{"id":1365,"date":"2019-09-15T19:40:11","date_gmt":"2019-09-15T19:40:11","guid":{"rendered":"http:\/\/matematika.besaba.com\/?p=1365"},"modified":"2023-11-02T18:52:23","modified_gmt":"2023-11-02T18:52:23","slug":"sutazne-ulohy-kategorii-z4-z9-3","status":"publish","type":"post","link":"https:\/\/matematika.besaba.com\/?p=1365","title":{"rendered":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9"},"content":{"rendered":"<p class=\"hh2\"><a name=\"top\"><\/a>69. ro\u010dn\u00edk matematickej olympi\u00e1dy 2019-2020<br \/>\nS\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9<\/p>\n<p><span class=\"povinne\">aktualizovan\u00e9 31.1.2020 13:00<\/span><\/p>\n\n<p><!--more--><\/p>\n<table style=\"width: 90%;\" border=\"1\" cellspacing=\"2\" cellpadding=\"5\" align=\"center\">\n<tbody>\n<tr align=\"center\" bgcolor=\"#c0c0c0\">\n<td><span class=\"a\">Z5<\/span><\/td>\n<td><span class=\"a\">Z6<\/span><\/td>\n<td><span class=\"a\">Z7<\/span><\/td>\n<td><span class=\"a\">Z8<\/span><\/td>\n<td><span class=\"a\">Z9<\/span><\/td>\n<\/tr>\n<tr align=\"center\">\n<td><span class=\"a\"><a href=\"#z51\">dom\u00e1ce<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z61\">dom\u00e1ce<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z71\">dom\u00e1ce<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z81\">dom\u00e1ce<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z91\">dom\u00e1ce<\/a><\/span><\/td>\n<\/tr>\n<tr align=\"center\">\n<td><span class=\"a\"><a href=\"#z52\">obvodn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z62\">obvodn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z72\">obvodn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z82\">obvodn\u00e9<\/a><\/span><\/td>\n<td><span class=\"a\"><a href=\"#z92\">obvodn\u00e9<\/a><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"4\" bgcolor=\"#c0c0c0\">\u00a0<\/td>\n<td align=\"center\"><span class=\"a\"><a href=\"#z93\">krajsk\u00e9<\/a><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"z51\"><\/a><\/p>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=3367\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt51z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=3398\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt51r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z5-I-1<\/strong><br \/>\nNa\u0161a star\u00e1 mama nakupovala v obchode, v ktorom mali iba jablk\u00e1, ban\u00e1ny a hru\u0161ky. Jablk\u00e1 boli po 50 centoch za kus, hru\u0161ky po 60 centoch a ban\u00e1ny boli lacnej\u0161ie ako hru\u0161ky. Star\u00e1 mama k\u00fapila p\u00e4\u0165 kusov ovocia, medzi ktor\u00fdmi bol pr\u00e1ve jeden ban\u00e1n, a zaplatila 2 eur\u00e1 a 75 centov. Ko\u013eko centov mohol st\u00e1\u0165 jeden ban\u00e1n? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Katar\u00edna Jasen\u010d\u00e1kov\u00e1)<\/div>\n<p><strong>Z5-I-2<\/strong><br \/>\nV\u0161etky cesty v parku s\u00fa meter \u0161irok\u00e9 a s\u00fa tvoren\u00e9 cel\u00fdmi \u0161tvorcov\u00fdmi dla\u017edicami s rozmermi meter kr\u00e1t meter, ktor\u00e9 k sebe natesno priliehaj\u00fa. Cesty, pri ktor\u00fdch sa maj\u00fa vymeni\u0165 v\u0161etky dla\u017edice, s\u00fa schematicky zn\u00e1zornen\u00e9 na obr\u00e1zku. Ko\u013eko dla\u017ed\u00edc sa m\u00e1 vymeni\u0165?<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"69\/69z5a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z5-I-3<\/strong><br \/>\nP\u00e1n kr\u00e1\u013e rozd\u00e1val svojim synom duk\u00e1ty. Najstar\u0161iemu synovi dal ur\u010dit\u00fd po\u010det duk\u00e1tov, mlad\u0161iemu dal o jeden duk\u00e1t menej, \u010fal\u0161iemu dal op\u00e4\u0165 o jeden duk\u00e1t menej a takto postupoval a\u017e k najmlad\u0161iemu. Potom sa vr\u00e1til k najstar\u0161iemu synovi, dal mu o jeden duk\u00e1t menej ako pred chv\u00ed\u013eou najmlad\u0161iemu a rovnak\u00fdm sp\u00f4sobom ako v prvom kole rozd\u00e1val \u010falej. V tomto kole vy\u0161iel na najmlad\u0161ieho syna jeden duk\u00e1t. Najstar\u0161\u00ed syn dostal celkom 21 duk\u00e1tov. Ur\u010dte, ko\u013eko mal kr\u00e1\u013e synov a ko\u013eko im celkom rozdal duk\u00e1tov.<\/p>\n<div class=\"pri\">(Karel Pazourek)<\/div>\n<p><strong>Z5-I-4<\/strong><br \/>\nVojto za\u010dal vypisova\u0165 do zo\u0161ita \u010d\u00edslo teraj\u0161ieho \u0161kolsk\u00e9ho roku 2019202020192020&#8230;a tak pokra\u010doval st\u00e1le \u010falej. Ke\u010f nap\u00edsal 2020 cifier, prestalo ho to bavi\u0165. Ko\u013eko tak nap\u00edsal dvojok?.<\/p>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\u00e1)<\/div>\n<p><strong>Z5-I-5<\/strong><br \/>\nDedko m\u00e1 v z\u00e1hrade tri jablone a na nich celkom 39 jab\u013ak. Jablk\u00e1 rast\u00fa iba na \u00f4smich kon\u00e1roch: na jednej jabloni plodia dva kon\u00e1re, na dvoch jabloniach plodia po tri kon\u00e1re. Na r\u00f4znych kon\u00e1roch s\u00fa r\u00f4zne po\u010dty jab\u013ak, ale na ka\u017edej jabloni je rovnak\u00fd po\u010det jab\u013ak. Ko\u013eko jab\u013ak mohlo by\u0165 na jednotliv\u00fdch kon\u00e1roch? Ur\u010dte aspo\u0148 jednu mo\u017enos\u0165.<\/p>\n<div class=\"pri\">(Al\u017ebeta Bohinikov\u00e1)<\/div>\n<p><strong>Z5-I-6<\/strong><br \/>\nObd\u013a\u017enikov\u00fd obrus je poskladan\u00fd z rovnako ve\u013ek\u00fdch \u0161tvorcov bielej, sivej a \u010diernej farby, a to tak, \u017ee<\/p>\n<ul>\n<li>\u0161tvorce so spolo\u010dnou stranou maj\u00fa r\u00f4zne farby,<\/li>\n<li>biele \u0161tvorce nemaj\u00fa spolo\u010dn\u00fd vrchol,<\/li>\n<li>\u010dierne \u0161tvorce nemaj\u00fa spolo\u010dn\u00fd vrchol,<\/li>\n<li>\u010diernych \u0161tvorcov je \u0161es\u0165,<\/li>\n<li>na ka\u017edej strane obrusu s\u00fa najmenej tri \u0161tvorce.<\/li>\n<\/ul>\n<p>Ako mohol obrus vyzera\u0165? N\u00e1jdite a nakreslite aspo\u0148 tri mo\u017enosti.