{"id":696,"date":"2012-09-17T19:45:32","date_gmt":"2012-09-17T19:45:32","guid":{"rendered":"http:\/\/matematika.okamzite.eu\/?p=696"},"modified":"2012-09-17T19:45:32","modified_gmt":"2012-09-17T19:45:32","slug":"sutazne-ulohy-kategorii-z5-z9","status":"publish","type":"post","link":"https:\/\/matematika.besaba.com\/?p=696","title":{"rendered":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9"},"content":{"rendered":"<p class=\"hh2\"><a name=\"top\"><\/a>62. ro\u010dn\u00edk matematickej olympi\u00e1dy 2012-2013<\/p>\n<p><span class=\"povinne\">aktualizovan\u00e9 29.4.2013 14:50<\/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=\"2\" 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\"><\/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=\"\/62\/62moz59.pdf\"><img loading=\"lazy\" decoding=\"async\" src=\"\/img\/down.png\" border=\"0\" alt=\"\" width=\"48\" height=\"48\" align=\"absmiddle\" \/><\/a> (ulo\u017ei\u0165 ako &#8230;)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"z51\"><\/a><\/p>\n<form>\n<input onclick=\"window.location='62\/62z5ir.pdf'\" name=\"butt51\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z5-I-1<\/strong><br \/>\nMami\u010dka zaplatila v kn\u00edhkupectve 270 \u20ac. Platila dvoma druhmi bankoviek, dvadsa\u0165eurov\u00fdmi a p\u00e4\u0165desiateurov\u00fdmi, a presne. Ko\u013eko ktor\u00fdch bankoviek mohla pou\u017ei\u0165? Uve\u010fte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(M. Krej\u010dov\u00e1)<\/div>\n<p><strong>Z5-I-2<\/strong><br \/>\nPat nap\u00edsal na tabu\u013eu \u010dudn\u00fd pr\u00edklad:<\/p>\n<p class=\"pc\">550 + 460 + 359 + 340 = 2012.<\/p>\n<p>Mat to chcel napravi\u0165, preto p\u00e1tral po nezn\u00e1mom \u010d\u00edsle, ktor\u00e9 by pripo\u010d\u00edtal ku ka\u017ed\u00e9mu z piatich uveden\u00fdch \u010d\u00edsel, aby potom bol pr\u00edklad vypo\u010d\u00edtan\u00fd spr\u00e1vne. Ak\u00e9 to bolo \u010d\u00edslo?<\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z5-I-3<\/strong><br \/>\nRudo dostal na narodeniny bud\u00edk. Mal z neho rados\u0165 a nastavil na \u0148om presn\u00fd \u010das. Odvtedy ka\u017ed\u00e9 r\u00e1no, ke\u010f vst\u00e1val (vr\u00e1tane sob\u00f4t, nedie\u013e a pr\u00e1zdnin), stla\u010dil presne na 4 sekundy tla\u010didlo, ktor\u00fdm sa osvet\u013euje cifern\u00edk. Pritom si v\u0161imol, \u017ee po\u010das stla\u010denia tla\u010didla je \u010das na bud\u00edku zastaven\u00fd. Inak sa ale bud\u00edk v\u00f4bec neoneskoruje. Popoludn\u00ed 11. decembra sa Rudo pozrel na svoj bud\u00edk a zistil, \u017ee ukazuje presne o 3 min\u00faty menej, ako by mal. Ak\u00fd je d\u00e1tum Rudov\u00fdch naroden\u00edn?<\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z5-I-4<\/strong><br \/>\n\u010cerv\u00edk sa sklad\u00e1 z bielej hlavy a nieko\u013ek\u00fdch \u010dl\u00e1nkov, pozri obr\u00e1zok.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/62\/62z5a1.gif\" alt=\"\" width=\"127\" height=\"41\" \/><\/p>\n<p>Ke\u010f sa \u010derv\u00edk narod\u00ed, m\u00e1 hlavu a jeden biely \u010dl\u00e1nok. Ka\u017ed\u00fd de\u0148 pribudne \u010derv\u00edkovi nov\u00fd \u010dl\u00e1nok jedn\u00fdm z nasleduj\u00facich sp\u00f4sobov:<br \/>\n\u2022\u00a0\u00a0bu\u010f sa niektor\u00fd biely \u010dl\u00e1nok rozdel\u00ed na biely a siv\u00fd:\u00a0\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"\/62\/62z5a2.gif\" alt=\"\" width=\"81\" height=\"22\" \/><br \/>\n\u2022\u00a0\u00a0alebo sa niektor\u00fd siv\u00fd \u010dl\u00e1nok rozdel\u00ed na siv\u00fd a biely:\u00a0\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"\/62\/62z5a3.gif\" alt=\"\" width=\"81\" height=\"22\" \/><\/p>\n<p>(V oboch pr\u00edpadoch opisujeme situ\u00e1ciu pri poh\u013eade na \u010derv\u00edka od hlavy.) Na \u0161tvrt\u00fd de\u0148 \u010derv\u00edk dospieva a \u010falej nerastie &#8211; jeho telo sa sklad\u00e1 z hlavy a \u0161tyroch \u010dl\u00e1nkov. Ko\u013eko najviac r\u00f4znych farebn\u00fdch variantov dospel\u00fdch \u010derv\u00edkov tohto druhu m\u00f4\u017ee existova\u0165?<\/p>\n<div class=\"pri\">(E. Novotn\u00e1)<\/div>\n<p><strong>Z5-I-5<\/strong><br \/>\nVypo\u010d\u00edtajte 3 . 15 + 20 : 4 + 1. Potom dopl\u0148te do zadania jeden alebo viac p\u00e1rov z\u00e1tvoriek tak, aby v\u00fdsledok bol:<br \/>\n1. \u010do najv\u00e4\u010d\u0161ie cel\u00e9 \u010d\u00edslo,<br \/>\n2. \u010do najmen\u0161ie cel\u00e9 \u010d\u00edslo.