{"id":1608,"date":"2023-10-05T20:44:00","date_gmt":"2023-10-05T20:44:00","guid":{"rendered":"https:\/\/matematika.besaba.com\/?p=1608"},"modified":"2024-07-09T21:44:45","modified_gmt":"2024-07-09T20:44:45","slug":"sutazne-ulohy-kategorii-z5-z9-11","status":"publish","type":"post","link":"https:\/\/matematika.besaba.com\/?p=1608","title":{"rendered":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9"},"content":{"rendered":"<p><span class=\"povinne\">aktualizovan\u00e9 08.07.2023 22:40<\/span><\/p>\n\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\">okresn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z62\">okresn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z72\">okresn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z82\">okresn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z92\">okresn\u00e9<\/a><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"4\" bgcolor=\"#c0c0c0\"><\/td>\n<td align=\"center\"><span class=\"a\"><a href=\"#z93\">krajsk\u00e9<\/a><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><!--more--><br \/>\n<a name=\"z51\"><\/a><\/p>\n<form><a href=\"https:\/\/skmo.sk\/dokument.php?id=4864\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt51z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=4888\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt51r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<div class=\"at\"><strong>Term\u00edny odovzdania ries\u030cen\u00ed:<\/strong><\/p>\n<ul>\n<li>\u00falohy 1, 2, 3: <strong><em>16. 11. 2022<\/em><\/strong><\/li>\n<li>\u00falohy 4, 5, 6: <strong><em>16. 12. 2022<\/em><\/strong><\/li>\n<\/ul>\n<\/div>\n<p><strong>Z5-I-1<\/strong><br \/>\nZajac sa z\u00fac\u030castnil na pretekoch dlhy\u0301ch 2024 metrov. Zo s\u030ctartovej c\u030ciary sa odrazil \u013eavou nohou a po cely\u0301 c\u030cas pretekov pravidelne striedal \u013eav\u00fa, prav\u00fa a obe nohy. Ke\u010f sa zajac odrazil \u013eavou nohou, skoc\u030cil 35 dm, ke\u010f sa odrazil pravou nohou, skoc\u030cil 15 dm, a ke\u010f sa odrazil oboma nohami, skoc\u030cil 61 dm. Ko\u013eko skokov zajac urobil, ky\u0301m dorazil do cie\u013ea? Ktorou nohou sa odrazil pri cie\u013eovom skoku?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z5-I-2<\/strong><br \/>\nZuzka postavila zo s\u030cestna\u0301stich rovnako ve\u013eky\u0301ch kociek kva\u0301der s p\u00f4dorysom 4 \u00d7 4. S \u010fal\u0161\u00edmi rovnako ve\u013eky\u0301mi kockami pokrac\u030covala v stavan\u00ed. Kocky ukladala tak, z\u030ce kaz\u030cde\u0301 dve susedne\u0301 mali spoloc\u030cn\u00fa cel\u00fa stenu. Vy\u0301sledna\u0301 stavba vyzerala z dvoch r\u00f4znych stra\u0301n ako na nasleduj\u00facom obra\u0301zku. Zistite, ko\u013eko najviac a ko\u013eko najmenej kociek mohla Zuzka na svoju stavbu pouz\u030ci\u0165. <\/p>\n<div class=\"pri\">(Erika Novotna\u0301)<\/div>\n<p><strong>Z5-I-3<\/strong><br \/>\nKatka mala v za\u0301hradke pa\u0308\u0165 za\u0301honov rozmiestneny\u0301ch ako na obra\u0301zku. Do za\u0301honov chcela zasadi\u0165 cesnak, mrkvu a re\u010fkvic\u030cku tak, aby na kaz\u030cdom za\u0301hone bol len jeden druh zeleniny a aby z\u030ciadne dva za\u0301hony s rovnakou zeleninou nesusedili. Ko\u013eky\u0301mi r\u00f4znymi sp\u00f4sobmi mohla Katka do za\u0301honov zasadi\u0165 cesnak, mrkvu a re\u010fkvic\u030cku?<\/p>\n<div class=\"pri\">(Libus\u030ce Hozova\u0301)<\/div>\n<p><strong>Z5-I-4<\/strong><br \/>\nV za\u0301hradka\u0301rskej osade mal pa\u0301n Jahoda vo svojom sude 16 litrov vody. Sused pa\u0301n Malina mal vo svojom sude trikra\u0301t viac vody ako pa\u0301n Jahoda. Zac\u030calo prs\u030ca\u0165 a do oboch sudov naprs\u030calo rovnake\u0301 mnoz\u030cstvo vody. Po daz\u030cdi pa\u0301n Malina zistil, z\u030ce ma\u0301 v sude dvakra\u0301t viac vody ako pa\u0301n Jahoda. Ko\u013eko litrov vody naprs\u030calo do kaz\u030cde\u0301ho suda? <\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z5-I-5<\/strong><br \/>\nZo s\u030ctyroch zhodny\u0301ch s\u030ctvorcov bol vytvoreny\u0301 ornament ako na obra\u0301zku. <\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"73\/73_z5a3.png\" alt=\"\" border=\"0\" \/><\/div>\n<p>Strany s\u030ctvorcov s\u00fa dlhe\u0301 4 cm, s\u00fa navza\u0301jom rovnobez\u030cne\u0301 alebo kolme\u0301 a pret\u00ednaj\u00fa sa bu\u010f vo svojich s\u030ctvrtina\u0301ch, alebo poloviciach. Libor chcel ornament vyfarbi\u0165 a zistil, z\u030ce farba na 1 cm2 kaz\u030cde\u0301ho s\u00favisle\u0301ho po\u013ea ho bude sta\u0301\u0165 to\u013eko eur, ko\u013eky\u0301m s\u030ctvorcom je toto pole spoloc\u030cne\u0301. Ko\u013eko eur bude sta\u0301\u0165 farba na vyfarbenie cele\u0301ho ornamentu? <\/p>\n<div class=\"pri\">(Karel Pazourek)<\/div>\n<p><strong>Z5-I-6<\/strong><br \/>\nLucka nap\u00edsala na l\u00edstok \u010d\u00edslo 