{"id":143,"date":"2010-09-28T20:07:01","date_gmt":"2010-09-28T20:07:01","guid":{"rendered":"http:\/\/matematika.okamzite.eu\/?p=143"},"modified":"2010-09-28T20:07:01","modified_gmt":"2010-09-28T20:07:01","slug":"kategorie-z4-z9-4","status":"publish","type":"post","link":"https:\/\/matematika.besaba.com\/?p=143","title":{"rendered":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z4-Z9"},"content":{"rendered":"<p class=\"hh2\"><a name=\"top\"><\/a>60. ro\u010dn\u00edk matematickej olympi\u00e1dy 2010-2011<\/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: 98%;\" border=\"1\" cellspacing=\"2\" cellpadding=\"2\" align=\"center\">\n<tbody>\n<tr align=\"center\" bgcolor=\"#c0c0c0\">\n<td><span class=\"a\">Z4<\/span><\/td>\n<td><span class=\"a\">Z5<\/span><\/td>\n<td><span class=\"a\">Z6<\/span><\/td>\n<td><span class=\"a\">Z7<\/span><\/td>\n<td><span class=\"a\">Z8<\/span><\/td>\n<td><span class=\"a\">Z9<\/span><\/td>\n<\/tr>\n<tr align=\"center\">\n<td><span class=\"a\"><a href=\"#z41\">dom\u00e1ce<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z51\">dom\u00e1ce<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z61\">dom\u00e1ce<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z71\">dom\u00e1ce<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z81\">dom\u00e1ce<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z91\">dom\u00e1ce<\/a><\/span><\/td>\n<\/tr>\n<tr align=\"center\">\n<td><span class=\"a\"><a href=\"#z42\">\u0161kolsk\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z52\">obvodn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z62\">obvodn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z72\">obvodn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z82\">obvodn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z92\">obvodn\u00e9<\/a><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"5\" bgcolor=\"#c0c0c0\">\u00a0<\/td>\n<td align=\"center\"><span class=\"a\"><a href=\"#z93\">krajsk\u00e9<\/a><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><!--more--><\/p>\n<p><a name=\"z41\"><\/a><\/p>\n<p><strong>Z4-I-1<\/strong><br \/>\nDopl\u0148 do pr\u00e1zdnych pol\u00ed\u010dok \u010d\u00edsla od 1 do 7 ka\u017ed\u00e9 raz tak, aby matematick\u00e9 oper\u00e1cie boli vypo\u010d\u00edtan\u00e9 spr\u00e1vne.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/60\/60z4a1.png\" alt=\"\" width=\"566\" height=\"46\" \/><\/p>\n<div class=\"pri\">(M. Smitkov\u00e1)<\/div>\n<p><strong>Z4-I-2<\/strong><br \/>\nMi\u0161ko a Jarka s\u00fa s\u00farodenci. Jarka m\u00e1 narodeniny niekedy v janu\u00e1ri. O Mi\u0161kovi vieme, \u017ee v roku 2010 bola od Jarkin\u00fdch naroden\u00edn po Mi\u0161kove narodeniny presne jedna sobota trin\u00e1steho. Zisti, v ktorom mesiaci sa narodil Mi\u0161ko. N\u00e1jdi v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(M. Dillingerov\u00e1)<\/div>\n<p><strong>Z4-I-3<\/strong><br \/>\nKo\u013eko trojcifern\u00fdch \u010d\u00edsiel m\u00e1 prv\u00fa \u010d\u00edslicu trikr\u00e1t v\u00e4\u010d\u0161iu ako druh\u00fa a tretiu \u010d\u00edslicu o\u00a04 men\u0161iu ako prv\u00fa? Vyp\u00ed\u0161 v\u0161etky tak\u00e9 \u010d\u00edsla.<\/p>\n<div class=\"pri\">(M. Smitkov\u00e1, M. Dillingerov\u00e1)<\/div>\n<p><strong>Z4-I-4<\/strong><br \/>\nJo\u017ekovi sa podarilo rozl\u00e1ma\u0165 \u010dokol\u00e1du na tak\u00e9to k\u00fasky:<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/60\/60z4a2.gif\" alt=\"\" width=\"294\" height=\"121\" \/><\/p>\n<p>Dala by sa t\u00e1to \u010dokol\u00e1da bez \u010fal\u0161ieho l\u00e1mania spravodlivo rozdeli\u0165 dvom kamar\u00e1tom? Ako?<br \/>\nDala by sa t\u00e1to \u010dokol\u00e1da spravodlivo rozdeli\u0165 bez \u010fal\u0161ieho l\u00e1mania trom kamar\u00e1tom? Ako?<br \/>\nAk sa to d\u00e1, n\u00e1jdi v\u017edy aspo\u0148 jeden sp\u00f4sob.<\/p>\n<div class=\"pri\">(M. Dillingerov\u00e1)<\/div>\n<p><strong>Z4-I-5<\/strong><br \/>\nNa st\u00f4l do kuchyne polo\u017eila mami\u010dka vyl\u00faskan\u00fd hrach v miske. Danka a Janka poch\u00fa\u0165ku objavili a za\u010dali hr\u00e1\u0161ky z misky vyjeda\u0165. Dohodli sa, \u017ee Danka si bude z misky bra\u0165 v\u017edy 2 gu\u013e\u00f4\u010dky hrachu. Janka si bude pravidelne bra\u0165 2, 4, 1 a 1 gu\u013e\u00f4\u010dku hrachu a potom za\u010dne op\u00e4\u0165 od za\u010diatku. Najsk\u00f4r si vzala z misky Danka 2 hr\u00e1\u0161ky, potom Janka 2, op\u00e4\u0165 Danka 2, Janka svoje 4, at\u010f. Zrazu pri\u0161la do kuchyne ich mami\u010dka a prekvapene zh\u00edkla: \u201dVe\u010f v miske u\u017e zostala iba polovica hrachu!\u201c Diev\u010dat\u00e1 za\u010dali by\u0165 zvedav\u00e9 a spo\u010d\u00edtali, \u017ee tam ostalo 45 gu\u013e\u00f4\u010dok hrachu.<br \/>\nAk sa mami\u010dka nem\u00fdlila a zvy\u0161n\u00fdch 45 gu\u013e\u00f4\u010dok bola naozaj polovica z toho, \u010do bolo v miske na za\u010diatku, zjedli potom diev\u010dat\u00e1 rovnako alebo niektor\u00e1 zjedla viac? Ko\u013eko hr\u00e1\u0161kov zjedla Danka? A ko\u013eko ich zjedla Janka?