<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z5-I-6<\/strong><br \/>\nSedem trpasl\u00edkov sa postavilo po obvode svojej z\u00e1hradky, do ka\u017ed\u00e9ho rohu jeden, a napli medzi sebou povraz okolo celej z\u00e1hrady. Snehulienka vy\u0161la od K\u00fdblika a i\u0161la pozd\u013a\u017e povrazu. Najsk\u00f4r i\u0161la \u0161tyri metre na v\u00fdchod, kde stretla Vedka. Od neho pokra\u010dovala dva metre na sever, k\u00fdm dorazila k Dudro\u0161ovi. Od Dudro\u0161a i\u0161la na z\u00e1pad a po dvoch metroch natrafila na Pla\u0161ka. \u010ealej pokra\u010dovala tri metre na sever, a\u017e do\u0161la k Smejkovi. Vydala sa na z\u00e1pad a po \u0161tyroch metroch stretla K\u00fdchala, odkia\u013e jej zost\u00e1vali tri metre na juh k Spachto\u0161ovi. Nakoniec pozd\u013a\u017e povr\u00e1zka do\u0161la najkrat\u0161ou cestou ku K\u00fdblikovi a t\u00fdm obi\u0161la cel\u00fa z\u00e1hradu. Ko\u013eko metrov \u0161tvorcov\u00fdch m\u00e1 cel\u00e1 z\u00e1hrada?<\/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 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<form>\n<input onclick=\"window.location='62\/62z5ii.pdf'\" name=\"butt521\" type=\"button\" value=\"Zadanie\" \/><\/form>\n<form>\n<input onclick=\"window.location='62\/62z5iir.pdf'\" name=\"butt522\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z5-II-1<\/strong><br \/>\nNa t\u00e1bore sa skauti v\u00e1\u017eili na starod\u00e1vnej v\u00e1he. Ved\u00faci ich upozornil, \u017ee v\u00e1ha s\u00edce nev\u00e1\u017ei spr\u00e1vne, ale rozdiel medzi skuto\u010dnou a nameranou hmotnos\u0165ou je v\u017edy rovnak\u00fd. Mi\u0161ovi v\u00e1ha uk\u00e1zala 30 kg, Emilovi 28 kg, ale ke\u010f si st\u00fapli na v\u00e1hu obaja naraz, uk\u00e1zala 56 kg. Ak\u00e1 bola skuto\u010dn\u00e1 hmotnos\u0165 Mi\u0161a a Emila?<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z5-II-2<\/strong><br \/>\nNa obr\u00e1zku je stavba zlepen\u00e1 zo 14 rovnak\u00fdch koc\u00f4\u010dok. Stavbu chceme zo v\u0161etk\u00fdch str\u00e1n ofarbi\u0165, teda aj zospodu. Ak\u00e1 bude celkov\u00e1 spotreba farby, ke\u010f 10 mililitrov farby sta\u010d\u00ed na ofarbenie jednej celej koc\u00f4\u010dky?<\/p>\n<div class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/62\/62z5b1.png\" alt=\"\" width=\"105\" height=\"119\" \/><\/div>\n<div class=\"pri\">(M. Mach)<\/div>\n<p><strong>Z5-II-3<\/strong><br \/>\nRadka dostala r\u00e1no v de\u0148 svojich naroden\u00edn ve\u013ek\u00e9 balenie lentiliek. Ka\u017ed\u00fd de\u0148 poobede lentilky ma\u0161krtila, a to tak, \u017ee v pracovn\u00fd de\u0148 (t. j. mimo v\u00edkendu) si vzala v\u017edy 3 lentilky a ka\u017ed\u00fa sobotu aj nede\u013eu si ich vzala 5. Ist\u00fd de\u0148 ve\u010der zistila, \u017ee zjedla pr\u00e1ve 111 lentiliek. Ktor\u00fd de\u0148 v t\u00fd\u017edni mohla ma\u0165 Radka narodeniny? N\u00e1jdite obe mo\u017enosti.<\/p>\n<div class=\"pri\">(E. Pat\u00e1kov\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 \/>\n\u013dubo\u0161 si mysl\u00ed trojcifern\u00e9 prirodzen\u00e9 \u010d\u00edslo, ktor\u00e9 m\u00e1 v\u0161etky svoje cifry nep\u00e1rne. Ak k nemu pripo\u010d\u00edta 421, dostane trojcifern\u00e9 \u010d\u00edslo, ktor\u00e9 nem\u00e1 ani jednu svoju cifru nep\u00e1rnu. N\u00e1jdite v\u0161etky \u010d\u00edsla, ktor\u00e9 si m\u00f4\u017ee \u013dubo\u0161 myslie\u0165.<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z6-I-2<\/strong><br \/>\nNa obr\u00e1zku s\u00fa vyzna\u010den\u00e9 mre\u017eov\u00e9 body \u0161tvor\u010dekovej siete, z ktor\u00fdch dva s\u00fa pomenovan\u00e9 <em>A<\/em> a <em>B<\/em>. Nech bod <em>C<\/em> je jeden zo zvy\u0161n\u00fdch mre\u017eov\u00fdch bodov. N\u00e1jdite v\u0161etky mo\u017en\u00e9 polohy bodu <em>C<\/em> tak, aby trojuholn\u00edk <em>ABC<\/em> mal obsah 4,5 \u0161tvor\u010deka.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/62\/62z6a1.png\" alt=\"\" width=\"74\" height=\"107\" \/><\/p>\n<div class=\"pri\">(E. Novotn\u00e1)<\/div>\n<p><strong>Z6-I-3<\/strong><br \/>\nObri Koloman a Bartolomej hovoria niektor\u00e9 dni iba pravdu a niekedy iba klam\u00fa. Koloman hovor\u00ed pravdu v pondelok, v piatok a v nede\u013eu, ostatn\u00e9 dni klame. Bartolomej hovor\u00ed pravdu v stredu, \u0161tvrtok a piatok, ostatn\u00e9 dni klame.<\/p>\n<p>1. Ur\u010dte, kedy m\u00f4\u017ee Koloman poveda\u0165: &#8222;V\u010dera som hovoril pravdu.&#8220;<br \/>\n2. Jedn\u00e9ho d\u0148a obaja povedali: &#8222;V\u010dera som klamal.