12345 a dvakra\u0301t ho medzi ciframi rozstrihla. Z\u00edskala tak tri mens\u030cie kartic\u030cky s tromi c\u030c\u00edslami. Tieto kartic\u030cky zoradila dvoma sp\u00f4sobmi, c\u030c\u00edm dostala dve r\u00f4zne pa\u0308\u0165ciferne\u0301 c\u030c\u00edsla. Rozdiel ty\u0301chto dvoch c\u030c\u00edsel bol 28 926. Medzi ktory\u0301mi ciframi Lucka l\u00edstok rozstrihla? <\/p>\n<div class=\"pri\">(Michaela Petrova\u0301)<\/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\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><a href=\"https:\/\/skmo.sk\/dokument.php?id=4893\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt52z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=4895\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt52r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z5-II-1<\/strong><br \/>\nDeti dostali \u00falohu s piatimi pr\u00e1zdnymi pol\u00ed\u010dkami:<\/p>\n<div class=\"rbig\">\u25a1 \u22c5 \u25a1 \u2212 \u25a1 \u2236 (\u25a1 + \u25a1) = ?<\/div>\n<p>Do ka\u017ed\u00e9ho pol\u00ed\u010dka mali vp\u00edsa\u0165 jedno z \u010d\u00edsel 1, 2, 3, 4, 5 tak, aby ka\u017ed\u00e9 \u010d\u00edslo pou\u017eili iba raz a aby delenie vych\u00e1dzalo bezo zvy\u0161ku.<br \/>\nN\u00e1jdite v\u0161etky v\u00fdsledky, ktor\u00e9 mohli deti dosta\u0165. <\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z5-II-2<\/strong><br \/>\nAlbert, Ben, Cyril, Daniel, Erik, Filip a Gabriel vst\u00fapili v tomto porad\u00ed po jednom za sebou do jaskyne so sto dverami. Svoje poradie menia len pri prechode dverami, a to takto: Prv\u00fd zo z\u00e1stupu otvor\u00ed dvere, podr\u017e\u00ed ich v\u0161etk\u00fdm ostatn\u00fdm a potom sa zarad\u00ed na koniec. Prv\u00e9 dvere teda otv\u00e1ra Albert, druh\u00e9 Ben a tak \u010falej.<br \/>\nKto otvor\u00ed ste\u0301 dvere? <\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z5-II-3<\/strong><br \/>\nNa obraze s\u00fa dva biele \u0161tvorce v sivom poli. Strany obrazu a strany oboch \u0161tvorcov s\u00fa orientovane\u0301 vodorovne alebo zvisle:<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"73\/73_z5b1.png\" alt=\"\" border=\"0\" \/><\/div>\n<p>Bodkovane vyzna\u010den\u00e9 vzdialenosti men\u0161ieho \u0161tvorca od str\u00e1n obrazu s\u00fa rovnak\u00e9 ako ve\u013ekos\u0165 strany tohto men\u0161ieho \u0161tvorca. \u010ciarkovane vyzna\u010den\u00e9 vzdialenosti v\u00e4\u010d\u0161ieho \u0161tvorca od str\u00e1n obrazu s\u00fa rovnak\u00e9 ako ve\u013ekos\u0165 strany tohto v\u00e4\u010d\u0161ieho \u0161tvorca. Obsah sivej \u010dasti obrazu je 62 cm<sup>2<\/sup> .<br \/>\nUr\u010dite obsah bielej \u010dasti obrazu. <\/p>\n<div class=\"pri\">()<\/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\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><a href=\"https:\/\/skmo.sk\/dokument.php?id=4865\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt61z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=4926\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt61r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z6-I-1<\/strong><br \/>\nKamara\u0301ti Jaro, Pavol a Ras\u0165o hrali gu\u013e\u00f4c\u030cky. Jarovi sa ve\u013emi nedarilo, takz\u030ce mal po hre najmenej gu\u013e\u00f4c\u030cok zo vs\u030cetky\u0301ch. Chlapcom to bolo \u013e\u00fato, preto dal Ras\u0165o Jarovi polovicu vs\u030cetky\u0301ch svojich gu\u013e\u00f4c\u030cok a Pavol tretinu ty\u0301ch svojich. Teraz mal najviac gu\u013e\u00f4c\u030cok Jaro, a tak svojim dvom kamara\u0301tom vra\u0301til kaz\u030cde\u0301mu sedem gu\u013e\u00f4c\u030cok. Po ty\u0301chto vy\u0301mena\u0301ch mali vs\u030cetci rovnako, a to po 25 gu\u013e\u00f4c\u030cok. Ko\u013eko gu\u013e\u00f4c\u030cok mal po hre (pred vy\u0301menami) Jaro?<\/p>\n<div class=\"pri\">(Michaela Petrova\u0301)<\/div>\n<p><strong>Z6-I-2<\/strong><br \/>\nKarol\u00edna narysovala s\u030ctvorec so stranou 6 cm. Na kaz\u030cdej strane s\u030ctvorca vyznac\u030cila modrou farbou dva body, ktory\u0301  mi rozdelila pr\u00edslus\u030cn\u00fa stranu na tri zhodne\u0301 c\u030casti. Potom zostrojila s\u030ctvoruholn\u00edk, ktory\u0301 mal vs\u030cetky vrcholy modrej farby a ktore\u0301ho z\u030ciadne dva vrcholy nelez\u030cali na rovnakej strane s\u030ctvorca. Ake\u0301 obsahy s\u030ctvoruholn\u00edkov mohla Karol\u00edna dosta\u0165? Na\u0301jdite vs\u030cetky moz\u030cnosti. <\/p>\n<div class=\"pri\">(Libus\u030ce Hozova\u0301)<\/div>\n<p><strong>Z6-I-3<\/strong><br \/>\nV osemcifernom c\u030c\u00edsle je kaz\u030cda\u0301 jeho cifra (okrem poslednej) va\u0308c\u030cs\u030cia ako nasleduj\u00faca cifra. Ko\u013eko je vs\u030cetky\u0301ch taky\u0301chto c\u030c\u00edsel? <\/p>\n<div class=\"pri\">(Iveta Janc\u030cigova\u0301)<\/div>\n<p><strong>Z6-I-4<\/strong><br \/>\nV nasleduj\u00facom p\u00edsomnom na\u0301soben\u00ed dvoch trojciferny\u0301ch c\u030c\u00edsel s\u00fa cifry zast\u00fapene\u0301 hviezdic\u030ckami. Na miesta hviezdic\u030ciek dopl\u0148te cifry tak, aby bol vy\u0301poc\u030cet spra\u0301vny:<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"73\/73_z6a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Libus\u030ce Hozova\u0301)<\/div>\n<p><strong>Z6-I-5<\/strong><br \/>\nZ navza\u0301jom zhodny\u0301ch trojuholn\u00edkov je zloz\u030ceny\u0301ch nieko\u013eko rovinny\u0301ch \u00fatvarov. Obvody prvy\u0301ch troch s\u00fa postupne 8 cm, 11,4 cm a 14,7 cm. Vypoc\u030c\u00edtajte obvod s\u030ctvrte\u0301ho \u00fatvaru. <\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"73\/73_z6a2.