<\/p>\n<div class=\"pri\">(Katar\u00edna Jasen\u010d\u00e1kov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z52\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=3396\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt52z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=3394\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt52r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z5-II-1<\/strong><br \/>\nZ\u00e1hradn\u00edk p\u00e1n Malina pred\u00e1val jahody. V posledn\u00fdch deviatich debni\u010dk\u00e1ch mal postupne 28, 51, 135, 67, 123, 29, 56, 38 a 79 saden\u00edc jah\u00f4d. Pred\u00e1val cel\u00e9 debni\u010dky, \u017eiadne sadenice z debni\u010diek nevy\u0165ahoval. Z\u00e1hradn\u00edk chcel rozpreda\u0165 debni\u010dky trom z\u00e1kazn\u00edkom tak, aby mu ni\u010d neostalo a aby v\u0161etci t\u00edto z\u00e1kazn\u00edci mali rovnak\u00fd po\u010det saden\u00edc. Ako to mohol urobi\u0165? Uve\u010fte dve mo\u017enosti.<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z5-II-2<\/strong><br \/>\nDo \u0161tvorca s\u00fa vp\u00edsan\u00e9 men\u0161ie \u0161tvorce, a to v\u017edy tak, \u017ee vrcholy men\u0161ieho \u0161tvorca s\u00fa v stredoch str\u00e1n v\u00e4\u010d\u0161ieho \u0161tvorca, pozri obr\u00e1zok. Sivo vyfarben\u00fd \u0161tvorec m\u00e1 obsah 1 cm<sup>2<\/sup>. Ur\u010dte obvod najv\u00e4\u010d\u0161ieho \u0161tvorca.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"69\/69z5b2.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\"> (Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z5-II-3<\/strong><br \/>\nNa obr\u00e1zku je \u010d\u00edseln\u00e1 os s vyzna\u010den\u00fdmi \u010d\u00edslami 10 a 30 a \u010fal\u0161\u00edmi bezmenn\u00fdmi bodmi predstavuj\u00facimi cel\u00e9 \u010d\u00edsla. Janko si na tejto osi bodkami vyzna\u010dil svoje ob\u013e\u00faben\u00e9 \u010d\u00edslo a \u010fal\u0161ie \u0161tyri \u010d\u00edsla, o ktor\u00fdch vieme, \u017ee<\/p>\n<ul>\n<li>jedno je polovicou Jankovho \u010d\u00edsla,<\/li>\n<li>jedno je o 6 v\u00e4\u010d\u0161ie ako Jankovo \u010d\u00edslo,<\/li>\n<li>jedno je o 10 men\u0161ie ako Jankovo \u010d\u00edslo,<\/li>\n<li>jedno je dvakr\u00e1t v\u00e4\u010d\u0161ie ako Jankovo \u010d\u00edslo.<\/li>\n<\/ul>\n<p>Zistite, ktor\u00e9 \u010d\u00edslo je Jankovo ob\u013e\u00faben\u00e9.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"69\/69z5b1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Svetlana Bedn\u00e1\u0159ov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z61\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=3366\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt61z\" type=\"button\" value=\"Zadanie\" \/><\/a><\/form>\n<p><strong>Z6-I-1<\/strong><br \/>\nStar\u00e1 mama povedala vn\u00fa\u010dat\u00e1m: \u201eDnes m\u00e1m 60 rokov a 50 mesiacov a 40 t\u00fd\u017ed\u0148ov a 30 dn\u00ed.\u201c Ko\u013ek\u00e9 narodeniny mala star\u00e1 mama naposledy?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z6-I-2<\/strong><br \/>\nNa obr\u00e1zku je trojuholn\u00edkov\u00e1 sie\u0165 a v nej \u0161tyri mnohouholn\u00edky. Obvody mnohouholn\u00edkov A, B a D s\u00fa postupne 56 cm, 34 cm a 42 cm. Ur\u010dte obvod trojuholn\u00edka C.