&#8220; V ktor\u00fd de\u0148 to bolo?<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z6-I-4<\/strong><br \/>\nEva m\u00e1 tri l\u00edsto\u010dky a na ka\u017edom z nich je nap\u00edsan\u00e9 jedno prirodzen\u00e9 \u010d\u00edslo. Ke\u010f vyn\u00e1sob\u00ed medzi sebou v\u0161etky mo\u017en\u00e9 dvojice \u010d\u00edsel z l\u00edsto\u010dkov, dostane v\u00fdsledky 48, 192 a 36. Ktor\u00e9 \u010d\u00edsla s\u00fa nap\u00edsan\u00e9 na Evin\u00fdch l\u00edsto\u010dkoch?<\/p>\n<div class=\"pri\">(E. Novotn\u00e1)<\/div>\n<p><strong>Z6-I-5<\/strong><br \/>\nNa obr\u00e1zku je \u00fatvar zlo\u017een\u00fd z dvan\u00e1stich zhodn\u00fdch kociek. Na ko\u013eko r\u00f4znych miest m\u00f4\u017eeme premiestni\u0165 tmav\u00fa kocku (ozna\u010den\u00fa \u0161\u00edpkou), ak chceme, aby sa povrch zostaven\u00e9ho telesa nezmenil?<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/62\/62z6a2.png\" alt=\"\" width=\"143\" height=\"100\" \/><\/p>\n<p>Rovnako ako pri p\u00f4vodnom telese sa aj pri novom telese musia kocky dot\u00fdka\u0165 cel\u00fdmi stenami. Poloha svetl\u00fdch kociek sa meni\u0165 nem\u00f4\u017ee.<\/p>\n<div class=\"pri\">(D. Reichmann)<\/div>\n<p><strong>Z6-I-6<\/strong><br \/>\nObsluhuj\u00faci v bufete U \u0160vindliara v\u017edy zapo\u010d\u00edtava platiacemu hos\u0165ovi do \u00fa\u010dtu aj d\u00e1tum: celkov\u00fa minut\u00fa sumu zv\u00e4\u010d\u0161\u00ed o to\u013eko centov, ko\u013ek\u00fd de\u0148 v mesiaci pr\u00e1ve je. V septembri sa v bufete dvakr\u00e1t zi\u0161la trojica priate\u013eov. Prv\u00fdkr\u00e1t platil ka\u017ed\u00fd z nich zvl\u00e1\u0161\u0165, obsluhuj\u00faci teda v\u017edy prip\u00edsal d\u00e1tum a \u017eiadal od ka\u017ed\u00e9ho 1,68 \u20ac. O \u0161tyri dni tam olovrantovali znova a dali si presne to ist\u00e9 \u010do minule. Tentoraz v\u0161ak jeden platil za v\u0161etk\u00fdch dokopy. Obsluhuj\u00faci teda prip\u00edsal d\u00e1tum do \u00fa\u010dtu iba raz a vyp\u00fdtal si 4,86 \u20ac. Priate\u013eom sa nezdalo, \u017ee aj ke\u010f sa ceny v jed\u00e1lnom l\u00edstku nezmenili, maj\u00fa olovrant lacnej\u0161\u00ed ako minule, a podvod odhalili. Ko\u013ek\u00e9ho septembra pr\u00e1ve bolo, ke\u010f podvod odhalili?<\/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 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><strong>Z6-II-1<\/strong><br \/>\nPat nap\u00edsal na tabu\u013eu pr\u00edklad:<\/p>\n<p class=\"r\">589 + 544 + 80 = 2 013.<\/p>\n<p>Mat chcel pr\u00edklad opravi\u0165, aby sa obe strany naozaj rovnali, a p\u00e1tral po nezn\u00e1mom \u010d\u00edsle, ktor\u00e9 potom k prv\u00e9mu s\u010d\u00edtancu na \u013eavej strane pripo\u010d\u00edtal, od druh\u00e9ho s\u010d\u00edtanca ho od\u010d\u00edtal a tretieho s\u010d\u00edtanca n\u00edm vyn\u00e1sobil. Po preveden\u00ed t\u00fdchto oper\u00e1ci\u00ed bol pr\u00edklad vypo\u010d\u00edtan\u00fd spr\u00e1vne. Ak\u00e9 \u010d\u00edslo Mat na\u0161iel? <\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z6-II-2<\/strong><br \/>\nLenka si mysl\u00ed dve dvojcifern\u00e9 \u010d\u00edsla. Jedno m\u00e1 obe cifry p\u00e1rne a druh\u00e9 obe nep\u00e1rne. Ke\u010f obe \u010d\u00edsla s\u010d\u00edta, dostane op\u00e4\u0165 dvojcifern\u00e9 \u010d\u00edslo, ktor\u00e9 m\u00e1 prv\u00fa cifru p\u00e1rnu a druh\u00fa nep\u00e1rnu. Navy\u0161e n\u00e1m Lenka prezradila, \u017ee v\u0161etky tri dvojcifern\u00e9 \u010d\u00edsla s\u00fa n\u00e1sobkami \u010d\u00edsla tri a jedna z nep\u00e1rnych cifier je 9. Ak\u00e9 \u010d\u00edsla si mohla Lenka myslie\u0165? N\u00e1jdite v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(V. Huc\u00edkov\u00e1)<\/div>\n<p><strong>Z6-II-3<\/strong><br \/>\n\u0160tvoruholn\u00edk <em>ABCD<\/em> m\u00e1 nasleduj\u00face vlastnosti:<\/p>\n<ul>\n<li>strany <em>AB<\/em> a <em>CD<\/em> s\u00fa rovnobe\u017en\u00e9,<\/li>\n<li>pri vrchole <em>B<\/em> je prav\u00fd uhol,<\/li>\n<li>trojuholn\u00edk <em>ADB<\/em> je rovnoramenn\u00fd so z\u00e1klad\u0148ou <em>AB<\/em>,<\/li>\n<li>strany <em>BC<\/em> a <em>CD<\/em> s\u00fa dlh\u00e9 10 cm.<\/li>\n<\/ul>\n<p>Ur\u010dte obsah tohto \u0161tvoruholn\u00edka.<\/p>\n<div class=\"pri\">(J. Maz\u00e1k)<\/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=\"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 \/>\nNa obr\u00e1zku je \u0161es\u0165 kr\u00fa\u017ekov, ktor\u00e9 tvoria vrcholy \u0161tyroch trojuholn\u00edkov. Nap\u00ed\u0161te do kr\u00fa\u017ekov navz\u00e1jom r\u00f4zne jednocifern\u00e9 prirodzen\u00e9 \u010d\u00edsla tak, aby v ka\u017edom trojuholn\u00edku platilo, \u017ee \u010d\u00edslo vn\u00fatri je s\u00fa\u010dtom \u010d\u00edsel nap\u00edsan\u00fdch v jeho vrcholoch. N\u00e1jdite v\u0161etky rie\u0161enia.