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Eva Semera\u0301dova\u0301)<\/div>\n<p><strong>Z6-I-6<\/strong><br \/>\nAlex, Barbora, Cyril, Dana, Eva, Frantis\u030cek a Gabika sa stali na svojich s\u030ckola\u0301ch v\u00ed\u0165azmi v stolnom tenise a zis\u030cli sa na dvojd\u0148ovom turnaji o celkove\u0301ho v\u00ed\u0165aza. Kaz\u030cde\u0301 z ty\u0301chto siedmich det\u00ed malo poc\u030cas turnaja zohra\u0165 jeden za\u0301pas s kaz\u030cdy\u0301m iny\u0301m. Prvy\u0301 de\u0148 turnaja hral Alex jeden za\u0301pas, Barbora hrala dva za\u0301pasy, Cyril tri, Dana s\u030ctyri, Eva pa\u0308\u0165 za\u0301pasov a Frantis\u030cek s\u030ces\u0165. Ko\u013eko za\u0301pasov hrala prvy\u0301 de\u0148 Gabika? <\/p>\n<div class=\"pri\">(Libus\u030ce Hozova\u0301)<\/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\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><a href=\"https:\/\/skmo.sk\/dokument.php?id=4932\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt62z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=4934\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt62r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z6-II-1<\/strong><br \/>\n\u017diaci dostali prirodzen\u00e9 \u010d\u00edslo men\u0161ie ako 100. Alex dan\u00e9 \u010d\u00edslo zaokr\u00fahlil na desiatky. Barbora dan\u00e9 \u010d\u00edslo zaokr\u00fahlila na stovky. Cyril dan\u00e9 \u010d\u00edslo vyn\u00e1sobil dvoma. Dana zaokr\u00fahlen\u00e9 Alexovo \u010d\u00edslo a Barborino \u010d\u00edslo s\u010d\u00edtala. Eva od Daninho \u010d\u00edsla odpo\u010d\u00edtala Cyrilovo \u010d\u00edslo. Franti\u0161ek ozn\u00e1mil Evin v\u00fdsledok a ten bol 30.<br \/>\nKtor\u00e9 \u010d\u00edslo mohli \u017eiaci na za\u010diatku dosta\u0165? Ur\u010dte v\u0161etky mo\u017enosti. <\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z6-II-2<\/strong><br \/>\nZo zhodn\u00fdch \u0161tvorcov so stranou d\u013a\u017eky 1 cm je zlo\u017een\u00fd \u00fatvar s nasleduj\u00facimi vlastnos\u0165ami: <\/p>\n<ul>\n<li>\u00datvar je tvoren\u00fd radmi susediacich \u0161tvorcov.<\/li>\n<li>Spodn\u00fd rad pozost\u00e1va z 12 \u0161tvorcov.<\/li>\n<li>Ka\u017ed\u00fd vy\u0161\u0161\u00ed rad za\u010d\u00edna z\u013eava rovnako ako rad pod n\u00edm, m\u00e1 v\u0161ak o jeden \u0161tvorec menej.<\/li>\n<li>Horn\u00fd rad tvoria 4 \u0161tvorce.<\/li>\n<\/ul>\n<p>Vrcholy spodnej strany \u00fatvaru s\u00fa ozna\u010den\u00e9 \ud835\udc34 a \ud835\udc35, prav\u00fd vrchol hornej strany je ozna\u010den\u00fd \ud835\udc36. Ur\u010dte obsah trojuholn\u00edka \ud835\udc34\ud835\udc35\ud835\udc36.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"73\/73_z6b1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">()<\/div>\n<p><strong>Z6-II-3<\/strong><br \/>\nV krajine Bin\u00e1ria \u017eij\u00fa matematici nieko\u013ek\u00fdch \u00farovn\u00ed. V\u0161etci matematici rovnakej \u00farovne vyrie\u0161ia za de\u0148 rovnak\u00fd po\u010det \u00faloh. Matematik vy\u0161\u0161ej \u00farovne vyrie\u0161i za de\u0148 dvojn\u00e1sobn\u00e9 mno\u017estvo \u00faloh ako matematik predch\u00e1dzaj\u00facej \u00farovne (teda napr\u00edklad matematik \u0161tvrtej \u00farovne vyrie\u0161i za de\u0148 dvakr\u00e1t viac \u00faloh ako matematik tretej \u00farovne).<br \/>\nTraja matematici piatej \u00farovne vyrie\u0161ia za de\u0148 o tis\u00edc \u00faloh viac ako \u0161tyria matematici druhej \u00farovne.<br \/>\nKo\u013eko \u00faloh vyrie\u0161i za de\u0148 jeden matematik druhej \u00farovne?