<\/p>\n<div class=\"pri\">(M. Dillingerov\u00e1)<\/div>\n<p><strong>Z4-I-6<\/strong><br \/>\nV na\u0161ej bytovke je 10 bytov. Niektor\u00e9 maj\u00fa 4, niektor\u00e9 3 a niektor\u00e9 2 okn\u00e1. Na na\u0161ej bytovke je celkom 27 okien. Bytov s dvomi oknami je v bytovke najviac. Ko\u013eko je ktor\u00fdch bytov?<\/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 \/>\nDopl\u0148 do pr\u00e1zdnych pol\u00ed\u010dok 8 za sebou id\u00facich jednocifern\u00fdch \u010d\u00edsel ka\u017ed\u00e9 raz tak, aby matematick\u00e9 oper\u00e1cie boli vypo\u010d\u00edtan\u00e9 spr\u00e1vne.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/60\/60z4b1.png\" alt=\"\" width=\"652\" height=\"46\" \/><\/p>\n<div class=\"pri\">(M. Dillingerov\u00e1, M. Smitkov\u00e1)<\/div>\n<p><strong>Z4-II-2<\/strong><br \/>\nPred \u0161kolou v Koc\u00farkove st\u00e1li bicykle a aut\u00e1. Keby sa tu zastavilo e\u0161te jedno auto, bolo by ich to\u013eko ako bicyklov. Keby sa tu zastavilo e\u0161te 5 bicyklov, mali by rovnako ve\u013ea kolies ako aut\u00e1. Ko\u013eko st\u00e1lo pred \u0161kolou \u00e1ut? Ko\u013eko tam st\u00e1lo bicyklov?<\/p>\n<div class=\"pri\">(M. Dillingerov\u00e1)<\/div>\n<p><strong>Z4-II-3<\/strong><br \/>\nJanko dostal na Vianoce knihu, ktor\u00fa hne\u010f v ten de\u0148 za\u010dal \u010d\u00edta\u0165. V posledn\u00fd janu\u00e1rov\u00fd de\u0148 nov\u00e9ho roka Janko zistil, \u017ee pre\u010d\u00edtal 60 str\u00e1n, \u010do je polovica knihy a povedal si, \u017ee ak chce knihu cel\u00fa do\u010d\u00edta\u0165 do svojich naroden\u00edn, mus\u00ed ka\u017ed\u00fd de\u0148 pre\u010d\u00edta\u0165 5 str\u00e1n. Kedy (presne ktor\u00fd de\u0148 a mesiac) m\u00e1 Janko narodeniny?<\/p>\n<div class=\"pri\">(M. Smitkov\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=\"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 \/>\nVlado m\u00e1 nap\u00edsan\u00e9 dve \u010d\u00edsla, 541 a 293. Zo \u0161iestich pou\u017eit\u00fdch cifer m\u00e1 najsk\u00f4r vy\u0161krtn\u00fa\u0165 dve tak, aby s\u00fa\u010det dvoch takto z\u00edskan\u00fdch \u010d\u00edsel bol najv\u00e4\u010d\u0161\u00ed mo\u017en\u00fd. Potom m\u00e1 z p\u00f4vodn\u00fdch \u0161iestich cifer vy\u0161krtn\u00fa\u0165 dve tak, aby rozdiel dvoch takto z\u00edskan\u00fdch \u010d\u00edsel bol najmen\u0161\u00ed mo\u017en\u00fd (od\u010d\u00edta men\u0161ie \u010d\u00edslo od v\u00e4\u010d\u0161ieho). Ktor\u00e9 cifry m\u00e1 v jednotliv\u00fdch pr\u00edpadoch vy\u0161krtn\u00fa\u0165?<\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z5-I-2<\/strong><br \/>\nV Trpasli\u010dom kr\u00e1\u013eovstve meraj\u00fa vzdialenosti v rozpr\u00e1vkov\u00fdch m\u00ed\u013each (rm), v rozpr\u00e1vkov\u00fdch siahach (rs) a v rozpr\u00e1vkov\u00fdch lak\u0165och (rl). Na vstupnej br\u00e1ne do Trpasli\u010dieho kr\u00e1\u013eovstva je nasleduj\u00faca tabu\u013eka na prevody medzi ich jednotkami a na\u0161imi:<br \/>\n\u2022\u00a0\u00a0<strong>1 rm = 385 cm<\/strong>,<br \/>\n\u2022\u00a0\u00a0<strong>1 rs = 105 cm<\/strong>,<br \/>\n\u2022\u00a0\u00a0<strong>1 rl = 250 mm<\/strong>.<br \/>\nKr\u00e1\u013e Trpasl\u00edk I. nechal premera\u0165 vzdialenos\u0165 od z\u00e1mockej br\u00e1ny k rozpr\u00e1vkov\u00e9mu jazierku. Traja pozvan\u00ed zememera\u010di dospeli k t\u00fdmto v\u00fdsledkom: prv\u00fd nameral <em>4\u00a0rm 4\u00a0rs 18\u00a0rl<\/em>, druh\u00fd <em>3\u00a0rm 2\u00a0rs 43\u00a0rl<\/em> a tret\u00ed <em>6\u00a0rm 1\u00a0rs 1\u00a0rl<\/em>. Jeden z nich sa v\u0161ak pom\u00fdlil. Ak\u00e1 je vzdialenos\u0165 v centimetroch od z\u00e1mockej br\u00e1ny k rozpr\u00e1vkov\u00e9mu jazierku? O ko\u013eko centimetrov sa pom\u00fdlil nepresn\u00fd zememera\u010d?<\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z5-I-3<\/strong><br \/>\n\u0160tyria kamar\u00e1ti Adam, Mojm\u00edr a dvoj\u010dat\u00e1 Peter a Pavol z\u00edskali na hodin\u00e1ch matematiky celkom 52\u00a0smajl\u00edkov, ka\u017ed\u00fd aspo\u0148 1. Pritom dvoj\u010dat\u00e1 dokopy maj\u00fa 33, ale naj\u00faspe\u0161nej\u0161\u00ed bol Mojm\u00edr. Ko\u013eko ich z\u00edskal Adam?<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z5-I-4<\/strong><br \/>\nP\u00e1n Tik a p\u00e1n Tak pred\u00e1vali bud\u00edky v predajniach &#8222;Pred Rohom&#8220; a &#8222;Za Rohom&#8220;. P\u00e1n Tik tvrdil, \u017ee &#8222;Pred Rohom&#8220; predali o\u00a030\u00a0bud\u00edkov viac ako &#8222;Za Rohom&#8220;, zatia\u013e \u010do p\u00e1n Tak tvrdil, \u017ee &#8222;Pred Rohom&#8220; predali trikr\u00e1t viac bud\u00edkov ako &#8222;Za Rohom&#8220;. Nakoniec sa uk\u00e1zalo, \u017ee Tik aj Tak mali pravdu. Ko\u013eko bud\u00edkov predali v oboch predajniach celkom?<\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z5-I-5<\/strong><br \/>\nDo kr\u00fa\u017ekov na obr\u00e1zku dopl\u0148te \u010d\u00edsla 1, 2, 3, 4, 5, 6 a 7 tak, aby s\u00fa\u010det \u010d\u00edsel na ka\u017edej vyzna\u010denej l\u00ednii bol rovnak\u00fd. \u017diadne \u010d\u00edslo pritom nesmie by\u0165 pou\u017eit\u00e9 viackr\u00e1t.