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/62\/62z7a1.png\" alt=\"\" width=\"133\" height=\"118\" \/><\/p>\n<div class=\"pri\">(E. Novotn\u00e1)<\/div>\n<p><strong>Z7-I-2<\/strong><br \/>\nPred na\u0161ou \u0161kolou je kvetinov\u00fd z\u00e1hon. Jednu p\u00e4tinu v\u0161etk\u00fdch kvetov tvoria tulip\u00e1ny, dve dev\u00e4tiny narcisy, \u0161tyri p\u00e4tn\u00e1stiny hyacinty a zvy\u0161ok s\u00fa sir\u00f4tky. Ko\u013eko kvetov je celkom na z\u00e1hone, ak zo \u017eiadneho druhu ich nie je viac ako 60 ani menej ako 30?<\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z7-I-3<\/strong><br \/>\nObri Bartolomej a Koloman hovoria niektor\u00e9 dni iba pravdu a niektor\u00e9 dni iba klam\u00fa. Bartolomej hovor\u00ed pravdu iba cez v\u00edkendy, ostatn\u00e9 dni klame. Koloman hovor\u00ed pravdu v pondelok, v piatok a v nede\u013eu, ostatn\u00e9 dni klame.<\/p>\n<p>Jedn\u00e9ho d\u0148a Bartolomej povedal: \u201eV\u010dera sme obaja klamali.\u201c<\/p>\n<p>Koloman v\u0161ak nes\u00fahlasil: \u201eAspo\u0148 jeden z n\u00e1s hovoril v\u010dera pravdu.\u201c<\/p>\n<p>Ktor\u00fd de\u0148 v t\u00fd\u017edni m\u00f4\u017eu obri vies\u0165 tak\u00fdto rozhovor?<\/p>\n<div class=\"pri\">(M. Volfov\u00e1, V. \u017d\u00e1dn\u00edk)<\/div>\n<p><strong>Z7-I-4<\/strong><br \/>\nPani u\u010dite\u013eka nap\u00edsala na tabu\u013eu nasleduj\u00face \u010d\u00edsla:<\/p>\n<p class=\"pc\">1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43.<\/p>\n<p>Dve susedn\u00e9 \u010d\u00edsla sa l\u00ed\u0161ia v\u017edy o rovnak\u00fa hodnotu, v tomto pr\u00edpade o 3. Potom z tabule zotrela v\u0161etky \u010d\u00edsla okrem 1, 19 a 43. \u010ealej medzi \u010d\u00edsla 1 a 43 dop\u00edsala nieko\u013eko cel\u00fdch \u010d\u00edsel tak, \u017ee sa ka\u017ed\u00e9 dve susedn\u00e9 \u010d\u00edsla op\u00e4\u0165 l\u00ed\u0161ili o rovnak\u00fa hodnotu a pritom \u017eiadne \u010d\u00edslo nebolo nap\u00edsan\u00e9 viackr\u00e1t. Ko\u013ek\u00fdmi sp\u00f4sobmi mohla pani u\u010dite\u013eka \u010d\u00edsla doplni\u0165?<\/p>\n<div class=\"pri\">(K. Pazourek)<\/div>\n<p><strong>Z7-I-5<\/strong><br \/>\nV \u0161portovom are\u00e1li je upraven\u00e1 plocha tvaru obd\u013a\u017enika <em>ABCD<\/em> s dlh\u0161ou stranou <em>AB<\/em>. Uhloprie\u010dky <em>AC<\/em> a <em>BD<\/em> zvieraj\u00fa uhol 60\u00b0. Be\u017eci tr\u00e9nuj\u00fa na ve\u013ekom okruhu <em>ACBDA<\/em> alebo na malej dr\u00e1he <em>ADA<\/em>. Mojm\u00edr be\u017eal desa\u0165kr\u00e1t po ve\u013ekom okruhu a Vojtech p\u00e4tn\u00e1s\u0165kr\u00e1t po malej dr\u00e1he, teda p\u00e4tn\u00e1s\u0165kr\u00e1t z <em>A<\/em> do <em>D<\/em> a p\u00e4tn\u00e1s\u0165kr\u00e1t z <em>D<\/em> do <em>A<\/em>. Dokopy ubehli celkom 4,5\u00a0km. Ak\u00e1 dlh\u00e1 je uhloprie\u010dka <em>AC<\/em>?<\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z7-I-6<\/strong><br \/>\nM\u00e1me \u0161tvor\u010dekov\u00fa sie\u0165 so 77 mre\u017eov\u00fdmi bodmi. Dva z nich s\u00fa ozna\u010den\u00e9 <em>A<\/em> a <em>B<\/em> ako na obr\u00e1zku. Bod <em>C<\/em> nech je jeden zo zvy\u0161n\u00fdch mre\u017eov\u00fdch bodov. N\u00e1jdite v\u0161etky mo\u017en\u00e9 polohy bodu <em>C<\/em> tak, aby trojuholn\u00edk <em>ABC<\/em> mal obsah 6 \u0161tvor\u010dekov.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/62\/62z7a2.png\" alt=\"\" width=\"98\" height=\"150\" \/><\/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 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><strong>Z7-II-1<\/strong><br \/>\nDedo Vendel\u00edn i\u0161iel so svojimi dvoma vnukmi Cyrilom a Metodom k\u00fapi\u0165 ryb\u00e1rske pr\u00faty a \u010fal\u0161ie potreby. Cena n\u00e1kupu zaujala Cyrila aj deda. I\u0161lo o \u0161tvorcifern\u00e9 \u010d\u00edslo, v ktorom prv\u00e1 cifra bola o jedna v\u00e4\u010d\u0161ia ako tretia cifra, ale o jedna men\u0161ia ako posledn\u00e1 cifra. S\u00fa\u010det v\u0161etk\u00fdch \u0161tyroch cifier bol 6. Cyril si e\u0161te v\u0161imol, \u017ee dvoj\u010d\u00edslie zlo\u017een\u00e9 z prv\u00fdch dvoch cifier predstavovalo dvojcifern\u00e9 \u010d\u00edslo o 7 v\u00e4\u010d\u0161ie ako dvoj\u010d\u00edslie zlo\u017een\u00e9 z posledn\u00fdch dvoch cifier. Deda v\u0161ak zaujalo \u010d\u00edslo preto, lebo bolo s\u00fa\u010dinom jeho veku a veku oboch vnukov, pritom ka\u017ed\u00fd z vnukov mal viac ako jeden rok. Ko\u013eko rokov mal dedo Vendel\u00edn a ko\u013eko jeho vnuci?