<\/p>\n<div class=\"pri\">()<\/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\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n<form><a href=\"https:\/\/skmo.sk\/dokument.php?id=4866\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt71z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=4927\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt71r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z7-I-1<\/strong><br \/>\nAjka, Barborka, Cilka a Daniel sa dohadovali o poc\u030cte zrniek piesku na ich pieskovisku. Daniel povedal kamara\u0301tkam svoj odhad a tie sa ho rozhodli overi\u0165. Ajka nara\u0301tala 873 451 230, Barborka 873 451 227 a Cilka 873 451 213 zrniek piesku. S\u00fac\u030cet (kladny\u0301ch) rozdielov ty\u0301chto troch vy\u0301sledkov od Danielovho odhadu bol 29. Ko\u013eko zrniek piesku mohol odhadn\u00fa\u0165 Daniel? Uve\u010fte vs\u030cetky moz\u030cnosti. <\/p>\n<div class=\"pri\">(Veronika Bachrata\u0301)<\/div>\n<p><strong>Z7-I-2<\/strong><br \/>\nPa\u0301n Delf\u00edn a pa\u0301n \u017dralok boli zdatn\u00ed ryba\u0301ri. Raz dokopy ulovili 70 ry\u0301b. Medzi rybami, ktore\u0301 ulovil pa\u0301n Delf\u00edn, bolo 5\/9 pstruhov. Medzi rybami, ktore\u0301 ulovil pa\u0301n \u017dralok, boli 2\/17 kaprov. Ko\u013eko ry\u0301b ulovil pa\u0301n Delf\u00edn?<\/p>\n<div class=\"pri\">(Libus\u030ce Hozova\u0301 )<\/div>\n<p><strong>Z7-I-3<\/strong><br \/>\nMysl\u00edm si tri c\u030c\u00edsla. Ke\u010f ich sc\u030c\u00edtam, dostanem 15. Ke\u010f od s\u00fac\u030ctu prvy\u0301ch dvoch c\u030c\u00edsel odc\u030c\u00edtam tretie, dostanem 10. Ke\u010f od s\u00fac\u030ctu prve\u0301ho a tretieho c\u030c\u00edsla odc\u030c\u00edtam druh\u00e9, dostanem 8. Ktore\u0301 c\u030c\u00edsla si mysl\u00edm? <\/p>\n<div class=\"pri\">(Eva Semera\u0301dova\u0301)<\/div>\n<p><strong>Z7-I-4<\/strong><br \/>\nAnetkin stry\u0301ko ma\u0301 narodeniny v ten isty\u0301 de\u0148 v roku ako Anetkina teta. Stry\u0301ko je stars\u030c\u00ed ako teta, nie vs\u030cak o viac ako desa\u0165 rokov, a obaja s\u00fa plnolet\u00ed. Na poslednej oslave si Anetka uvedomila, z\u030ce ke\u010f vyna\u0301sob\u00ed ich oslavovane\u0301 veky a vy\u0301sledny\u0301 s\u00fac\u030cin es\u030cte vyna\u0301sob\u00ed poc\u030ctom psov, ktor\u00ed sa na oslave zis\u030cli, dostane c\u030c\u00edslo 2024. Ko\u013eko psov mohlo by\u0165 na tejto oslave? Urc\u030cte vs\u030cetky moz\u030cnosti.<\/p>\n<div class=\"pri\">(Michaela Petrova\u0301)<\/div>\n<p><strong>Z7-I-5<\/strong><br \/>\nPravouhly\u0301 trojuholn\u00edk ma\u0301 obsah 36 m<sup>2<\/sup>. V \u0148om je umiestneny\u0301 s\u030ctvorec tak, z\u030ce dve strany s\u030ctvorca s\u00fa c\u030cas\u0165ami dvoch stra\u0301n trojuholn\u00edka a jeden vrchol s\u030ctvorca je v tretine najdlhs\u030cej strany. Urc\u030cte obsah tohto s\u030ctvorca.<\/p>\n<div class=\"pri\">(Erika Novotna\u0301)<\/div>\n<p><strong>Z7-I-6<\/strong><br \/>\nTrpasl\u00edci poc\u030c\u00edtaj\u00fa svoje veky v d\u0148och, takz\u030ce kaz\u030cdy\u0301 de\u0148 oslavuj\u00fa narodeniny. U trpasl\u00edka Nos\u030cteka sa zis\u030clo sedem trpasl\u00edkov s vekmi 105, 120, 140, 168, 210, 280 a 420 dn\u00ed. Poc\u030cas oslavy sa vs\u030cetky\u0301m \u00f4smim trpasl\u00edkom podarilo rozdeli\u0165 do dvoch skup\u00edn s rovnaky\u0301mi s\u00fac\u030ctami vekov. Ko\u013eko najmenej a ko\u013eko najviac dn\u00ed mohol ma\u0165 trpasl\u00edk Nos\u030ctek? <\/p>\n<div class=\"pri\">(Erika 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=\"z72\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><a href=\"https:\/\/skmo.sk\/dokument.php?id=4935\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt72z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=4937\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt72r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z7-II-1<\/strong><br \/>\nDeti h\u00e1dzali klasickou hracou kockou s \u010d\u00edslami od 1 do 6. Ke\u010f padlo p\u00e1rne \u010d\u00edslo, tak zatlieskali. Ke\u010f padlo \u010d\u00edslo delite\u013en\u00e9 tromi, tak zadupali. Ke\u010f nastali oba pr\u00edpady, urobili oboje, v ostatn\u00fdch pr\u00edpadoch nerobili ni\u010d. Po\u010das piatich hodov deti celkom trikr\u00e1t zatlieskali a trikr\u00e1t zadupali. S\u00fa\u010det \u010d\u00edsel, ktor\u00e9 postupne padli, bol 20.<br \/>\nN\u00e1jdite v\u0161etky mo\u017en\u00e9 p\u00e4tice \u010d\u00edsel, ktor\u00e9 mohli padn\u00fa\u0165, bez oh\u013eadu na ich poradie. <\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z7-II-2<\/strong><br \/>\nZo zhodn\u00fdch \u0161tvorcov so stranou d\u013a\u017eky 1 cm je zlo\u017een\u00fd \u00fatvar s nasleduj\u00facimi vlastnos\u0165ami: <\/p>\n<ul>\n<li>Spodn\u00fd a horn\u00fd rad pozost\u00e1va z jedn\u00e9ho \u0161tvorca.<\/li>\n<li>Ostatn\u00e9 rady maj\u00fa po dvoch susediacich \u0161tvorcoch.<\/li>\n<li>V\u0161etky st\u013apce s\u00fa tvoren\u00e9 dvoma susediacimi \u0161tvorcami.<\/li>\n<li>Obvod \u00fatvaru je 178 cm.<\/li>\n<\/ul>\n<p>Vrcholy spodnej strany \u00fatvaru s\u00fa ozna\u010den\u00e9 \ud835\udc34 a \ud835\udc35, prav\u00fd vrchol hornej strany je ozna\u010den\u00fd \ud835\udc36. Ur\u010dte obsah trojuholn\u00edka \ud835\udc34\ud835\udc35\ud835\udc36.