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"69\/69z6a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Karel Pazourek)<\/div>\n<p><strong>Z6-I-3<\/strong><br \/>\nNa p\u00edsomke bolo 25 \u00faloh trojak\u00e9ho druhu: \u013eahk\u00e9 po 2 bodoch, stredne \u0165a\u017ek\u00e9 po 3 bodoch a \u0165a\u017ek\u00e9 po 5 bodoch. Spr\u00e1vne vyrie\u0161en\u00e9 \u00falohy boli hodnoten\u00e9 uveden\u00fdm po\u010dtom bodov pod\u013ea stup\u0148a ob\u0165a\u017enosti, inak 0. Najlep\u0161ie mo\u017en\u00e9 celkov\u00e9 hodnotenie p\u00edsomky bolo 84 bodov. Peter spr\u00e1vne vyrie\u0161il v\u0161etky \u013eahk\u00e9 \u00falohy, polovicu stredne \u0165a\u017ek\u00fdch a tretinu \u0165a\u017ek\u00fdch. Ko\u013eko bodov z\u00edskal Peter za svoju p\u00edsomku?<\/p>\n<div class=\"pri\">(Al\u017ebeta Bohinikov\u00e1)<\/div>\n<p><strong>Z6-I-4<\/strong><br \/>\nRaz si kr\u00e1\u013e zavolal v\u0161etky svoje p\u00e1\u017eat\u00e1 a postavil ich do radu. Prv\u00e9mu p\u00e1\u017ea\u0165u dal ur\u010dit\u00fd po\u010det duk\u00e1tov, druh\u00e9mu dal o dva duk\u00e1ty menej, tretiemu op\u00e4\u0165 o dva duk\u00e1ty menej a tak \u010falej. Ke\u010f do\u0161iel k posledn\u00e9mu p\u00e1\u017ea\u0165u, dal mu pr\u00edslu\u0161n\u00fd po\u010det duk\u00e1tov, oto\u010dil sa a obdobn\u00fdm sp\u00f4sobom postupoval na za\u010diatok radu (t. j. predposledn\u00e9mu p\u00e1\u017ea\u0165u dal o dva duk\u00e1ty menej ako pred chv\u00ed\u013eou posledn\u00e9mu at\u010f.). Na prv\u00e9 p\u00e1\u017ea v tomto kole vy\u0161li dva duk\u00e1ty. Potom jedno z p\u00e1\u017eat zistilo, \u017ee m\u00e1 32 duk\u00e1tov. Ko\u013eko mohol ma\u0165 kr\u00e1\u013e p\u00e1\u017eat a ko\u013eko celkom im mohol rozda\u0165 duk\u00e1tov? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Karel Pazourek)<\/div>\n<p><strong>Z6-I-5<\/strong><br \/>\n\u00datvar na obr\u00e1zku vznikol tak, \u017ee z ve\u013ek\u00e9ho kr\u00ed\u017ea bol vystrihnut\u00fd mal\u00fd kr\u00ed\u017e. Ka\u017ed\u00fd z t\u00fdchto kr\u00ed\u017eov m\u00f4\u017ee by\u0165 zlo\u017een\u00fd z piatich zhodn\u00fdch \u0161tvorcov, pri\u010dom strany mal\u00fdch \u0161tvorcov s\u00fa polovi\u010dn\u00e9 vzh\u013eadom na strany ve\u013ek\u00fdch \u0161tvorcov. Obsah vystrihnut\u00e9ho \u00fatvaru je 45 cm<sup>2<\/sup>. Ak\u00fd je obsah ve\u013ek\u00e9ho kr\u00ed\u017ea?<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"69\/69z6a2.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\u00e1)<\/div>\n<p><strong>Z6-I-6<\/strong><br \/>\nMajka sk\u00famala viaccifern\u00e9 \u010d\u00edsla, v ktor\u00fdch sa po jednej striedaj\u00fa nep\u00e1rne a p\u00e1rne cifry. Tie, ktor\u00e9 za\u010d\u00ednaj\u00fa nep\u00e1rnou cifrou, nazvala komick\u00e9 a tie, ktor\u00e9 za\u010d\u00ednaj\u00fa p\u00e1rnou cifrou, nazvala vesel\u00e9. (Napr. \u010d\u00edslo 32387 je komick\u00e9, \u010d\u00edslo 4529 je vesel\u00e9.) Medzi trojcifern\u00fdmi \u010d\u00edslami ur\u010dte, \u010di je viac komick\u00fdch alebo vesel\u00fdch, a o ko\u013eko.