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/60\/60z5a1.gif\" alt=\"\" width=\"181\" height=\"129\" \/><\/p>\n<div class=\"pri\">(M. Smitkov\u00e1)<\/div>\n<p><strong>Z5-I-6<\/strong><br \/>\nPani \u0160ikovn\u00e1 \u010dakala ve\u010der host\u00ed. Najsk\u00f4r pre nich pripravila 25\u00a0chleb\u00ed\u010dkov. Potom spo\u010d\u00edtala, \u017ee by si ka\u017ed\u00fd hos\u0165 mohol zobra\u0165 dva, ale po troch by u\u017e na v\u0161etk\u00fdch nevy\u0161lo. Povedala si, \u017ee keby vyrobila e\u0161te 10\u00a0chleb\u00ed\u010dkov, mohol by si ka\u017ed\u00fd hos\u0165 vzia\u0165 tri, ale \u0161tyri nie ka\u017ed\u00fd. To sa jej zdalo st\u00e1le m\u00e1lo. Na koniec prichystala dokopy 52 chleb\u00ed\u010dkov. Ka\u017ed\u00fd hos\u0165 by si teda mohol vzia\u0165 \u0161tyri chleb\u00ed\u010dky, ale po p\u00e4\u0165 by u\u017e v\u0161etk\u00fdch nevy\u0161lo. Ko\u013eko host\u00ed pani \u0160ikovn\u00e1 \u010dakala? Ona sama dr\u017e\u00ed di\u00e9tu a ve\u010der nikdy neje.<\/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=\"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 \/>\nMiro nap\u00edsal do radu v\u0161etky n\u00e1sobky \u010d\u00edsla sedem po\u010dn\u00fac 7 a kon\u010diac 70. Medzi \u010d\u00edslami nep\u00edsal \u010diarky ani medzery a \u010d\u00edsla nap\u00edsal od najmen\u0161ieho po najv\u00e4\u010d\u0161ie. V tomto rade \u010d\u00edsel potom \u0161krtol jeden\u00e1s\u0165 cifier. Zistite, ak\u00e9 najv\u00e4\u010d\u0161ie a ak\u00e9 najmen\u0161ie \u010d\u00edslo mohol dosta\u0165.<\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z5-II-2<\/strong><br \/>\nRytier Miloslav sa chystal na turnaj do Vesel\u00edna. Turnaj sa konal v stredu. Ke\u010f\u017ee cesta z Rytierova, kde b\u00fdva, do Vesel\u00edna trv\u00e1 a\u017e dva dni, vyrazil u\u017e v pondelok. Cesta vedie cez \u010fal\u0161ie dve mest\u00e1, Kost\u00edn a Zub\u00edn. Prv\u00fd de\u0148 jazdy pre\u0161iel Miloslav 25m\u00ed\u013e a prenocoval v Zub\u00edne. Druh\u00fd de\u0148, v utorok, \u0161\u0165astne do\u0161iel do Vesel\u00edna. Turnaj s preh\u013eadom vyhral, tak\u017ee ke\u010f sa vo \u0161tvrtok vracal sp\u00e4\u0165, i\u0161iel r\u00fdchlej\u0161ie. Pre\u0161iel o 6 m\u00ed\u013e viac ako v pondelok a prenocoval v Kost\u00edne. V piatok pre\u0161iel zvy\u0161n\u00fdch 11 m\u00ed\u013e a bol doma. Zisti vzdialenos\u0165 medzi Zub\u00ednom a Vesel\u00ednom.<\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z5-II-3<\/strong><br \/>\nSpr\u00e1vca k\u00fape\u013eov p\u00e1n Slnie\u010dko k\u00fapil pre k\u00fape\u013en\u00fdch host\u00ed 58 slne\u010dn\u00edkov. Niektor\u00e9 boli \u010derven\u00e9 a niektor\u00e9 \u017elt\u00e9. \u010cerven\u00e9 boli balen\u00e9 v krabiciach po deviatich kusoch. \u017dlt\u00e9 boli balen\u00e9 v krabiciach po \u0161tyroch kusoch. Oba druhy slne\u010dn\u00edkov nakupoval po cel\u00fdch baleniach. Ko\u013eko mohlo by\u0165 \u017elt\u00fdch slne\u010dn\u00edkov?<\/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=\"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><\/p>\n<p>Ke\u010f Bernard natieral dvere gar\u00e1\u017ee, pretrel omylom aj stupnicu n\u00e1stenn\u00e9ho vonkaj\u0161ieho teplomera. Trubi\u010dka s ortu\u0165ou v\u0161ak zostala nepo\u0161koden\u00e1, a tak Bernard p\u00f4vodn\u00fa stupnicu prelepil p\u00e1sikom vlastnej v\u00fdroby. Na nej starostlivo narysoval dieliky, v\u0161etky boli rovnako ve\u013ek\u00e9 a ozna\u010den\u00e9 \u010d\u00edslami. Jeho dielik mal v\u0161ak in\u00fa ve\u013ekos\u0165 ako p\u00f4vodn\u00fd dielik, ktor\u00fd predstavoval jeden stupe\u0148 Celzia, a aj nulu Bernard umiestnil inde, ako bolo 0\u00b0C. Takto za\u010dal Bernard mera\u0165 teplotu vo vlastn\u00fdch jednotk\u00e1ch: bernardoch. Ke\u010f by mal teplomer ukazova\u0165 teplotu 18\u00b0C, ukazoval 23\u00a0bernardov. Ke\u010f by mal ukazova\u0165 9\u00b0C, ukazoval 8\u00a0bernardov. Ak\u00e1 je teplota v\u00a0\u00b0C, ak vid\u00ed Bernard na svojom teplomere teplotu 13\u00a0bernardov?<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z6-I-2<\/strong><\/p>\n<p>Firma vyr\u00e1baj\u00faca mikrovlnn\u00e9 r\u00fary pred\u00e1vala na trhu v\u017edy po kr\u00e1tkej prezent\u00e1cii svoje modely. Vo \u0161tvrtok predala osem rovnak\u00fdch mikrovlniek. De\u0148 nato u\u017e pon\u00fakala aj svoj nov\u00fd model a \u013eudia si tak mohli k\u00fapi\u0165 ten ist\u00fd ako vo \u0161tvrtok alebo nov\u00fd. V sobotu chceli v\u0161etci z\u00e1ujemcovia nov\u00fd model a firma ich predala v ten de\u0148 \u0161es\u0165. V jednotliv\u00fdch d\u0148och ut\u0155\u017eila 590\u20ac, 720\u20ac a 840\u20ac, neprezrad\u00edme v\u0161ak, ktor\u00e1 suma patr\u00ed ku ktor\u00e9mu d\u0148u.<br \/>\n\u2022\u00a0\u00a0Ko\u013eko st\u00e1l star\u0161\u00ed model mikrovlnky?<br \/>\n\u2022\u00a0\u00a0Ko\u013eko nov\u00fdch modelov predala firma v piatok?<br \/>\n<em><strong>Pozn\u00e1mka:<\/strong> Cena ka\u017edej mikrovlnky bola v cel\u00fdch eur\u00e1ch.<\/em><\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z6-I-3<\/strong><br \/>\nVojto nap\u00edsal \u010d\u00edslo 2010 stokr\u00e1t bez medzier za sebou. Ko\u013eko \u0161tvorcifern\u00fdch a ko\u013eko p\u00e4\u0165cifern\u00fdch s\u00famern\u00fdch \u010d\u00edsel bolo skryt\u00fdch v tomto z\u00e1pise?