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"73\/73_z7b1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">()<\/div>\n<p><strong>Z7-II-3<\/strong><br \/>\nNa pretekoch sa spolu stretlo p\u00e4\u0165 det\u00ed, diev\u010dat\u00e1 Anna a Fiona a chlapci Jurko, Luk\u00e1\u0161 a Tom\u00e1\u0161. O cie\u013eovom porad\u00ed m\u00e1me tieto inform\u00e1cie: <\/p>\n<ul>\n<li>\u017diadne dve deti sa neumiestnili na rovnakom mieste.<\/li>\n<li>Priemern\u00e9 umiestnenie diev\u010dat bolo rovnak\u00e9 ako priemern\u00e9 umiestnenie chlapcov.<\/li>\n<li>Obe diev\u010dat\u00e1 sa umiestnili pred Tom\u00e1\u0161om.<\/li>\n<li>Jurko sa umiestnil medzi Annou a Luk\u00e1\u0161om.<\/li>\n<\/ul>\n<p>Ur\u010dite cie\u013eov\u00e9 poradie det\u00ed.<\/p>\n<div class=\"pri\">()<\/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\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><a href=\"https:\/\/skmo.sk\/dokument.php?id=4867\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt81z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=4928\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt81r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<div class=\"at\"><strong>Term\u00edny odovzdania ries\u030cen\u00ed:<\/strong><\/p>\n<ul>\n<li>\u00falohy 1, 2, 3: <strong><em>19. 1. 2024<\/em><\/strong><\/li>\n<li>\u00falohy 4, 5, 6: <strong><em>1. 3. 2024<\/em><\/strong><\/li>\n<\/ul>\n<\/div>\n<p><strong>Z8-I-1<\/strong><br \/>\nV minulom roku bolo v nas\u030com skautskom oddiele o 30 chlapcov viac ako dievc\u030cat. Tento rok sa poc\u030cet det\u00ed v oddiele zva\u0308c\u030cs\u030cil o 10 %, pric\u030com poc\u030cet chlapcov sa zva\u0308c\u030cs\u030cil o 5 % a poc\u030cet dievc\u030cat sa zva\u0308c\u030cs\u030cil o 20 %.<br \/>Ko\u013eko det\u00ed ma\u0301me tento rok v oddiele? <\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z8-I-2<\/strong><br \/>\nAdam mal papier, ktory\u0301 bol nato\u013eko ve\u013ek\u00fd, z\u030ce by sa z neho dalo natrha\u0165 nieko\u013eko desiatok tis\u00edc k\u00faskov. Najprv papier roztrhal na s\u030ctyri k\u00fasky. Kaz\u030cdy\u0301 z ty\u0301chto k\u00faskov vzal a roztrhal bu\u010f na s\u030ctyri, alebo na desa\u0165 k\u00faskov. Rovnaky\u0301m sp\u00f4sobom pokrac\u030coval \u010falej: kaz\u030cdy\u0301 novovzniknuty\u0301 k\u00fasok roztrhal bu\u010f na s\u030ctyri, alebo na desa\u0165 mens\u030c\u00edch k\u00faskov. Rozhodnite a vysvetlite, c\u030ci m\u00f4z\u030ce Adam ty\u0301mto sp\u00f4sobom natrha\u0165 presne 20 000 k\u00faskov. <\/p>\n<div class=\"pri\">(Iveta Janc\u030cigova\u0301)<\/div>\n<p><strong>Z8-I-3<\/strong><br \/>\nV s\u030cportovom area\u0301li tvorili stanovis\u030ctia vrcholy pravidelne\u0301ho pa\u0308\u0165uholn\u00edka \ud835\udc34\ud835\udc35\ud835\udc36\ud835\udc37\ud835\udc38. Tieto stanovis\u030ctia boli pospa\u0301jane\u0301 priamymi cestami. Na ceste z \ud835\udc34 do \ud835\udc35 bola fonta\u0301na \ud835\udc39, ktor\u00fa so stanovis\u030c\u0165om \ud835\udc36 spa\u0301jala cesta kolma\u0301 na cestu z \ud835\udc35 do \ud835\udc38. Pat a Mat sa zis\u030cli na stanovis\u030cti \ud835\udc38 a rozhodli sa zamies\u0165 niektore\u0301 cesty. Pat pozametal cestu z \ud835\udc38 do \ud835\udc35. Mat pozametal cestu z \ud835\udc38 do \ud835\udc34 a es\u030cte z \ud835\udc34 do \ud835\udc39.<br \/> Urc\u030cte, ktory\u0301 z nich zametal dlhs\u030c\u00ed \u00fasek. <\/p>\n<div class=\"pri\">(Libus\u030ce Hozova\u0301)<\/div>\n<p><strong>Z8-I-4<\/strong><br \/>\nHynek nap\u00edsal nasleduj\u00faci s\u00fac\u030cet s piatimi za\u0301hadny\u0301mi sc\u030c\u00edtancami: <\/p>\n<div class=\"r\">@ + ## + \u2217 \u2217 \u2217 + &#038;&#038;&#038;&#038; + $$$$$.<\/div>\n<p> Prezradil, z\u030ce znaky @, #, \u2217, &#038;, $ predstavuj\u00fa navza\u0301jom r\u00f4zne cifry 1, 2, 3, 4, 5 a z\u030ce vy\u0301sledny\u0301 s\u00fac\u030cet je delite\u013eny\u0301 jedena\u0301stimi.<br \/>Ktor\u00e9 najmen\u0161ie a ktor\u00e9 najva\u0308c\u030cs\u030cie c\u030c\u00edslo m\u00f4z\u030ce by\u0165 vy\u0301sledkom Hynkovho s\u00fac\u030ctu? <\/p>\n<div class=\"pri\">(Erika Novotna\u0301)<\/div>\n<p><strong>Z8-I-5<\/strong><br \/>\nTrojuholn\u00edk \ud835\udc34\ud835\udc35\ud835\udc36 je rozdeleny\u0301 \u00fasec\u030ckami ako na obra\u0301zku. \u00dasec\u030cky <em>DE<\/em> a <em>AB<\/em> s\u00fa rovnobez\u030cne\u0301. Trojuholn\u00edky \ud835\udc36\ud835\udc37\ud835\udc3b, \ud835\udc36\ud835\udc3b\ud835\udc3c, \ud835\udc36\ud835\udc3c\ud835\udc38, \ud835\udc39\ud835\udc3c\ud835\udc3b maj\u00fa rovnaky\u0301 obsah, a to 8 dm<sup>2<\/sup>. Zistite obsah s\u030ctvoruholn\u00edka <em>AFHD<\/em>.