\"><\/a><a href=\"#top\"><img decoding=\"async\" alt=\"\" src=\"\/img\/top.gif\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><input onclick=\"window.location='http:\/\/skmo.sk\/dokument.php?id=1002'\" type=\"button\" name=\"butt92\" value=\"Zadanie\" \/><\/form>\n<form><input onclick=\"window.location='http:\/\/skmo.sk\/dokument.php?id=1003'\" type=\"button\" name=\"butt92\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z9-II-1<\/strong><br \/> Jana mala za dom\u00e1cu \u00falohu vypo\u010d\u00edta\u0165 s\u00fa\u010din dvoch \u0161es\u0165cifern\u00fdch \u010d\u00edsel. Pri prepisovan\u00ed druh\u00e9ho z nich z tabule zabudla odp\u00edsa\u0165 jednu cifru a tak namiesto cel\u00e9ho \u0161es\u0165cifern\u00e9ho \u010d\u00edsla nap\u00edsala iba p\u00e4\u0165cifern\u00e9 \u010d\u00edslo 85522. Ke\u010f bola doma, zistila svoj omyl. Pam\u00e4tala si v\u0161ak, \u017ee \u010d\u00edslo, ktor\u00e9 zle odp\u00edsala, bolo delite\u013en\u00e9 tromi. Rozhodla sa, \u017ee sa pok\u00fasi ur\u010di\u0165, ak\u00e9 mohlo by\u0165 p\u00f4vodn\u00e9 \u010d\u00edslo. Zistite, ko\u013eko tak\u00fdch \u0161es\u0165cifern\u00fdch \u010d\u00edsel existuje.<\/p>\n<div class=\"pri\">(M. Dillingerov\u00e1)<\/div>\n<p><strong>Z9-II-2<\/strong><br \/> Ren\u00e1ta zostrojila lichobe\u017en\u00edk <em>PRST<\/em> so z\u00e1klad\u0148ami <em>PR<\/em> a <em>ST<\/em>, v ktorom s\u00fa\u010dasne plat\u00ed:<\/p>\n<ul>\n<li>lichobe\u017en\u00edk <em>PRST<\/em> nie je pravouhl\u00fd,<\/li>\n<li>trojuholn\u00edk <em>TRP<\/em> je rovnostrann\u00fd,<\/li>\n<li>trojuholn\u00edk <em>TRS<\/em> je pravouhl\u00fd,<\/li>\n<li>jeden z trojuholn\u00edkov <em>TRS<\/em> a <em>TRP<\/em> m\u00e1 obsah 10cm<sup>2<\/sup>.<\/li>\n<\/ul>\n<p>Ur\u010dte obsah druh\u00e9ho z t\u00fdchto dvoch trojuholn\u00edkov; n\u00e1jdite v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z9-II-3<\/strong><br \/> Lenka mala papierov\u00fd kvietok s \u00f4smimi okvetn\u00fdmi l\u00edstkami. Na ka\u017edom l\u00edstku bola nap\u00edsan\u00e1 pr\u00e1ve jedna cifra a \u017eiadna z cifier sa na \u017eiadnom inom l\u00edstku neopakovala. Ke\u010f sa Lenka s kvietkom hrala, uvedomila si nieko\u013eko vec\u00ed:<\/p>\n<ul>\n<li>Z kvietka bolo mo\u017en\u00e9 odtrhn\u00fa\u0165 \u0161tyri l\u00edstky tak, \u017ee s\u00fa\u010det na nich nap\u00edsan\u00fdch \u010d\u00edsel by bol rovnak\u00fd ako s\u00fa\u010det \u010d\u00edsel na neodtrhnut\u00fdch l\u00edstkoch.<\/li>\n<li>Tie\u017e bolo mo\u017en\u00e9 odtrhn\u00fa\u0165 \u0161tyri l\u00edstky tak, \u017ee s\u00fa\u010det na nich nap\u00edsan\u00fdch \u010d\u00edsel by bol dvakr\u00e1t v\u00e4\u010d\u0161\u00ed ako s\u00fa\u010det \u010d\u00edsel na neodtrhnut\u00fdch l\u00edstkoch.<\/li>\n<li>Dokonca bolo mo\u017en\u00e9 odtrhn\u00fa\u0165 \u0161tyri l\u00edstky tak, \u017ee s\u00fa\u010det na nich nap\u00edsan\u00fdch \u010d\u00edsel by bol \u0161tyrikr\u00e1t v\u00e4\u010d\u0161\u00ed ako s\u00fa\u010det \u010d\u00edsel na neodtrhnut\u00fdch l\u00edstkoch.<\/li>\n<\/ul>\n<p>Ur\u010dte, ak\u00e9 cifry mohli by\u0165 nap\u00edsan\u00e9 na okvetn\u00fdch l\u00edstkoch.<\/p>\n<div class=\"pri\">(E. Novotn\u00e1)<\/div>\n<p><strong>Z9-II-4<\/strong><br \/> V hostinci U temn\u00e9ho lesa obsluhuj\u00fa obrie dvoj\u010dat\u00e1 Pravdoslav a Krivom\u00edr. Pravdoslav je poctiv\u00fd a \u00fa\u010dtuje v\u017edy presne, Krivom\u00edr je ne\u010destn\u00fd a ku ka\u017ed\u00e9mu kol\u00e1\u010du a ka\u017ed\u00e9mu d\u017eb\u00e1nku medoviny v\u017edy pripo\u010d\u00edta dva grajciare. Raz tento hostinec nav\u0161t\u00edvilo sedem trpasl\u00edkov, ktor\u00ed si sadli k dvom stolom. Trpasl\u00edci zaplatili za \u0161tyri kol\u00e1\u010de u jedn\u00e9ho obra rovnako ve\u013ea ako za tri d\u017eb\u00e1nky medoviny u druh\u00e9ho. Inokedy trpasl\u00edci platili za \u0161tyri d\u017eb\u00e1nky medoviny u Pravdoslava o 21 grajciarov viac ako za tri kol\u00e1\u010de u Krivom\u00edra. Ur\u010dte, ko\u013eko st\u00e1l jeden kol\u00e1\u010d a ko\u013eko jeden d\u017eb\u00e1nok medoviny. V\u0161etky ceny s\u00fa v cel\u00fdch grajciaroch a medzi oboma n\u00e1v\u0161tevami sa tieto ceny nijako nemenili.