<br \/>\n<em>(S\u00famern\u00e9 \u010d\u00edslo je tak\u00e9 \u010d\u00edslo, ktor\u00e9 je rovnak\u00e9, \u010di ho \u010d\u00edtame spredu alebo zozadu, napr. 39193.)<\/em><\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z6-I-4<\/strong><br \/>\nS\u00fa\u010din vekov deda Vendel\u00edna a jeho vn\u00fa\u010dat je 2010. S\u00fa\u010det vekov v\u0161etk\u00fdch vn\u00fa\u010dat je\u00a012 a \u017eiadne dve vn\u00fa\u010dat\u00e1 nemaj\u00fa rovnako ve\u013ea rokov. Ko\u013eko vn\u00fa\u010dat m\u00e1 dedo Vendel\u00edn?<\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z6-I-5<\/strong><br \/>\nNa t\u00e1bore sa dvaja ved\u00faci s dvoma t\u00e1born\u00edkmi a psom potrebovali dosta\u0165 cez rieku a k dispoz\u00edcii mali iba jednu lo\u010fku s nosnos\u0165ou 65\u00a0kg. Na\u0161\u0165astie v\u0161etci (okrem psa) dok\u00e1zali lo\u010fku cez rieku priviez\u0165. Ka\u017ed\u00fd ved\u00faci v\u00e1\u017eil pribli\u017ene 60\u00a0kg, ka\u017ed\u00fd t\u00e1born\u00edk 30\u00a0kg a pes 12\u00a0kg. Ako si mali po\u010d\u00edna\u0165? Ko\u013ekokr\u00e1t najmenej musela lo\u010fka prekona\u0165 rieku?<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z6-I-6<\/strong><br \/>\nKarol obstaval krabicu s obd\u013a\u017enikov\u00fdm dnom obrubou z koc\u00f4\u010dok. Pou\u017eil pr\u00e1ve 22 koc\u00f4\u010dok s hranou 1\u00a0dm, ktor\u00e9 staval tesne ved\u013ea seba v jednej vrstve. Medzi obrubou a stenami krabice nebola medzera a cel\u00e1 t\u00e1to stavba mala obd\u013a\u017enikov\u00fd p\u00f4dorys. Ak\u00e9 rozmery mohlo ma\u0165 dno krabice?<\/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=\"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 \/>\nPani Hundrav\u00e1 mala 1.j\u00fala 2010 na svojom mobile kredit 3,14\u20ac. Z kreditu sa postupne odpo\u010d\u00edtavaj\u00fa \u010diastky za hovory a to tak, \u017ee za ka\u017ed\u00fa za\u010dat\u00fa min\u00fatu sa od\u010d\u00edta 9 centov. Textov\u00e9 spr\u00e1vy pani Hundrav\u00e1 nep\u00ed\u0161e a nevyu\u017e\u00edva ani \u017eiadne \u010fal\u0161ie platen\u00e9 slu\u017eby. Svoj kredit dob\u00edja pod\u013ea potreby a to v\u017edy sumou 8\u20ac. D\u0148a 31.decembra 2010 bol jej kredit 7,06\u20ac. Ko\u013ekokr\u00e1t minim\u00e1lne dob\u00edjala pani Hundrav\u00e1\u00a0za uveden\u00fd polrok svojkredit?<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z6-II-2<\/strong><br \/>\nV obd\u013a\u017eniku <em>KLMN<\/em> je vzdialenos\u0165 priese\u010dn\u00edka jeho uhloprie\u010dok od priamky <em>KL <\/em>o 2 cm men\u0161ia ako jeho vzdialenos\u0165 od priamky <em>LM<\/em>. Obvod obd\u013a\u017enika je 56 cm. Ak\u00fd je obsah obd\u013a\u017enika <em>KLMN<\/em>?<\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z6-II-3<\/strong><br \/>\nV lete sa u babi\u010dky stretlo v\u0161etk\u00fdch jej \u0161es\u0165 vn\u00fa\u010dat. O vn\u00fa\u010dat\u00e1ch n\u00e1m babi\u010dka prezradila, \u017ee<\/p>\n<p>\u2022 Martinka sa niekedy mus\u00ed stara\u0165 o bra\u010deka Tom\u00e1\u0161ka, ktor\u00fd je o 8 rokov mlad\u0161\u00ed,<br \/>\n\u2022 Vierka, ktor\u00e1 je o 7 rokov star\u0161ia ako Ivana, rada rozpr\u00e1va stra\u0161ideln\u00e9 pr\u00edbehy,<br \/>\n\u2022 s Martinkou sa \u010dasto ha\u0161ter\u00ed o rok mlad\u0161\u00ed Jarom\u00edr,<br \/>\n\u2022 Tom\u00e1\u0161ko je o 11 rokov mlad\u0161\u00ed ako Katka,<br \/>\n\u2022 Ivana \u010dasto hnev\u00e1 svojho o 4 roky star\u0161ieho brata Jarom\u00edra,<br \/>\n\u2022 chlapci maj\u00fa dokopy 13 rokov.<\/p>\n<p>Ko\u013eko rokov maj\u00fa jednotliv\u00e9 deti?<\/p>\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=\"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 \/>\nS\u00fa\u010din cifier \u013eubovo\u013en\u00e9ho viaccifern\u00e9ho \u010d\u00edsla je v\u017edy men\u0161\u00ed ako toto \u010d\u00edslo. Ak po\u010d\u00edtame s\u00fa\u010din cifier dan\u00e9ho \u010d\u00edsla, potom s\u00fa\u010din cifier tohto s\u00fa\u010dinu, potom znova s\u00fa\u010din cifier nov\u00e9ho s\u00fa\u010dinu at\u010f., nutne po nejakom po\u010dte krokov dospejeme k\u00a0jednocifern\u00e9mu \u010d\u00edslu. Tento po\u010det krokov naz\u00fdvame <em>perzistencia<\/em> \u010d\u00edsla. Napr. \u010d\u00edslo 723 m\u00e1 perzistenciu 2, lebo 7\u00a0\u00b7\u00a02\u00a0\u00b7\u00a03\u00a0=\u00a042 (1. krok) a 4\u00a0\u00b7\u00a02\u00a0=\u00a08 (2. krok).<br \/>\n\u2022\u00a0\u00a0N\u00e1jdite najv\u00e4\u010d\u0161ie nep\u00e1rne \u010d\u00edslo, ktor\u00e9 m\u00e1 navz\u00e1jom r\u00f4zne cifry a perzistenciu 1.<br \/>\n\u2022\u00a0\u00a0N\u00e1jdite najv\u00e4\u010d\u0161ie p\u00e1rne \u010d\u00edslo, ktor\u00e9 m\u00e1 navz\u00e1jom r\u00f4zne nenulov\u00e9 cifry a perzistenciu 1.<br \/>\n\u2022\u00a0\u00a0N\u00e1jdite najmen\u0161ie prirodzen\u00e9 \u010d\u00edslo, ktor\u00e9 m\u00e1 perzistenciu 3.