<\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z7-II-2<\/strong><br \/>\nPetra m\u00e1 rovnostrann\u00fd trojuholn\u00edk <em>ABC<\/em>. Najsk\u00f4r trojuholn\u00edk prehla tak, aby bod <em>A<\/em> splynul s bodom <em>C<\/em>. Potom vzniknut\u00fd \u0307\u00fatvar prehla tak, \u017ee bod <em>B<\/em> splynul s bodom <em>C<\/em>. Tento \u0307\u00fatvar potom obkreslila na papier a zistila, \u017ee jeho obsah je 12 cm<sup>2<\/sup>. Ur\u010dte obsah p\u00f4vodn\u00e9ho trojuholn\u00edka.<\/p>\n<div class=\"pri\">(E. Novotn\u00e1)<\/div>\n<p><strong>Z7-II-3<\/strong><br \/>\nDo skladu priviezli cement vo vreciach po 25 kg a po 40 kg. Men\u0161\u00edch vriec bolo dvakr\u00e1t viac ako t\u00fdch v\u00e4\u010d\u0161\u00edch. Skladn\u00edk nahl\u00e1sil ved\u00facemu po\u010det v\u0161etk\u00fdch privezen\u00fdch vriec, ale nespomenul, ko\u013eko je ktor\u00fdch. Ved\u00faci si myslel, \u017ee v\u0161etky vrecia v\u00e1\u017eia 25 kg. Nahl\u00e1sen\u00fd po\u010det vriec teda vyn\u00e1sobil \u010d\u00edslom 25 a v \u00fdsledok zadokumentoval ako hmotnos\u0165 dod\u00e1vky cementu. Cez noc zlodeji ukradli 60 v\u00e4\u010d\u0161\u00edch vriec, a tak na sklade ostalo presne to\u013eko kg cementu, ko\u013eko ved\u00faci zap\u00edsal. Ko\u013eko kg cementu ostalo?<\/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 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 \/>\nS\u00fa\u010din troch prirodzen\u00fdch \u010d\u00edsel je 600. Keby sme jedn\u00e9ho \u010dinite\u013ea zmen\u0161ili o 10, zmen\u0161il by sa s\u00fa\u010din o 400. Keby sme namiesto toho jedn\u00e9ho \u010dinite\u013ea zv\u00e4\u010d\u0161ili o 5, zv\u00e4\u010d\u0161il by sa s\u00fa\u010din na dvojn\u00e1sobok p\u00f4vodnej hodnoty. Ktor\u00e9 tri prirodzen\u00e9 \u010d\u00edsla maj\u00fa t\u00fato vlastnos\u0165?<\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z8-I-2<\/strong><br \/>\nStano zlo\u017eil 7 zhodn\u00fdch \u00fatvarov, ka\u017ed\u00fd zlepen\u00fd z 8 rovnak\u00fdch siv\u00fdch kociek s hranou 1 cm tak ako na obr\u00e1zku. Potom v\u0161etky ponoril do bielej farby a n\u00e1sledne ka\u017ed\u00fd z \u00fatvarov rozobral na p\u00f4vodn\u00fdch 8 dielov, ktor\u00e9 tak mali niektor\u00e9 steny siv\u00e9 a in\u00e9 biele. Pridal k nim e\u0161te 8 nov\u00fdch kociek, ktor\u00e9 boli rovnak\u00e9 ako ostatn\u00e9, akur\u00e1t cel\u00e9 biele. Zo v\u0161etk\u00fdch kociek dokopy zlo\u017eil jednu ve\u013ek\u00fa kocku a sna\u017eil sa pritom, aby \u010do najv\u00e4\u010d\u0161ia \u010das\u0165 povrchu vzniknutej kocky bola siv\u00e1. Ko\u013eko cm<sup>2<\/sup> povrchu bude ur\u010dite bielych?<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/62\/62z8a1.png\" alt=\"\" width=\"90\" height=\"88\" \/><\/p>\n<div class=\"pri\">(M. Mach)<\/div>\n<p><strong>Z8-I-3<\/strong><br \/>\nDedo zabudol \u0161tvorcifern\u00fd k\u00f3d svojho mobilu. Vedel len, \u017ee na prvom mieste nebola nula, \u017ee uprostred boli bu\u010f dve \u0161tvorky alebo dve sedmi\u010dky alebo tie\u017e \u0161tvorka so sedmi\u010dkou (v nezn\u00e1mom porad\u00ed) a \u017ee sa jednalo o \u010d\u00edslo delite\u013en\u00e9 \u010d\u00edslom 15. Ko\u013eko je mo\u017enost\u00ed pre zabudnut\u00fd k\u00f3d? Ak\u00e1 cifra mohla by\u0165 na prvom mieste?<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z8-I-4<\/strong><br \/>\nDan\u00e1 je pravideln\u00e1 sedemc\u00edpa hviezda ako na obr\u00e1zku. Ak\u00e1 je ve\u013ekos\u0165 vyzna\u010den\u00e9ho uhla?<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/62\/62z8a2.png\" alt=\"\" width=\"101\" height=\"102\" \/><\/p>\n<div class=\"pri\">(E. Pat\u00e1kov\u00e1)<\/div>\n<p><strong>Z8-I-5<\/strong><br \/>\nD\u0148a 1. septembra 2007 bola zalo\u017een\u00e1 jazykov\u00e1 \u0161kola, v ktorej vyu\u010dovalo sedem pedag\u00f3gov. D\u0148a 1. septembra 2010 k t\u00fdmto siedmim u\u010dite\u013eom pribudol nov\u00fd kolega, ktor\u00fd mal pr\u00e1ve 25 rokov. Do 1. septembra 2012 jeden z u\u010dite\u013eov zo \u0161koly odi\u0161iel, a tak ich zostalo op\u00e4\u0165 sedem. Priemern\u00fd vek pedag\u00f3gov na \u0161kole bol vo v\u0161etk\u00fdch troch spomenut\u00fdch d\u00e1tumoch rovnak\u00fd. Ko\u013eko rokov mal 1. septembra 2012 u\u010dite\u013e, ktor\u00fd v \u0161kole u\u017e nepracoval? Ak\u00fd bol v ten de\u0148 priemern\u00fd vek u\u010dite\u013eov na \u0161kole?