<\/p>\n<div class=\"pri\">(Monika Dillingerov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"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><input name=\"butt62z\" type=\"button\" value=\"Zadanie\" \/><input name=\"butt62r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z6-II-1<\/strong><\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z6-II-2<\/strong><\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z6-II-3<\/strong><\/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=\"http:\/\/skmo.sk\/dokument.php?id=3365\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt71z\" type=\"button\" value=\"Zadanie\" \/><\/a><input name=\"butt71r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z7-I-1<\/strong><br \/>\nSnehulienka so siedmimi trpasl\u00edkmi nazbierali \u0161i\u0161ky na t\u00e1bor\u00e1k. Snehulienka povedala, \u017ee po\u010det v\u0161etk\u00fdch \u0161i\u0161iek je \u010d\u00edslo delite\u013en\u00e9 dvoma. Prv\u00fd trpasl\u00edk prehl\u00e1sil, \u017ee je to \u010d\u00edslo delite\u013en\u00e9 tromi, druh\u00fd trpasl\u00edk povedal, \u017ee je to \u010d\u00edslo delite\u013en\u00e9 \u0161tyrmi, tret\u00ed trpasl\u00edk povedal, \u017ee je to \u010d\u00edslo delite\u013en\u00e9 piatimi, \u0161tvrt\u00fd trpasl\u00edk povedal, \u017ee je to \u010d\u00edslo delite\u013en\u00e9 \u0161iestimi, piaty trpasl\u00edk povedal, \u017ee je to \u010d\u00edslo delite\u013en\u00e9 siedmimi, \u0161iesty trpasl\u00edk povedal, \u017ee je to \u010d\u00edslo delite\u013en\u00e9 \u00f4smimi, siedmy trpasl\u00edk povedal, \u017ee je to \u010d\u00edslo delite\u013en\u00e9 deviatimi. Dvaja z \u00f4smich zbera\u010dov, ktor\u00ed sa k po\u010dtu \u0161i\u0161iek vyjadrovali bezprostredne po sebe, nemali pravdu, ostatn\u00ed \u00e1no. Ko\u013eko \u0161i\u0161iek bolo na hromade, ak ich ur\u010dite bolo menej ako 350?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z7-I-2<\/strong><br \/>\nV ostrouhlom trojuholn\u00edku KLM je V priese\u010dn\u00edk jeho v\u00fd\u0161ok a X je p\u00e4ta v\u00fd\u0161ky na stranu KL. Os uhla XV L je rovnobe\u017en\u00e1 so stranou LM a uhol MKL m\u00e1 ve\u013ekos\u0165 70\u00b0. Ak\u00fa ve\u013ekos\u0165 maj\u00fa uhly KLM a KML?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z7-I-3<\/strong><br \/>\nRoman m\u00e1 r\u00e1d k\u00fazla a matematiku. Naposledy \u010daroval s trojcifern\u00fdmi alebo \u0161tvorcifern\u00fdmi \u010d\u00edslami takto:<\/p>\n<ul>\n<li>z dan\u00e9ho \u010d\u00edsla vytvoril dve nov\u00e9 \u010d\u00edsla tak, \u017ee ho rozdelil medzi ciframi na mieste stoviek a desiatok (napr. z \u010d\u00edsla 581 by dostal 5 a 81),<\/li>\n<li>nov\u00e9 \u010d\u00edsla s\u010d\u00edtal a zap\u00edsal v\u00fdsledok (v uvedenom pr\u00edklade by dostal 86),<\/li>\n<li>od v\u00e4\u010d\u0161ieho z nov\u00fdch \u010d\u00edsel od\u010d\u00edtal men\u0161ie a v\u00fdsledok zap\u00edsal za predch\u00e1dzaj\u00faci s\u00fa\u010det, \u010d\u00edm vy\u010daroval v\u00fdsledn\u00e9 \u010d\u00edslo (v uvedenom pr\u00edklade by dostal 8676).