<\/p>\n<div class=\"pri\">(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z7-I-2<\/strong><br \/>\nOndro na v\u00fdlete utratil 2\/3 pe\u0148az\u00ed a zo zvy\u0161ku dal e\u0161te 2\/3 na \u0161kolu pre deti z Tibetu. Za 2\/3 nov\u00e9ho zvy\u0161ku k\u00fapil mal\u00fd dar\u010dek pre mami\u010dku. Z derav\u00e9h ovrecka stratil 4\/5 zvy\u0161n\u00fdch pe\u0148az\u00ed, a ke\u010f zo zvy\u0161n\u00fdch dal polovicu malej sestri\u010dke, ostalo mu pr\u00e1ve jedno euro. S akou sumou i\u0161iel Ondro na v\u00fdlet?<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z7-I-3<\/strong><br \/>\nSilvia prehl\u00e1sila:<br \/>\n&#8222;Sme tri sestry, ja som najmlad\u0161ia, L\u00edvia je star\u0161ia o\u00a0tri roky a Edita o\u00a0osem. Na\u0161a mamka rada po\u010duje, \u017ee v\u0161etky (aj s\u00a0\u0148ou) m\u00e1me v priemere 21\u00a0rokov. Pritom ke\u010f som sa narodila, mala mamka u\u017e 29.&#8220;<br \/>\nPred ko\u013ek\u00fdmi rokmi sa Silvia narodila?<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z7-I-4<\/strong><br \/>\nJuro mal nap\u00edsan\u00e9 \u0161tvorcifern\u00e9 \u010d\u00edslo. Toto \u010d\u00edslo zaokr\u00fahlil na desiatky, na stovky a na tis\u00edcky a v\u0161etky tri v\u00fdsledky zap\u00edsal pod p\u00f4vodn\u00e9 \u010d\u00edslo. V\u0161etky \u0161tyri \u010d\u00edsla spr\u00e1vne s\u010d\u00edtal a dostal 5\u00a0443. Ktor\u00e9 \u010d\u00edslo mal Juro nap\u00edsan\u00e9?<\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z7-I-5<\/strong><br \/>\nLaco narysoval kru\u017enicu so stredom <em>S<\/em> a body <em>A, B, C, D<\/em>, ako ukazuje obr\u00e1zok. Zistil, \u017ee \u00fase\u010dky <em>SC<\/em> a <em>BD<\/em> s\u00fa rovnako dlh\u00e9. V akom pomere s\u00fa ve\u013ekosti uhlov <em>ASC<\/em> a <em>SCD<\/em>?<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/60\/60z7a1.gif\" alt=\"\" width=\"323\" height=\"152\" \/><\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z7-I-6<\/strong><br \/>\nN\u00e1jdite v\u0161etky trojcifern\u00e9 prirodzen\u00e9 \u010d\u00edsla, ktor\u00e9 s\u00fa bezo zvy\u0161ku delite\u013en\u00e9 \u010d\u00edslom\u00a06 a v ktor\u00fdch m\u00f4\u017eeme vy\u0161krtn\u00fa\u0165 ktor\u00fako\u013evek cifru a v\u017edy dostaneme dvojcifern\u00e9 prirodzen\u00e9 \u010d\u00edslo, ktor\u00e9 je tie\u017e bezo zvy\u0161ku delite\u013en\u00e9 \u010d\u00edslom\u00a06.<\/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=\"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 \/>\nM\u00e1m karti\u010dku, na ktorej je nap\u00edsan\u00e9 \u0161tvorcifern\u00e9 prirodzen\u00e9 \u010d\u00edslo. V tomto \u010d\u00edsle m\u00f4\u017eeme vy\u0161krtn\u00fa\u0165 ak\u00e9ko\u013evek dve cifry a v\u017edy dostaneme dvojcifern\u00e9 prirodzen\u00e9 \u010d\u00edslo, ktor\u00e9 je bezo zvy\u0161ku delite\u013en\u00e9 \u010d\u00edslom 5. Ko\u013eko tak\u00fdch \u0161tvorcifern\u00fdch prirodzen\u00fdch \u010d\u00edsel existuje?<br \/>\n<em>(Pozor, napr. 06 nie je dvojcifern\u00e9 \u010d\u00edslo!)<\/em><\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z7-II-2<\/strong><br \/>\nKarol a Vojto zistili, \u017ee kuchynsk\u00e9 hodiny na chalupe ka\u017ed\u00fa hodinu nadbehn\u00fa o 1,5 min\u00faty a hodiny v sp\u00e1lni ka\u017ed\u00fa hodinu pol min\u00faty me\u0161kaj\u00fa. Druh\u00e9ho apr\u00edla na prav\u00e9 poludnie nastavili hodiny na rovnak\u00fd a spr\u00e1vny \u010das. Ur\u010di, kedy op\u00e4\u0165 bud\u00fa (bez \u010fal\u0161ieho opravovania)<\/p>\n<p>\u2022\u00a0\u00a0kuchynsk\u00e9 hodiny ukazova\u0165 spr\u00e1vny \u010das;<br \/>\n\u2022\u00a0\u00a0hodiny v sp\u00e1lni ukazova\u0165 spr\u00e1vny \u010das;<br \/>\n\u2022\u00a0\u00a0oboje hodiny ukazova\u0165 rovnak\u00fd (aj ke\u010f nie nutne spr\u00e1vny) \u010das.<\/p>\n<p><em>(Hodiny v kuchyni aj v sp\u00e1lni maj\u00fa dvan\u00e1s\u0165 hodinov\u00fd cifern\u00edk.)<\/em><\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z7-II-3<\/strong><br \/>\nV trojuholn\u00edku <em>ABC<\/em> ozna\u010d\u00edme stredy str\u00e1n <em>CB<\/em> a <em>CA<\/em> p\u00edsmenami <em>K<\/em> a <em>L<\/em>. Vieme, \u017ee \u0161tvoruholn\u00edk <em>ABKL<\/em> m\u00e1 obvod 10 cm a trojuholn\u00edk <em>KLC<\/em> m\u00e1 obvod 6 cm. Vypo\u010d\u00edtaj d\u013a\u017eku \u00fase\u010dky <em>KL<\/em>.<\/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=\"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 \/>\nMartin m\u00e1 na papieri nap\u00edsan\u00e9 p\u00e4\u0165cifern\u00e9 \u010d\u00edslo s piatimi r\u00f4znymi ciframi a nasleduj\u00facimi vlastnos\u0165ami:<br \/>\n\u2022\u00a0\u00a0\u0161krtnut\u00edm druhej cifry z\u013eava (t.j. cifry na mieste tis\u00edcok) dostane \u010d\u00edslo, ktor\u00e9 je delite\u013en\u00e9 dvoma,<br \/>\n\u2022\u00a0\u00a0\u0161krtnut\u00edm tretej cifry z\u013eava dostane \u010d\u00edslo, ktor\u00e9 je delite\u013en\u00e9 tromi,<br \/>\n\u2022\u00a0\u00a0\u0161krtnut\u00edm \u0161tvrtej cifry z\u013eava dostane \u010d\u00edslo, ktor\u00e9 je delite\u013en\u00e9 \u0161tyrmi,<br \/>\n\u2022\u00a0\u00a0\u0161krtnut\u00edm piatej cifry z\u013eava dostane \u010d\u00edslo, ktor\u00e9 je delite\u013en\u00e9 piatimi,<br \/>\n\u2022\u00a0\u00a0ak ne\u0161krtne \u017eiadnu cifru, m\u00e1 \u010d\u00edslo delite\u013en\u00e9 \u0161iestimi.