<\/p>\n<div class=\"pri\">(M. Petrov\u00e1, M. 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=\"z93\"><\/a><a href=\"#top\"><img decoding=\"async\" alt=\"\" src=\"\/img\/top.gif\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><input onclick=\"window.location='http:\/\/skmo.sk\/dokument.php?id=1068'\" type=\"button\" name=\"butt93\" value=\"Zadanie\" \/><\/form>\n<form><input onclick=\"window.location='http:\/\/skmo.sk\/dokument.php?id=1069'\" type=\"button\" name=\"butt93\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z9-III-1<\/strong><br \/> Hviezdi\u010dky v sch\u00e9me predstavuj\u00fa 16 bezprostredne po sebe id\u00facich prirodzen\u00fdch n\u00e1sobkov \u010d\u00edsla tri nap\u00edsan\u00fdch z\u013eava doprava od najmen\u0161ieho po najv\u00e4\u010d\u0161\u00ed. Pritom \u010d\u00edsla v prvom r\u00e1miku maj\u00fa rovnak\u00fd s\u00fa\u010det ako \u010d\u00edsla v druhom r\u00e1miku. Ur\u010dte najmen\u0161ie z t\u00fdchto 16 \u010d\u00edsel.<\/p>\n<p class=\"pc\"><img decoding=\"async\" alt=\"\" src=\"..\/63\/63z9c1.png\" \/><\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z9-III-2<\/strong><br \/> V rovnostrannom trojuholn\u00edku <em>ABC<\/em> je vp\u00edsan\u00fd rovnostrann\u00fd trojuholn\u00edk <em>DEF<\/em>. Vrcholy <em>D<\/em>, <em>E<\/em> a <em>F<\/em> le\u017eia na stran\u00e1ch <em>AB<\/em>, <em>BC<\/em> a <em>AC<\/em> tak, \u017ee strany trojuholn\u00edka <em>DEF<\/em> s\u00fa kolm\u00e9 na strany trojuholn\u00edka <em>ABC<\/em> (tak ako na obr\u00e1zku). \u010ealej plat\u00ed, \u017ee \u00fase\u010dka <em>DG<\/em> je \u0165a\u017enicou v trojuholn\u00edku <em>DEF<\/em> a bod <em>H<\/em> je priese\u010dn\u00edkom priamok <em>DG<\/em> a <em>BC<\/em>. Ur\u010dte pomer obsahov trojuholn\u00edkov <em>HGC<\/em> a <em>DBE<\/em>.<\/p>\n<p class=\"pc\"><img decoding=\"async\" alt=\"\" src=\"..\/63\/63z9c2.png\" \/><\/p>\n<div class=\"pri\">(E. Pat\u00e1kov\u00e1)<\/div>\n<p><strong>Z9-III-3<\/strong><br \/> Danka mala papierov\u00fd kvietok s desiatimi okvetn\u00fdmi l\u00edstkami. Na ka\u017edom l\u00edstku bola nap\u00edsan\u00e1 pr\u00e1ve jedna cifra a \u017eiadna z cifier sa na \u017eiadnom inom l\u00edstku neopakovala. Danka odtrhla dva l\u00edstky tak, \u017ee s\u00fa\u010det \u010d\u00edsel na zvy\u0161n\u00fdch l\u00edstkoch bol n\u00e1sobkom deviatich. Potom odtrhla \u010fal\u0161ie dva l\u00edstky tak, \u017ee s\u00fa\u010det \u010d\u00edsel na zvy\u0161n\u00fdch l\u00edstkoch bol n\u00e1sobkom \u00f4smich. Nakoniec odtrhla \u010fal\u0161ie dva l\u00edstky tak, \u017ee s\u00fa\u010det \u010d\u00edsel na zvy\u0161n\u00fdch l\u00edstkoch bol n\u00e1sobkom desiatich. Ur\u010dte, ak\u00e9 mohli by\u0165 s\u00fa\u010dty \u010d\u00edsel po ka\u017edom odtrh\u00e1van\u00ed; n\u00e1jdite v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(E. Novotn\u00e1)<\/div>\n<p><strong>Z9-III-4<\/strong><br \/> \u0160tyri diev\u010dat\u00e1 hrali na s\u00fastreden\u00ed nieko\u013eko z\u00e1pasov. Ke\u010f s\u010d\u00edtame po\u010dty v\u00fdhier dvoch diev\u010dat dokopy (pre v\u0161etky mo\u017en\u00e9 dvojice diev\u010dat), dostaneme \u010d\u00edsla 8, 10, 12, 12, 14 a 16. Ur\u010dte, ko\u013eko v\u00fdhier vybojovalo ka\u017ed\u00e9 z diev\u010dat.<\/p>\n<div class=\"pri\">(M. 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