<\/li>\n<\/ul>\n<p>Z ktor\u00fdch \u010d\u00edsel mohol Roman vy\u010darova\u0165 a) 171, b) 1513? Ur\u010dte v\u0161etky mo\u017enosti. Ak\u00e9 najv\u00e4\u010d\u0161ie \u010d\u00edslo mo\u017eno takto vy\u010darova\u0165 a z ktor\u00fdch \u010d\u00edsel m\u00f4\u017ee vznikn\u00fa\u0165? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Monika Dillingerov\u00e1)<\/div>\n<p><strong>Z7-I-4<\/strong><br \/>\nJanka a Marienku zaujalo vodn\u00e9 dielo, ktor\u00e9ho \u010das\u0165 je zn\u00e1zornen\u00e1 na obr\u00e1zku. Koryt\u00e1 sa postupne rozde\u013euj\u00fa a zasa sp\u00e1jaj\u00fa v na\u010drtnut\u00fdch bodoch, v ka\u017edom rade je o jeden tak\u00fd bod viac ako v rade predch\u00e1dzaj\u00facom. Voda pr\u00fadi v nazna\u010den\u00fdch smeroch a pri ka\u017edom vetven\u00ed sa vodn\u00fd tok rozdel\u00ed do dvoch kor\u00fdt s rovnak\u00fdm prietokom. Janka zauj\u00edmalo, ko\u013eko vody pretek\u00e1 v s\u00fa\u010dte \u0161tyrmi miestami zv\u00fdraznen\u00fdmi \u010dierno. Marienku zauj\u00edmalo, ko\u013eko vody pretek\u00e1 v s\u00fa\u010dte v\u0161etk\u00fdmi miestami, ktor\u00e9 s\u00fa v 2019. rade. Porovnajte celkov\u00e9 prietoky Jankov\u00fdmi a Marienkin\u00fdmi miestami.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"69\/69z7a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Katar\u00edna Jasen\u010d\u00e1kov\u00e1)<\/div>\n<p><strong>Z7-I-5<\/strong><br \/>\nHviezdny \u0161tvorec je tak\u00e1 \u0161tvorcov\u00e1 tabu\u013eka \u010d\u00edsel, pre ktor\u00fa plat\u00ed, \u017ee s\u00fa\u010dty \u010d\u00edsel v jednotliv\u00fdch riadkoch a st\u013apcoch s\u00fa st\u00e1le rovnak\u00e9. Na obr\u00e1zku je pozostatok hviezdneho \u0161tvorca, v ktorom boli \u010d\u00edsla v jednom riadku a jednom st\u013apci zotret\u00e9. Dopl\u0148te ch\u00fdbaj\u00face \u010d\u00edsla tak, aby v\u0161etky boli cel\u00e9 a pr\u00e1ve \u0161tyri z\u00e1porn\u00e9. Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pc\">\n<table align=\"center\" height=\"100\" width=\"100\" border-collapse: collapse; border-style: solid; border-color: #000000;\" border=\"2\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 25%\">1<\/td>\n<td style=\"width: 25%\">2<\/td>\n<td style=\"width: 25%\">3<\/td>\n<td style=\"width: 25%;\">\u00a0<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>8<\/td>\n<td>9<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>&nbsp;<\/td>\n<td>&nbsp;<\/td>\n<td>&nbsp;<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z7-I-6<\/strong><br \/>\nZ pravideln\u00e9ho \u0161es\u0165uholn\u00edka bol vystrihnut\u00fd \u00fatvar ako na obr\u00e1zku. Pritom vyzna\u010den\u00e9 body ako na obvode, tak vn\u00fatri \u0161es\u0165uholn\u00edka delia prisl\u00fachaj\u00face \u00fase\u010dky na \u0161tvrtiny. Ak\u00fd je pomer obsahov p\u00f4vodn\u00e9ho \u0161es\u0165uholn\u00edka a vystrihnut\u00e9ho \u00fatvaru?<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"69\/69z7a2.