<br \/>\nKtor\u00e9 najv\u00e4\u010d\u0161ie \u010d\u00edslo m\u00f4\u017ee ma\u0165 Martin nap\u00edsan\u00e9 na papieri?<\/p>\n<div class=\"pri\">(M. Petrov\u00e1)<\/div>\n<p><strong>Z8-I-2<\/strong><br \/>\nKarol sa sna\u017eil do pr\u00e1zdnych pol\u00ed\u010dok na obr\u00e1zku vp\u00edsa\u0165 prirodzen\u00e9 \u010d\u00edsla od\u00a01 do\u00a014 tak, aby \u017eiadne \u010d\u00edslo nebolo pou\u017eit\u00e9 viackr\u00e1t a s\u00fa\u010det v\u0161etk\u00fdch \u010d\u00edsel na ka\u017edej priamej l\u00ednii bol rovnak\u00fd. Po chv\u00edli si uvedomil, \u017ee to nie je mo\u017en\u00e9. Ako by ste Karolovo pozorovanie zd\u00f4vodnili vy?<br \/>\n(<em>Pod priamou l\u00edniou rozumieme skupinu v\u0161etk\u00fdch susediacich pol\u00ed\u010dok, ktor\u00fdch stredy le\u017eia na jednej priamke.<\/em>)<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/60\/60z8a1.gif\" alt=\"\" width=\"131\" height=\"148\" \/><\/p>\n<div class=\"pri\">(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n<p><strong>Z8-I-3<\/strong><br \/>\nCena encyklop\u00e9die &#8222;H\u00e1danky, r\u00e9busy a hlavolamy&#8220; bola zn\u00ed\u017een\u00e1 o\u00a062,5%. Matej zistil, \u017ee obe ceny (pred zn\u00ed\u017een\u00edm aj po \u0148om) s\u00fa dvojcifern\u00e9 \u010d\u00edsla a daj\u00fa sa vyjadri\u0165 rovnak\u00fdmi ciframi, len v r\u00f4znom porad\u00ed. O ko\u013eko \u20ac bola encyklop\u00e9dia zlacnen\u00e1?<\/p>\n<div class=\"pri\">(M.Volfov\u00e1)<\/div>\n<p><strong>Z8-I-4<\/strong><br \/>\nRozde\u013ete kocku s hranou 8\u00a0cm na men\u0161ie zhodn\u00e9 koc\u00f4\u010dky tak, aby s\u00fa\u010det ich povrchov bol p\u00e4\u0165kr\u00e1t v\u00e4\u010d\u0161\u00ed ako povrch p\u00f4vodnej kocky. Ak\u00fd bude objem malej koc\u00f4\u010dky a ko\u013eko centimetrov bude mera\u0165 jej hrana?<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z8-I-5<\/strong><br \/>\nKl\u00e1ra, Lenka a Matej si precvi\u010dovali p\u00edsomn\u00e9 delenie so zvy\u0161kom. Ako delenca mal ka\u017ed\u00fd zadan\u00e9 in\u00e9 prirodzen\u00e9 \u010d\u00edslo, ako delite\u013ea v\u0161ak mali v\u0161etci rovnak\u00e9 prirodzen\u00e9 \u010d\u00edslo. Lenkin delenec bol o\u00a030 v\u00e4\u010d\u0161\u00ed ako Kl\u00e1rin. Matejov delenec bol o\u00a050 v\u00e4\u010d\u0161\u00ed ako Lenkin. Kl\u00e1re vy\u0161iel vo v\u00fdsledku zvy\u0161ok\u00a08, Lenke zvy\u0161ok\u00a02 a Matejovi zvy\u0161ok\u00a04. V\u0161etci po\u010d\u00edtali bez chyby. Ak\u00fd delite\u013e mali \u017eiaci zadan\u00fd?<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z8-I-6<\/strong><br \/>\nV rovnoramennom lichobe\u017en\u00edku <em>ABCD<\/em> s\u00fa uhloprie\u010dky <em>AC<\/em> a <em>DB<\/em> na seba kolm\u00e9, ich d\u013a\u017eka je 8\u00a0cm a d\u013a\u017eka najdlh\u0161ej strany <em>AB<\/em> je tie\u017e 8\u00a0cm. Vypo\u010d\u00edtajte 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=\"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 \/>\nMysl\u00edm si dvojcifern\u00e9 prirodzen\u00e9 \u010d\u00edslo. S\u00fa\u010det ci fier tohto \u010d\u00edsla je delite\u013en\u00fd tromi. Ke\u010f od\u010d\u00edtam od myslen\u00e9ho \u010d\u00edsla \u010d\u00edslo 27, dostanem in\u00e9 dvojcifern\u00e9 prirodzen\u00e9 \u010d\u00edslo, zap\u00edsan\u00e9 pomocou t\u00fdch ist\u00fdch cifier, ale v opa\u010dnom porad\u00ed. Ak\u00e9 \u010d\u00edsla si m\u00f4\u017eem myslie\u0165?<\/p>\n<div class=\"pri\">(L. Hozov\u00e1)<\/div>\n<p><strong>Z8-II-2<\/strong><br \/>\nMartina si vymyslela postup na v\u00fdrobu \u010d\u00edselnej postupnosti. Za\u010dala \u010d\u00edslom 52. Z neho odvodila \u010fal\u0161\u00ed \u010dlen postupnosti takto: 2<sup>2<\/sup> +2.5=4+10=14. Potom pokra\u010dovala rovnak\u00fdm sp\u00f4sobom \u010falej a z \u010d\u00edsla 14 dostala 4<sup>2<\/sup>+2.1=16+2=18. V\u017edy teda vezme \u010d\u00edslo, odtrhne z neho cifru na mieste jednotiek, t\u00fato odtrhnut\u00fa cifru umocn\u00ed na druh\u00fa a\u00a0 k v\u00fdslednej mocnine pripo\u010d\u00edta dvojn\u00e1sobok \u010d\u00edsla, ktor\u00e9 ostalo po odtrhnut\u00ed poslednej cifry. Ak\u00e9 je 2011. \u010d\u00edslo takto vytvorenej postupnosti?<\/p>\n<div class=\"pri\">(M. Dillingerov\u00e1)<\/div>\n<p><strong>Z8-II-3<\/strong><br \/>\nV kru\u017enici <em>k<\/em> so stredom <em>S<\/em> a polomerom 52 mm s\u00fa dan\u00e9 dve na seba kolm\u00e9 tetivy <em>AB<\/em> a <em>CD<\/em>. Ich priese\u010dn\u00edk <em>X<\/em> je od stredu <em>S<\/em> vzdialen\u00fd 25 mm. Ak\u00e1 dlh\u00e1 je tetiva <em>CD<\/em>, ak d\u013a\u017eka tetivy <em>AB<\/em> je 96 mm?