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"73\/73_z8a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Eva Semera\u0301dova\u0301)<\/div>\n<p><strong>Z8-I-6<\/strong><br \/>\nAdam vp\u0131\u0301sal do tabu\u013eky 3 \u00d7 3 c\u030c\u00edsla od 1 po 9 ako na obra\u0301zku:<\/p>\n<table border=\"1\" bordercolor=\"#000000\">\n<tr>\n<td width=\"30\">7<\/td>\n<td width=\"30\">6<\/td>\n<td width=\"30\">4<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>2<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>3<\/td>\n<td>5<\/td>\n<\/tr>\n<\/table>\n<p>Pre toto vyplnenie plat\u00ed, z\u030ce s\u00fac\u030cet c\u030c\u00edsel troch pol\u00edc\u030cok pozd\u013az\u030c kaz\u030cdej strany je sta\u0301le rovnak\u00fd. Adam zistil, z\u030ce c\u030c\u00edsla do tabu\u013eky je moz\u030cne\u0301 vyplni\u0165 aj inak, bez toho, aby pokazil vlastnos\u0165 s rovnaky\u0301mi s\u00fac\u030ctami pozd\u013az\u030c stra\u0301n. Ak\u00fa najmens\u030ciu hodnotu m\u00f4z\u030ce ma\u0165 tento s\u00fac\u030cet? Uve\u010fte pr\u00edklad tabu\u013eky s najmens\u030c\u00edm s\u00fac\u030ctom pozd\u013az\u030c stra\u0301n a vysvetlite, prec\u030co mens\u030c\u0131\u0301 s\u00fac\u030cet by\u0165 nem\u00f4z\u030ce. <\/p>\n<div class=\"pri\">(Josef Tkadlec)<\/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\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><a href=\"https:\/\/skmo.sk\/dokument.php?id=4938\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt82z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=4940\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt82r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z8-II-1<\/strong><br \/>\nPo\u010d\u00edta\u010dov\u00fd program vypisoval po riadkoch \u010d\u00edsla tvoren\u00e9 ciframi od 1 do 9. Cifry boli pou\u017eit\u00e9 opakovane v prirodzenom porad\u00ed. Na ka\u017edom riadku bolo o jednu cifru viac ako na riadku predch\u00e1dzaj\u00facom, na prvom riadku bola 1:<\/p>\n<table width=\"200\" style=\"border-style: none;\">\n<tr>\n<td class=\"pri\" style=\"border-style: none;\"><b>1<br \/>12<br \/>123<br \/>\u22ee<br \/>123456789<br \/>1234567891<br \/>12345678912<br \/>\u22ee<\/td>\n<\/tr>\n<\/table>\n<p>V\u00fdpis bol ukon\u010den\u00fd \u010d\u00edslom na 2024. riadku.<br \/>\nZistite, na ko\u013ek\u00fdch riadkoch boli \u010d\u00edsla delite\u013en\u00e9<br \/>\na) tromi,<br \/>b) \u0161tyrmi.<\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z8-II-2<\/strong><br \/>\nMedzi hra\u010dkami v obchode s\u00fa iba lode a aut\u00e1. Lode tvoria \u0161tvrtinu hra\u010diek. 75 % lod\u00ed a 40 % \u00e1ut je \u010derven\u00fdch. \u010cerven\u00fdch hra\u010diek je o 10 menej ako t\u00fdch s inou farbou.<br \/>\nKo\u013eko hra\u010diek je v obchode? <\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z8-II-3<\/strong><br \/>\nZostrojte deltoid <em>ABCD<\/em> so stranami <em>AB<\/em> a <em>AD<\/em> d\u013a\u017eky 11 cm, stranami <em>CB<\/em> a <em>CD<\/em> d\u013a\u017eky 6 cm a uhloprie\u010dkou <em>AC<\/em> d\u013a\u017eky 15 cm. N\u00e1jdite aspo\u0148 jedno jeho rozdelenie na \u0161tyri \u0161tvoruholn\u00edky tak, aby dva z nich boli deltoidy a druh\u00e9 dva zhodn\u00e9 koso\u0161tvorce. Kon\u0161trukciu op\u00ed\u0161te a zd\u00f4vodnite.<br \/>\n<i>(Deltoidom naz\u00fdvame tak\u00fd konvexn\u00fd \u0161tvoruholn\u00edk, ktor\u00fd m\u00e1 dve dvojice zhodn\u00fdch susedn\u00fdch str\u00e1n.)<\/i><\/p>\n<div class=\"pri\">()<\/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\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><a href=\"https:\/\/skmo.sk\/dokument.php?id=4868\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt91z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=4889\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt91r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<div class=\"at\"><strong>Term\u00edny odovzdania ries\u030cen\u00ed:<\/strong><\/p>\n<ul>\n<li>\u00falohy 1, 2, 3: <strong><em>15. 11. 2023<\/em><\/strong><\/li>\n<li>\u00falohy 4, 5, 6: <strong><em>15. 12. 2023<\/em><\/strong><\/li>\n<\/ul>\n<\/div>\n<p><strong>Z9-I-1<\/strong><br \/>\nPat a Mat si dali preteky v behu do svojej chal\u00fapky. V istom okamihu platilo, z\u030ce keby Mat mal zdolan\u00fa polovicu vzdialenosti, ktor\u00fa doteraz ubehol, chy\u0301bal by mu do chal\u00fapky trojna\u0301sobok tejto polovic\u030cnej vzdialenosti. V tom istom okamihu platilo, z\u030ce keby Pat mal zdolany\u0301 dvojna\u0301sobok vzdialenosti, ktor\u00fa doteraz ubehol, chy\u0301bala by mu do chal\u00fapky tretina tejto dvojna\u0301sobnej vzdialenosti.