<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z8-I-6<\/strong><br \/>\nAni\u010dka a Hanka chodili v labyrinte po \u0161pir\u00e1lovitej cesti\u010dke, ktorej za\u010diatok je schematicky zn\u00e1zornen\u00fd na obr\u00e1zku. Strana \u0161tvor\u010deka v \u0161tvor\u010dekovej sieti m\u00e1 d\u013a\u017eku 1\u00a0m a cel\u00e1 cesti\u010dka od stredu bludiska a\u017e k v\u00fdchodu je dlh\u00e1 210\u00a0m.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/62\/62z8a3.png\" alt=\"\" width=\"122\" height=\"123\" \/><\/p>\n<p>Diev\u010dat\u00e1 vy\u0161li zo stredu bludiska, nikde sa nevracali a po \u010dase ka\u017ed\u00e1 zastavila v niektorom z rohov. Ani\u010dka pritom pre\u0161la o 24\u00a0m viac ako Hanka. V ktor\u00fdch rohoch mohli diev\u010dat\u00e1 st\u00e1\u0165? Ur\u010dte v\u0161etky rie\u0161enia.<\/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 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><strong>Z8-II-1<\/strong><br \/>\nJ\u00falia m\u00e1 na papieri nap\u00edsan\u00e9 \u0161tvorcifern\u00e9 \u010d\u00edslo. Ke\u010f vymen\u00ed cifry na mieste stoviek a jednotiek a s\u010d\u00edta toto nov\u00e9 \u010d\u00edslo s \u010d\u00edslom p\u00f4vodn\u00fdm, dostane v\u00fdsledok 3332. Keby v\u0161ak vymenila cifry na mieste tis\u00edcok a desiatok a s\u010d\u00edtala by toto \u010d\u00edslo s p\u00f4vodn\u00fdm, dostala by v\u00fdsledok 7886. Zistite, ak\u00e9 \u010d\u00edslo mala J\u00falia nap\u00edsan\u00e9 na papieri.<\/p>\n<div class=\"pri\">(E. Novotn\u00e1)<\/div>\n<p><strong>Z8-II-2<\/strong><br \/>\nP\u00e1n Zeler mal na z\u00e1hradke dve rovnak\u00e9 n\u00e1dr\u017ee tvaru \u0161tvorbok\u00e9ho hranola so \u0161tvorcov\u00fdm dnom, v oboch dokopy mal 300 litrov vody. V prvej n\u00e1dr\u017ei tvorila voda presn\u00fa kocku a vyplnila 62,5% n\u00e1dr\u017ee, v druhej n\u00e1dr\u017ei bolo vody o 50 litrov viac. Ak\u00e9 rozmery mali n\u00e1dr\u017ee p\u00e1na Zelera? <\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z8-II-3<\/strong><br \/>\n\u0160ifrovacej hry sa z\u00fa\u010dastnilo 168 hr\u00e1\u010dov v 50 t\u00edmoch, ktor\u00e9 mali dva a\u017e p\u00e4\u0165 \u010dlenov. Najviac bolo \u0161tvor\u010dlenn\u00fdch t\u00edmov, troj\u010dlenn\u00fdch t\u00edmov bolo 20 a hry sa z\u00fa\u010dastnil aspo\u0148 jeden p\u00e4\u0165\u010dlenn\u00fd t\u00edm. Ko\u013eko bolo dvoj\u010dlenn\u00fdch, \u0161tvor\u010dlenn\u00fdch a p\u00e4\u0165\u010dlenn\u00fdch t\u00edmov?<\/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 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<form>\n<input onclick=\"window.location='62\/62z9ir.pdf'\" name=\"butt91\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z9-I-1<\/strong><br \/>\nNa tabuli bolo nap\u00edsan\u00e9 trojcifern\u00e9 prirodzen\u00e9 \u010d\u00edslo. Prip\u00edsali sme k nemu v\u0161etky \u010fal\u0161ie trojcifern\u00e9 \u010d\u00edsla, ktor\u00e9 mo\u017eno z\u00edska\u0165 zmenou poradia jeho cifier. Na tabuli tak boli okrem p\u00f4vodn\u00e9ho \u010d\u00edsla e\u0161te tri nov\u00e9. S\u00fa\u010det najmen\u0161\u00edch dvoch zo v\u0161etk\u00fdch \u0161tyroch \u010d\u00edsel je 1088. Ak\u00e9 cifry obsahuje p\u00f4vodn\u00e9 \u010d\u00edslo?<\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z9-I-2<\/strong><br \/>\nTrojuholn\u00edk m\u00e1 dve strany, ktor\u00fdch d\u013a\u017eky sa l\u00ed\u0161ia o 12 cm, a dve strany, ktor\u00fdch d\u013a\u017eky sa l\u00ed\u0161ia o 15 cm. Obvod tohto trojuholn\u00edka je 75 cm. Ur\u010dte d\u013a\u017eky jeho str\u00e1n. N\u00e1jdite v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z9-I-3<\/strong><br \/>\nPri horskej chate n\u00e1m tr\u00e9ner povedal: \u201eAk p\u00f4jdeme \u010falej t\u00fdmto pohodln\u00fdm tempom 4\u00a0km za hodinu, pr\u00eddeme na stanicu 45 min\u00fat po odchode n\u00e1\u0161ho vlaku.\u201c Potom uk\u00e1zal na skupinu, ktor\u00e1 n\u00e1s pr\u00e1ve m\u00ed\u0148ala: \u201eT\u00ed maj\u00fa lep\u0161iu obuv, a tak dosahuj\u00fa priemern\u00fa r\u00fdchlos\u0165 6\u00a0km za hodinu. Na stanici bud\u00fa u\u017e pol hodiny pred odchodom n\u00e1\u0161ho vlaku.\u201c<\/p>\n<p>Ako \u010faleko bola stanica od horskej chaty?