<\/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=\"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 \/>\nP\u00e1n Vlk \u010dakal na zast\u00e1vke pred \u0161kolou na autobus. Z okna po\u010dul slov\u00e1 u\u010dite\u013ea:<br \/>\n&#8222;Ak\u00fd povrch m\u00f4\u017ee ma\u0165 pravideln\u00fd \u0161tvorbok\u00fd hranol, ak viete, \u017ee d\u013a\u017eky v\u0161etk\u00fdch jeho hr\u00e1n s\u00fa v centimetroch vyjadren\u00e9 cel\u00fdmi \u010d\u00edslami a \u017ee jeho objem je &#8230;&#8220;<br \/>\nToto d\u00f4le\u017eit\u00e9 \u010d\u00edslo p\u00e1n Vlk nepo\u010dul, preto\u017ee pr\u00e1ve pre\u0161lo okolo auto. Za chv\u00ed\u013eu po\u010dul \u017eiaka oznamuj\u00faceho v\u00fdsledok 918\u00a0cm<sup>2<\/sup>. U\u010dite\u013e na to povedal:<br \/>\n&#8222;\u00c1no, ale \u00faloha m\u00e1 celkom \u0161tyri rie\u0161enia. H\u013eadajte \u010falej.&#8220;<br \/>\nViac sa p\u00e1n Vlk u\u017e nedozvedel, lebo nast\u00fapil do svojho autobusu. Ke\u010f\u017ee matematika bola v\u017edy jeho hobby, vybral si v autobuse ceruzku a papier a po \u010dase ur\u010dil aj zvy\u0161n\u00e9 tri rie\u0161enia u\u010dite\u013eovej \u00falohy. Spo\u010d\u00edtajte ich aj vy.<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z9-I-2<\/strong><br \/>\nNa obr\u00e1zku s\u00fa bodkovanou \u010diarou zn\u00e1zornen\u00e9 hranice \u0161tyroch rovnako ve\u013ek\u00fdch obd\u013a\u017enikov\u00fdch parciel. Sivou farbou je v\u00fdzna\u010dn\u00e1 zastavan\u00e1 plocha. T\u00e1 m\u00e1 tvar obd\u013a\u017enika, ktor\u00e9ho jedna strana tvor\u00ed z\u00e1rove\u0148 hranice parciel. Zap\u00edsan\u00e9 \u010d\u00edsla vyjadruj\u00fa obsah nezastavanej plochy na jednotliv\u00fdch parcel\u00e1ch, a to v\u00a0m<sup>2<\/sup>. Vypo\u010d\u00edtajte obsah celkovej zastavanej plochy.<\/p>\n<p class=\"pc\"><img loading=\"lazy\" decoding=\"async\" src=\"\/60\/60z9a1.gif\" alt=\"\" width=\"331\" height=\"185\" \/><\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z9-I-3<\/strong><br \/>\nV\u013a\u010dkovci lisovali jablkov\u00fd mu\u0161t. Mali ho v dvoch rovnako objemn\u00fdch s\u00fadkoch, v oboch takmer rovnak\u00e9 mno\u017estvo. Keby z prv\u00e9ho preliali do druh\u00e9ho 1\u00a0liter, mali by v oboch rovnako, ale to by ani jeden s\u00fadok nebol pln\u00fd. Tak rad\u0161ej preliali 9\u00a0litrov z druh\u00e9ho do prv\u00e9ho. Potom bol prv\u00fd s\u00fadok \u00faplne pln\u00fd a mu\u0161t v druhom zap\u013a\u0148al pr\u00e1ve tretinu objemu. Ko\u013eko litrov mu\u0161tu vylisovali, ak\u00fd bol objem s\u00fadkov a ko\u013eko mu\u0161tu v nich bolo p\u00f4vodne?<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z9-I-4<\/strong><br \/>\nP\u00e1n R\u00fdchly a p\u00e1n \u0164arb\u00e1k v rovnakom \u010dase vy\u0161tartovali na t\u00fa ist\u00fa turistick\u00fa trasu, len p\u00e1n R\u00fdchly ju i\u0161iel zhora z\u00a0horskej chaty a p\u00e1n \u0164arb\u00e1k naopak od autobusu dolu v meste\u010dku na chatu smerom nahor. Ke\u010f bolo 10\u00a0hod\u00edn, stretli sa na trase. P\u00e1n R\u00fdchly sa pon\u00e1h\u013eal a u\u017e o\u00a012:00 bol v cieli. Naopak p\u00e1n \u0164arb\u00e1k postupoval pomaly, a tak dorazil na chatu a\u017e o\u00a018:00. O ko\u013ekej p\u00e1ni vyrazili na cestu, ak vieme, \u017ee ka\u017ed\u00fd z nich i\u0161iel cel\u00fd \u010das svojou st\u00e1lou r\u00fdchlos\u0165ou?<\/p>\n<div class=\"pri\">(M. Volfov\u00e1)<\/div>\n<p><strong>Z9-I-5<\/strong><br \/>\nKru\u017enici so stredom <em>S<\/em> a polomerom 12\u00a0cm sme op\u00edsali pravideln\u00fd \u0161es\u0165uholn\u00edk <em>ABCDEF<\/em> a vp\u00edsali pravideln\u00fd \u0161es\u0165uholn\u00edk <em>TUVXYZ<\/em> tak, aby bod <em>T<\/em> bol stredom strany <em>BC<\/em>. Vypo\u010d\u00edtajte obsah a obvod \u0161tvoruholn\u00edka <em>TCUS<\/em>.<\/p>\n<div class=\"pri\">(M. Krej\u010dov\u00e1)<\/div>\n<p><strong>Z9-I-6<\/strong><br \/>\nPeter a Pavol oberali v sade jablk\u00e1 a hru\u0161ky. V pondelok zjedol Peter o\u00a02\u00a0hru\u0161ky viac ako Pavol a o\u00a02\u00a0jablk\u00e1 menej ako Pavol. V utorok Peter zjedol o\u00a04\u00a0hru\u0161ky menej ako v pondelok. Pavol zjedol v utorok o\u00a03\u00a0hru\u0161ky viac ako Peter a o\u00a03\u00a0jablk\u00e1 menej ako Peter. Pavol zjedol za oba dni 12\u00a0jab\u013ak a v utorok zjedol rovnak\u00fd po\u010det jab\u013ak ako hru\u0161iek. V utorok ve\u010der obaja chlapci zistili, \u017ee po\u010det jab\u013ak, ktor\u00e9 spolu za oba dni zjedli, je rovnako ve\u013ek\u00fd ako po\u010det spolo\u010dne zjeden\u00fdch hru\u0161iek. Ko\u013eko jab\u013ak zjedol Peter v pondelok a ko\u013eko hru\u0161iek zjedol Pavol v utorok?<\/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=\"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 \/>\nKo\u013eko existuje dvoj\u00edc \u0161tvorcifern\u00fdch palindr\u00f3mov, ktor\u00fdch rozdiel je 3674? Palindr\u00f3m je \u010d\u00edslo, ktor\u00e9 ostane rovnak\u00e9, ke\u010f ho nap\u00ed\u0161eme odzadu. \u0160tvorcifern\u00fd palindr\u00f3m je teda tak\u00e9 \u0161tvorcifern\u00e9 prirodzen\u00e9 \u010d\u00edslo, ktor\u00e9 m\u00e1 na mieste jednotiek rovnak\u00fa cifru ako na mieste tis\u00edcok a na mieste desiatok rovnak\u00fa cifru ako na mieste stoviek.