<br \/>Kto bol v danom okamihu bliz\u030cs\u030cie k chal\u00fapke? <\/p>\n<div class=\"pri\">(Libus\u030ce Hozova\u0301)<\/div>\n<p><strong>Z9-I-2<\/strong><br \/>\nZostrojte kosos\u030ctvorec \ud835\udc34\ud835\udc35\ud835\udc36\ud835\udc37 tak\u00fd, z\u030ce plat\u00ed |\ud835\udc34\ud835\udc36| = 8 cm a |\ud835\udc34\ud835\udc46| = 7 cm, kde \ud835\udc46 je stredom strany \ud835\udc36\ud835\udc37. <\/p>\n<div class=\"pri\">(Karel Pazourek)<\/div>\n<p><strong>Z9-I-3<\/strong><br \/>\nV za\u0301kladnej s\u030ckole, kam chod\u00ed aj \u017digmund, kaz\u030cdoroc\u030cne organizuj\u00fa vedomostn\u00fa s\u00fa\u0165a\u017e, v ktorej kaz\u030cdy\u0301 s\u00fa\u0165az\u030ciaci m\u00f4z\u030ce z\u00edska\u0165 najviac 15 bodov. Tento rok bol priemerny\u0301 bodovy\u0301 zisk s\u00fa\u0165az\u030ciacich zaokr\u00fahleny\u0301 na desatiny rovny\u0301 10,4. \u017digmund si po s\u00fa\u0165az\u030ci uvedomil, z\u030ce niektore\u0301 ota\u0301zky si zle prec\u030c\u00edtal a odpovedal na niec\u030co in\u00e9. Mohol tak ma\u0165 o 4 body viac a priemerny\u0301 bodovy\u0301 zisk zaokr\u00fahleny\u0301 na desatiny by sa ty\u0301m zvy\u0301s\u030cil na 10,6.<br \/>Urc\u030cte, ko\u013eko najmenej a ko\u013eko najviac det\u0131\u0301 mohlo tento rok s\u00fa\u0165az\u030ci\u0165. <\/p>\n<div class=\"pri\">(Michaela Petrova\u0301 )<\/div>\n<p><strong>Z9-I-4<\/strong><br \/>\nKarol mal vyna\u0301sobi\u0165 dve dvojciferne\u0301 c\u030c\u00edsla. Z nepozornosti vymenil poradie cifier v jednom z c\u030cinite\u013eov a dostal s\u00fac\u030cin, ktory\u0301 bol o 4 248 mens\u030c\u0131\u0301 ako spra\u0301vny vy\u0301sledok.<br \/>\nKo\u013eko malo Karolovi spra\u0301vne vyjs\u0165? <\/p>\n<div class=\"pri\">(Libus\u030ce Hozova\u0301)<\/div>\n<p><strong>Z9-I-5<\/strong><br \/>\nTrojuholn\u0131\u0301k \ud835\udc34\ud835\udc35\ud835\udc36 je pravouhly\u0301 s pravy\u0301m uhlom pri vrchole \ud835\udc36. Body \ud835\udc34\u2032 , \ud835\udc35\u2032 , \ud835\udc36\u2032 s\u00fa obrazy bodov \ud835\udc34, \ud835\udc35, \ud835\udc36 postupne v stredovy\u0301ch su\u0301mernostiach so stredmi \ud835\udc36, \ud835\udc34, \ud835\udc35. Doka\u0301z\u030cte, z\u030ce plat\u00ed<\/p>\n<div class=\"r\"> |\ud835\udc34\u2032\ud835\udc35\u2032|<sup>2<\/sup> + |\ud835\udc35\u2032\ud835\udc36\u2032|<sup>2<\/sup> + |\ud835\udc36\u2032\ud835\udc34\u2032|<sup>2<\/sup> = 14.|\ud835\udc34\ud835\udc35|<sup>2<\/sup>. <\/div>\n<div class=\"pri\">(Jaroslav Zhouf)<\/div>\n<p><strong>Z9-I-6<\/strong><br \/>\nUrc\u030cte poc\u030cet s\u030ctvorcov, ktory\u0301ch vs\u030cetky vrcholy s\u00fa mrez\u030covy\u0301mi bodmi s\u030ctvorcovej siete pozosta\u0301vaj\u00facej zo 4 riadkov a 2023 st\u013apcov. <\/p>\n<div class=\"pri\">(Karel Pazourek)<\/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\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><a href=\"https:\/\/skmo.sk\/dokument.php?id=4896\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt92z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=4898\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt92r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-II-1<\/strong><br \/>\nNa prehliadke v miestnosti m\u00fazea bola skupina chlapcov a diev\u010dat. Po prehliadke odi\u0161lo 15 diev\u010dat, a v miestnosti tak zostalo dvakr\u00e1t viac chlapcov ako diev\u010dat. N\u00e1sledne odi\u0161lo 45 chlapcov, a v miestnosti tak zostalo p\u00e4\u0165kr\u00e1t viac diev\u010dat ako chlapcov.<br \/>\nKo\u013eko diev\u010dat bolo v miestnosti m\u00fazea po\u010das prehliadky? <\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z9-II-2<\/strong><br \/>\nObd\u013a\u017enik na obr\u00e1zku je rozdelen\u00fd na dva zhodn\u00e9 \u0161tvorce, \u0161tyri zhodn\u00e9 men\u0161ie pravouhl\u00e9 trojuholn\u00edky a \u0161tyri zhodn\u00e9 v\u00e4\u010d\u0161ie pravouhl\u00e9 trojuholn\u00edky. Ve\u013ekosti niektor\u00fdch str\u00e1n s\u00fa vyzna\u010den\u00e9 na obr\u00e1zku.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"73\/73_z9b1.png\" alt=\"\" border=\"0\" \/><\/div>\n<p>Vypo\u010d\u00edtajte rozmery obd\u013a\u017enika. <\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z9-II-3<\/strong><br \/>\nIveta postupne vypisovala prirodzen\u00e9 \u010d\u00edsla tvoren\u00e9 \u010d\u00edslicami 1, 3, 5, 7. \u017diadne in\u00e9 \u010d\u00edslice nepou\u017eila, postupovala vzostupne od najmen\u0161ieho \u010d\u00edsla a \u017eiadne \u010d\u00edslo nezabudla. \u010c\u00edsla p\u00edsala bezprostredne za sebou, \u010d\u00edm zostavila jedno mimoriadne dlh\u00e9 \u010d\u00edslo:<\/p>\n<div class=\"r\">1357111315173133&#8230;<\/div>\n<p>Ktor\u00e1 \u010d\u00edslica je v tomto \u010d\u00edsle na 1286. mieste? <\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z9-II-4<\/strong><br \/>\nV piatich vrec\u00fa\u0161kach je spolu 52 gu\u013e\u00f4\u010dok. V \u017eiadnych dvoch vrec\u00fa\u0161kach nie je rovnak\u00fd po\u010det gu\u013e\u00f4\u010dok, niektor\u00e9 vrec\u00fa\u0161ko m\u00f4\u017ee by\u0165 i pr\u00e1zdne. V\u0161etky gu\u013e\u00f4\u010dky z ak\u00e9hoko\u013evek (nepr\u00e1zdneho) vrec\u00fa\u0161ka je mo\u017en\u00e9 premiestni\u0165 do ostatn\u00fdch \u0161tyroch vrec\u00fa\u0161ok tak, \u017ee v nich bud\u00fa rovnak\u00e9 po\u010dty gu\u013e\u00f4\u010dok.