<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z9-I-4<\/strong><br \/>\nDo kru\u017enice s polomerom 5 cm je vp\u00edsan\u00fd pravideln\u00fd osemuholn\u00edk <em>ABCDEFGH<\/em>. Zostrojte trojuholn\u00edk <em>ABX<\/em> tak, aby bod <em>D<\/em> bol ortocentrom (priese\u010dn\u00edkom v\u00fd\u0161ok) trojuholn\u00edka <em>ABX<\/em>.<\/p>\n<div class=\"pri\">(M. Mach)<\/div>\n<p><strong>Z9-I-5<\/strong><br \/>\nDo ka\u017ed\u00e9ho pol\u00ed\u010dka sch\u00e9my na obr\u00e1zku m\u00e1me zap\u00edsa\u0165 \u0161tvorcifern\u00e9 prirodzen\u00e9 \u010d\u00edslo tak, aby v\u0161etky nazna\u010den\u00e9 v\u00fdpo\u010dtov\u00e9 oper\u00e1cie boli spr\u00e1vne. Ko\u013ek\u00fdmi r\u00f4znymi sp\u00f4sobmi mo\u017eno sch\u00e9mu vyplni\u0165?<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/62\/62z9a1.png\" alt=\"\" width=\"545\" height=\"104\" \/><\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z9-I-6<\/strong><br \/>\nDan\u00fd je pravouhl\u00fd lichobe\u017en\u00edk <em>ABCD<\/em> s prav\u00fdm uhlom pri vrchole <em>B<\/em> a s rovnobe\u017en\u00fdmi stranami <em>AB<\/em> a <em>CD<\/em>. Uhloprie\u010dky lichobe\u017en\u00edka s\u00fa na seba kolm\u00e9 a maj\u00fa d\u013a\u017eky |<em>AC<\/em>| = 12 cm, |<em>BD<\/em>| = 9 cm. Vypo\u010d\u00edtajte obvod a obsah tohto lichobe\u017en\u00edka.<\/p>\n<div class=\"pri\">(M. Krej\u010dov\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<form>\n<input onclick=\"window.location='62\/62z9ii.pdf'\" name=\"butt921\" type=\"button\" value=\"Zadanie\" \/><\/form>\n<form>\n<input onclick=\"window.location='62\/62z9iir.pdf'\" name=\"butt922\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z9-II-1<\/strong><br \/>\nDo triedy chod\u00ed 33 \u017eiakov. Pred Vianocami boli s h\u00e1jnikom v lese plni\u0165 k\u0155midl\u00e1. Diev\u010dat\u00e1 si rozobrali bal\u00edky sena. Chlapci sa rozdelili na dve skupiny: t\u00ed z prvej skupiny vzali ka\u017ed\u00fd 4 vreck\u00e1 mrkvy a 3 vreck\u00e1 orechov (teda ka\u017ed\u00fd z nich vzal 7 vreciek) a t\u00ed z druhej skupiny vzali ka\u017ed\u00fd jedno vrecko jab\u013ak a jedno vrecko orechov (teda ka\u017ed\u00fd\u00a0z nich vzal 2 vreck\u00e1). Pomer po\u010dtu diev\u010dat, chlapcov z prvej skupiny a chlapcov z druhej skupiny bol rovnak\u00fd ako pomer celkov\u00e9ho po\u010dtu vreciek orechov, jab\u013ak a mrkvy. Ko\u013eko bolo v triede diev\u010dat, ko\u013eko chlapcov nieslo vreck\u00e1 s mrkvou a ko\u013eko ich nieslo vreck\u00e1 s jablkami?<\/p>\n<div class=\"pri\">(M. Mach)<\/div>\n<p><strong>Z9-II-2<\/strong><br \/>\nNa \u010dist\u00fa tabu\u013eu sme \u017eltou kriedou nap\u00edsali trojcifern\u00e9 prirodzen\u00e9 \u010d\u00edslo tvoren\u00e9 navz\u00e1jom r\u00f4znymi nenulov\u00fdmi ciframi. Potom sme na tabu\u013eu bielou kriedou vyp\u00edsali v\u0161etky \u010fal\u0161ie trojcifern\u00e9 \u010d\u00edsla, ktor\u00e9 mo\u017eno z\u00edska\u0165 zmenou poradia cifi\fer \u017elt\u00e9ho \u010d\u00edsla. Aritmetick\u00fd priemer v\u0161etk\u00fdch \u010d\u00edsel na tabuli bol 370. Ka\u017ed\u00e9 \u010d\u00edslo men\u0161ie ako \u017elt\u00e9 sme pod\u010diarkli. Pod\u010diarknut\u00e9 \u010d\u00edsla boli pr\u00e1ve tri a ich aritmetick\u00fd priemer bol 205. Ur\u010dte \u017elt\u00e9 \u010d\u00edslo.<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z9-II-3<\/strong><br \/>\nVyzna\u010dme vo v\u0161eobecnom trojuholn\u00edku <em>ABC<\/em> nasleduj\u00face body pod\u013ea obr\u00e1zka. Body <em>A<\/em><sub>1<\/sub> a <em>B<\/em><sub>1<\/sub> s\u00fa stredy str\u00e1n <em>BC<\/em> a <em>AC<\/em>, body <em>C<\/em><sub>1<\/sub>, <em>C<\/em><sub>2<\/sub> a <em>C<\/em><sub>3<\/sub> delia stranu <em>AB<\/em> na \u0161tyri rovnak\u00e9 \u010dasti. Spoj\u00edme body <em>A<\/em><sub>1<\/sub> a <em>B<\/em><sub>1<\/sub> s bodmi <em>C<\/em><sub>1<\/sub> a <em>C<\/em><sub>3<\/sub>, tak\u017ee vznikne ma\u0161\u013ea ohrani\u010den\u00e1 t\u00fdmito spojnicami. Ak\u00fa \u010das\u0165 obsahu cel\u00e9ho trojuholn\u00edka ma\u0161\u013ea zaber\u00e1?<\/p>\n<div class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/62\/62z9b1.png\" alt=\"\" width=\"159\" height=\"125\" \/><\/div>\n<div class=\"pri\">(M. Mach)<\/div>\n<p><strong>Z9-II-4<\/strong><br \/>\nN\u00e1jdite v\u0161etky sedemcifern\u00e9 \u010d\u00edsla, ktor\u00e9 obsahuj\u00fa ka\u017ed\u00fa z cifi\fer 0 a\u017e 6 pr\u00e1ve raz a pre ktor\u00e9 plat\u00ed, \u017ee ich prv\u00e9 aj posledn\u00e9 dvoj\u010d\u00edslie je delite\u013en\u00e9 2, prv\u00e9 aj posledn\u00e9 troj\u010d\u00edslie je delite\u013en\u00e9 3, prv\u00e9 aj posledn\u00e9 \u0161tvor\u010d\u00edslie je delite\u013en\u00e9 4, prv\u00e9 aj posledn\u00e9 p\u00e4\u0165\u010d\u00edslie je delite\u013en\u00e9 5 a prv\u00e9 aj posledn\u00e9 \u0161es\u0165\u010d\u00edslie je delite\u013en\u00e9 6.