<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z9-II-2<\/strong><br \/>\nNa obr\u00e1zku s\u00fa rovnostrann\u00e9 trojuholn\u00edky <em>ABC<\/em>, <em>DBE<\/em>, <em>IEF<\/em> a <em>HIG<\/em>. Obsahy trojuholn\u00edkov <em>DBE<\/em>, <em>IEF<\/em> a <em>HIG<\/em> s\u00fa v pomere <strong>9:16:4<\/strong>. V akom pomere s\u00fa<br \/>\n1. d\u013a\u017eky \u00fase\u010diek <em>HI<\/em> a <em>IE<\/em>,<br \/>\n2. obsahy trojuholn\u00edkov <em>ABC<\/em> a <em>HEC<\/em>?<\/p>\n<div class=\"pri\">(K. Pazourek)<\/div>\n<p><strong>Z9-II-3<\/strong><br \/>\nDan\u00e9 s\u00fa \u0161tvorce <em>ABCD<\/em> a <em>KLMN<\/em>. D\u013a\u017eky str\u00e1n oboch \u0161tvorcov s\u00fa v centimetroch vyjadren\u00e9 cel\u00fdm \u010d\u00edslom. Bod <em>K<\/em> je vn\u00fatorn\u00fdm bodom \u00fase\u010dky <em>AB<\/em>, bod <em>L<\/em> le\u017e\u00ed v bode <em>B<\/em> a bod <em>M<\/em> je vn\u00fatorn\u00fdm bodom \u00fase\u010dky <em>BC<\/em>. Obsah \u0161es\u0165uholn\u00edka <em>AKNMCD<\/em> je 225\u00a0cm<sup>2<\/sup>.<br \/>\nAk\u00fd m\u00f4\u017ee by\u0165 obvod tohto \u0161es\u0165uholn\u00edka? N\u00e1jdite v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z9-II-4<\/strong><br \/>\nMartina si vymyslela postup na v\u00fdrobu \u010d\u00edselnej postupnosti. Za\u010dala \u010d\u00edslom <strong>128<\/strong>. Z neho odvodila \u010fal\u0161\u00ed \u010dlen postupnosti takto:<strong>8<sup>2<\/sup> + 5 = 64 + 5 = 69<\/strong>. Potom pokra\u010dovala rovnak\u00fdm sp\u00f4sobom a z \u010d\u00edsla <strong>69<\/strong> dostala <strong>9<sup>2<\/sup> + 5 = 81 + 5 = 86<\/strong>. V\u017edy teda z predch\u00e1dzaj\u00faceho \u010dlena postupnosti vezme cifru na mieste jednotiek, umocn\u00ed ju na druh\u00fa a k tejto mocnine pripo\u010d\u00edta kon\u0161tantu <strong>5<\/strong>.<\/p>\n<p>1. Ak\u00e9 je 2011. \u010d\u00edslo tejto postupnosti?<br \/>\n2. Martina op\u00e4\u0165 za\u010dala \u010d\u00edslom <strong>128<\/strong>, ale namiesto \u010d\u00edsla <strong>5<\/strong> zvolila ako kon\u0161tantu in\u00e9<br \/>\nprirodzen\u00e9 \u010d\u00edslo. Tentoraz jej na 2011. mieste vy\u0161lo \u010d\u00edslo <strong>16<\/strong>. Ak\u00fa kon\u0161tantu zvolila v tomto pr\u00edpade?<\/p>\n<div class=\"pri\">(M. Dillingerov\u00e1)<\/div>\n<p><br class=\"spacer_\" \/><\/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=\"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 \/>\nUsporiadate\u013eom v\u00fdstavy &#8222;Na Mesiac a e\u0161te \u010falej&#8220; sa po prvom v\u00fdstavnom dni zdalo, \u017ee m\u00e1lo \u013eud\u00ed si k\u00fapilo na pamiatku let\u00e1k o rakete Apollo\u00a011. Preto zn\u00ed\u017eili jeho cenu o\u00a012\u00a0centov. T\u00fdm sa s\u00edce druh\u00fd de\u0148 zv\u00fd\u0161il po\u010det kupcov let\u00e1ku o\u00a010%, ale celkov\u00e1 denn\u00e1 tr\u017eba za let\u00e1ky sa zn\u00ed\u017eila o\u00a05%. Ko\u013eko centov st\u00e1l let\u00e1k Apollo\u00a011 po z\u013eave?<\/p>\n<div class=\"pri\">(M.Petrov\u00e1)<\/div>\n<p><br class=\"spacer_\" \/><\/p>\n<p><strong>Z9-III-2<\/strong><br \/>\nLichobe\u017en\u00edk <em>ABCD<\/em>, v ktorom strana <em>AB<\/em> je rovnobe\u017en\u00e1 so stranou <em>CD<\/em>, je rozdelen\u00fd uhloprie\u010dkami, ktor\u00e9 sa pret\u00ednaj\u00fa v bode <em>M<\/em>, na \u0161tyri \u010dasti. Ur\u010dte jeho obsah, ke\u010f viete, \u017ee trojuholn\u00edk <em>AMD<\/em> m\u00e1 obsah\u00a08cm<sup>2<\/sup> a trojuholn\u00edk <em>DCM<\/em> m\u00e1 obsah\u00a04cm<sup>2<\/sup>.<\/p>\n<div class=\"pri\">(M.Volfov\u00e1)<\/div>\n<p><br class=\"spacer_\" \/><\/p>\n<p><strong>Z9-III-3<\/strong><br \/>\nCyril a Mirka po\u010d\u00edtali zo zbierky t\u00fa ist\u00fa \u00falohu. Zadan\u00e9 boli d\u013a\u017eky hr\u00e1n kv\u00e1dra v milimetroch a \u00falohou bolo vypo\u010d\u00edta\u0165 jeho objem a povrch. Cyril najsk\u00f4r previedol zadan\u00e9 d\u013a\u017eky na centimetre. Po\u010d\u00edtalo sa mu tak \u013eah\u0161ie, preto\u017ee aj po prevode boli v\u0161etky d\u013a\u017eky vyjadren\u00e9 cel\u00fdmi \u010d\u00edslami. Obom vy\u0161li spr\u00e1vne v\u00fdsledky, Mirke v mm<sup>3<\/sup> a mm<sup>2<\/sup>, Cyrilovi v cm<sup>3<\/sup> a cm<sup>2<\/sup>. Mirkin v\u00fdsledok v mm<sup>3<\/sup> bol o 17982 v\u00e4\u010d\u0161\u00ed ako Cyrilov v\u00fdsledok v cm<sup>3<\/sup>. Mirkin v\u00fdsledok v mm<sup>2<\/sup> bol o 5742 v\u00e4\u010d\u0161\u00ed ako Cyrilov v\u00fdsledok v cm<sup>2<\/sup>. Ur\u010dte d\u013a\u017eky hr\u00e1n kv\u00e1dra.<\/p>\n<div class=\"pri\">(L. \u0160im\u016fnek)<\/div>\n<p><strong>Z9-III-4<\/strong><br \/>\nNa tabuli s\u00fa nap\u00edsan\u00e9 \u010d\u00edsla 1, [pmath size=7]1\/2[\/pmath], [pmath size=8]1\/3[\/pmath], [pmath size=8]1\/4[\/pmath], [pmath size=8]1\/5[\/pmath] a [pmath size=8]1\/6[\/pmath]. Na tabu\u013eu m\u00f4\u017eeme prip\u00edsa\u0165 s\u00fa\u010det alebo s\u00fa\u010din \u013eubovo\u013en\u00fdch dvoch \u010d\u00edsel z tabule. Je mo\u017en\u00e9 tak\u00fdmto pripisovan\u00edm dosiahnu\u0165, aby sa na tabuli objavilo \u010d\u00edslo<\/p>\n<p><strong>a) [pmath size=8]1\/60[\/pmath]<\/strong><br \/>\n<strong>b) [pmath size=8]2011\/375[\/pmath]<\/strong><br \/>\n<strong>c) [pmath size=8]1\/7[\/pmath]<\/strong>?<\/p>\n<div class=\"pri\">(V. Bachrat\u00e1, J. 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