<br \/>\na) N\u00e1jdite nejak\u00e9 rozdelenie gu\u013e\u00f4\u010dok do vrec\u00fa\u0161ok, ktor\u00e9 m\u00e1 v\u0161etky uveden\u00e9 vlastnosti.<br \/>b) Uk\u00e1\u017ete, \u017ee pri akomko\u013evek rozdelen\u00ed s uveden\u00fdmi vlastnos\u0165ami je v niektorom vrec\u00fa\u0161ku pr\u00e1ve 12 gu\u013e\u00f4\u010dok.<\/p>\n<div class=\"pri\">()<\/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\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><a href=\"https:\/\/skmo.sk\/dokument.php?id=4929\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt93z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"https:\/\/skmo.sk\/dokument.php?id=4931\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt93r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-III-1<\/strong><br \/>\nV z\u00e1kladnej \u0161kole, kam chod\u00ed aj Luk\u00e1\u0161, organizuj\u00fa vedomostn\u00fa s\u00fa\u0165a\u017e s vopred zoraden\u00fdmi \u00falohami. Ka\u017ed\u00e1 spr\u00e1vne vyrie\u0161en\u00e1 \u00faloha je hodnoten\u00e1 to\u013ek\u00fdmi bodmi, ak\u00e9 je jej poradie. Ka\u017ed\u00e1 nerie\u0161en\u00e1 \u010di nie celkom spr\u00e1vne vyrie\u0161en\u00e1 \u00faloha je hodnoten\u00e1 0 bodmi. Luk\u00e1\u0161 spr\u00e1vne vyrie\u0161il len prv\u00fdch 12 \u00faloh. Ak by spr\u00e1vne vyrie\u0161il len posledn\u00fdch 12 \u00faloh, z\u00edskal by o 708 bodov viac.<br \/>Ko\u013eko bolo s\u00fa\u0165a\u017en\u00fdch \u00faloh? <\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z9-III-2<\/strong><br \/>\nNech <em>ABC<\/em> je trojuholn\u00edk a <em>D<\/em> priese\u010dn\u00edk os\u00ed jeho vn\u00fatorn\u00fdch uhlov. Nech <em>DEF<\/em>, <em>DGH<\/em>, <em>DIJ<\/em> s\u00fa rovnostrann\u00e9 trojuholn\u00edky tak\u00e9, \u017ee vrcholy <em>A, B, C<\/em> s\u00fa postupne stredy str\u00e1n <em>EF<\/em>, <em>GH<\/em>, <em>IJ<\/em> a body <em>E, G, I<\/em> le\u017eia postupne v uhloch <em>CDA<\/em>, <em>ADB<\/em>, <em>BDC<\/em>. Ve\u013ekos\u0165 uhla <em>EDJ<\/em> je 51\u00b0, ve\u013ekos\u0165 uhla <em>HDI<\/em> je 66\u00b0. <\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"73\/73_z9c1.png\" alt=\"\" border=\"0\" \/><\/div>\n<p>Ur\u010dte ve\u013ekosti vn\u00fatorn\u00fdch uhlov trojuholn\u00edka <em>ABC<\/em>.<\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z9-III-3<\/strong><br \/>\nNa stran\u00e1ch trojuholn\u00edka <em>ABC<\/em> s\u00fa dan\u00e9 body <em>D, E, F, G, H<\/em> tak\u00e9, \u017ee plat\u00ed: <\/p>\n<ul>\n<li>bod <em>D<\/em> je stredom strany <em>AC<\/em>,<\/li>\n<li>bod <em>E<\/em> je stredom strany <em>BC<\/em>,<\/li>\n<li>body <em>F<\/em> a <em>G<\/em> le\u017eia postupne na \u00fase\u010dk\u00e1ch <em>CD<\/em> a <em>CE<\/em> tak, \u017ee <em>FG<\/em> \u2225 <em>DE<\/em>,<\/li>\n<li>bod <em>H<\/em> le\u017e\u00ed na \u00fase\u010dke <em>AB<\/em> tak, \u017ee |<em>AH<\/em>| \u2236 |<em>AB<\/em>| = 3 \u2236 7,<\/li>\n<li>pomer obsahov mnohouholn\u00edkov <em>DEGF<\/em> a <em>DHEGF<\/em> je 3 \u2236 7.<\/li>\n<\/ul>\n<p>Ur\u010dte pomer ve\u013ekost\u00ed \u00fase\u010diek <em>FG<\/em> a <em>DE<\/em>. <\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z9-III-4<\/strong><br \/>\nN\u00e1jdite v\u0161etky trojice trojcifern\u00fdch prirodzen\u00fdch \u010d\u00edsel (<em>a, b, c<\/em>) tak\u00e9, \u017ee<\/p>\n<div class=\"r\">b<sup>2<\/sup> = a \u22c5 c,&nbsp;&nbsp;&nbsp;&nbsp;b = \ud835\udc4e + 34.<\/div>\n<div class=\"pri\">()<\/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 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<p><span style=\"border-radius: 2px; text-indent: 20px; width: auto; padding: 0px 4px 0px 0px; text-align: center; font: bold 11px\/20px 'Helvetica Neue',Helvetica,sans-serif; color: #ffffff; background: #bd081c no-repeat scroll 3px 50% \/ 14px 14px; position: absolute; opacity: 1; z-index: 8675309; display: none; cursor: pointer;\">Ulo\u017ei\u0165<\/span><\/p>\n<p><span style=\"border-radius: 2px; text-indent: 20px; width: auto; padding: 0px 4px 0px 0px; text-align: center; font: bold 11px\/20px 'Helvetica Neue',Helvetica,sans-serif; color: #ffffff; background: #bd081c no-repeat scroll 3px 50% \/ 14px 14px; position: absolute; opacity: 1; z-index: 8675309; display: none; cursor: pointer;\">Ulo\u017ei\u0165<\/span><\/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>aktualizovan\u00e9 08.07.2023 22:40 Z5 Z6 Z7 Z8 Z9 dom\u00e1ce dom\u00e1ce dom\u00e1ce dom\u00e1ce dom\u00e1ce okresn\u00e9 okresn\u00e9 okresn\u00e9 okresn\u00e9 okresn\u00e9 krajsk\u00e9<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","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":[49],"tags":[],"class_list":["post-1608","post","type-post","status-publish","format-standard","hentry","category-73-rocnik-2023-24"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.6 - 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