<\/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 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<form>\n<input onclick=\"window.location='62\/62z9iii.pdf'\" name=\"butt931\" type=\"button\" value=\"Zadanie\" \/><\/form>\n<form>\n<input onclick=\"window.location='62\/62z9iiir.pdf'\" name=\"butt932\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z9-III-1<\/strong><br \/>\nV malom kr\u00e1\u013eovstve pri\u0161li poddan\u00ed pozdravi\u0165 nov\u00e9ho kr\u00e1\u013ea a jeho ministra. Ka\u017ed\u00fd priniesol nejak\u00fd dar\u010dek: 60 najchudobnej\u0161\u00edch prinieslo len vlastnoru\u010dne vyroben\u00fa dreven\u00fa so\u0161ku, t\u00ed bohat\u0161\u00ed bu\u010f 2 zlatky, alebo 3 strieborniaky. Pritom strieborniakov bolo darovan\u00fdch viac ako 40, ale menej ako 100. V\u0161etky so\u0161ky dostal minister a k tomu e\u0161te sedminu v\u0161etk\u00fdch zlatiek a tretinu v\u0161etk\u00fdch strieborniakov. Kr\u00e1\u013e aj jeho minister dostali rovnak\u00fd po\u010det predmetov. Zistite, ko\u013eko mohlo by\u0165 poddan\u00fdch, ko\u013eko mohlo by\u0165 darovan\u00fdch zlatiek a ko\u013eko strieborniakov.<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z9-III-2<\/strong><br \/>\nMatej mal v riadku v zo\u0161ite nap\u00edsan\u00fdch \u0161es\u0165 r\u00f4znych prirodzen\u00fdch \u010d\u00edsel. Druh\u00e9 z nich bolo dvojn\u00e1sobkom prv\u00e9ho, tretie bolo dvojn\u00e1sobkom druh\u00e9ho a podobne ka\u017ed\u00e9 \u010fal\u0161ie \u010d\u00edslo bolo dvojn\u00e1sobkom predch\u00e1dzaj\u00faceho. Matej v\u0161etky tieto \u010d\u00edsla op\u00edsal do nasleduj\u00facej tabu\u013eky, a to v \u013eubovo\u013enom porad\u00ed, do ka\u017ed\u00e9ho pol\u00ed\u010dka jedno:<\/p>\n<table style=\"width: 180px;\" border=\"1\" cellspacing=\"0\" cellpadding=\"8\" align=\"center\">\n<tbody>\n<tr>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>S\u00fa\u010det oboch \u010d\u00edsel v prvom st\u013apci tabu\u013eky bol 136 a s\u00fa\u010det \u010d\u00edsel v druhom st\u013apci bol dvojn\u00e1sobn\u00fd, teda 272. Ur\u010dte s\u00fa\u010det \u010d\u00edsel v tre\u0165om st\u013apci tabu\u013eky.<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z9-III-3<\/strong><br \/>\nDan\u00fd je pravideln\u00fd osemuholn\u00edk <em>ABCDEFGH<\/em> a bod <em>X<\/em> tak\u00fd, \u017ee bod <em>A<\/em> je ortocentrom (t. j. priese\u010dn\u00edkom v\u00fd\u0161ok) trojuholn\u00edka <em>BDX<\/em>. Vypo\u010d\u00edtajte ve\u013ekosti vn\u00fatorn\u00fdch uhlov tohto trojuholn\u00edka.<\/p>\n<div class=\"pri\">(Vojt\u011bch \u017d\u00e1dn\u00edk)<\/div>\n<p><strong>Z9-III-4<\/strong><br \/>\nV slove TESTOVANIE maj\u00fa by\u0165 nahraden\u00e9 rovnak\u00e9 p\u00edsmenk\u00e1 rovnak\u00fdmi nenulov\u00fdmi ciframi a r\u00f4zne p\u00edsmenk\u00e1 r\u00f4znymi nenulov\u00fdmi ciframi. Pritom s\u00fa\u010din cifier v\u00fdsledn\u00e9ho \u010d\u00edsla m\u00e1 by\u0165 druhou mocninou nejak\u00e9ho prirodzen\u00e9ho \u010d\u00edsla. N\u00e1jdite najv\u00e4\u010d\u0161ie \u010d\u00edslo, ktor\u00e9 mo\u017eno nahraden\u00edm p\u00edsmen pri splnen\u00ed uveden\u00fdch podmienok z\u00edska\u0165.<\/p>\n<div class=\"pri\">(Eva Pat\u00e1kov\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 href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><script type=\"text\/javascript\"><!--\r\ngoogle_ad_client = \"pub-0508472359151885\";\r\n\/* 468x60, bola vytvoren\u00e1 12.11.2010 *\/\r\ngoogle_ad_slot = \"3127737040\";\r\ngoogle_ad_width = 468;\r\ngoogle_ad_height = 60;\r\n\/\/-->\r\n<\/script>\r\n<script type=\"text\/javascript\"\r\nsrc=\"http:\/\/pagead2.googlesyndication.com\/pagead\/show_ads.js\">\r\n<\/script><\/p>\n<!--CusAds0-->\n<div style=\"font-size: 0px; height: 0px; line-height: 0px; margin: 0; padding: 0; clear: both;\"><\/div>","protected":false},"excerpt":{"rendered":"<p>62. ro\u010dn\u00edk matematickej olympi\u00e1dy 2012-2013 aktualizovan\u00e9 29.4.2013 14:50<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[11],"tags":[],"class_list":["post-696","post","type-post","status-publish","format-standard","hentry","category-62-rocnik"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.6 - 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