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Al\u017ebeta Bohinikov\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><input name=\"butt72z\" type=\"button\" value=\"Zadanie\" \/><input name=\"butt72r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z7-II-1<\/strong><\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z7-II-2<\/strong><\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z7-II-3<\/strong><\/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=\"http:\/\/skmo.sk\/dokument.php?id=3364\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt81z\" type=\"button\" value=\"Zadanie\" \/><\/a><input name=\"butt81r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z8-I-1<\/strong><br \/>\nZostrojte koso\u0161tvorec ABCD tak, aby jeho uhloprie\u010dka BD mala ve\u013ekos\u0165 8 cm a vzdialenos\u0165 vrcholu B od priamky AD bola 5 cm. Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n<div class=\"pri\">(Karel Pazourek)<\/div>\n<p><strong>Z8-I-2<\/strong><br \/>\nRichard sa pohr\u00e1val s dvoma p\u00e4\u0165cifern\u00fdmi \u010d\u00edslami. Ka\u017ed\u00e9 pozost\u00e1valo z navz\u00e1jom r\u00f4znych cifier, ktor\u00e9 pri jednom boli v\u0161etky nep\u00e1rne a pri druhom v\u0161etky p\u00e1rne. Po chv\u00edli zistil, \u017ee s\u00fa\u010det t\u00fdchto dvoch \u010d\u00edsel za\u010d\u00edna dvoj\u010d\u00edsl\u00edm 11 a kon\u010d\u00ed \u010d\u00edslom 1 a \u017ee ich rozdiel za\u010d\u00edna \u010d\u00edslom 2 a kon\u010d\u00ed dvoj\u010d\u00edsl\u00edm 11. Ur\u010dte Richardove \u010d\u00edsla.<\/p>\n<div class=\"pri\">(Monika Dillingerov\u00e1)<\/div>\n<p><strong>Z8-I-3<\/strong><br \/>\nVendel\u00edn b\u00fdva medzi dvoma zast\u00e1vkami autobusu, a to v troch osmin\u00e1ch ich vzdialenosti. Dnes vyrazil z domu a zistil, \u017ee \u010di by utekal k jednej, alebo druhej zast\u00e1vke, dorazil by na zast\u00e1vku s\u00fa\u010dasne s autobusom. Priemern\u00e1 r\u00fdchlos\u0165 autobusu je 60 km\/h. Akou priemernou r\u00fdchlos\u0165ou dnes be\u017e\u00ed Vendel\u00edn? <\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z8-I-4<\/strong><br \/>\nPre p\u00e4ticu cel\u00fdch \u010d\u00edsel plat\u00ed, \u017ee ke\u010f k prv\u00e9mu pripo\u010d\u00edtame jednotku, druh\u00e9 umocn\u00edme na druh\u00fa, od tretieho od\u010d\u00edtame trojku, \u0161tvrt\u00e9 vyn\u00e1sob\u00edme \u0161tyrmi a piate vydel\u00edme piatimi, dostaneme zaka\u017ed\u00fdm ten ist\u00fd v\u00fdsledok. N\u00e1jdite v\u0161etky tak\u00e9 p\u00e4tice \u010d\u00edsel, ktor\u00fdch s\u00fa\u010det je 122.<\/p>\n<div class=\"pri\">(Lenka Dedkov\u00e1)<\/div>\n<p><strong>Z8-I-5<\/strong><br \/>\nPre osem navz\u00e1jom r\u00f4znych bodov ako na obr\u00e1zku plat\u00ed, \u017ee body C, D, E le\u017eia na priamke rovnobe\u017enej s priamkou AB, F je stredom \u00fase\u010dky AD, G je stredom \u00fase\u010dky AC a H je priese\u010dn\u00edkom priamok AC a BE. Obsah trojuholn\u00edka BCG je 12 cm<sup>2<\/sup> a obsah \u0161tvoruholn\u00edka DFHG je 8 cm<sup>2<\/sup>. Ur\u010dte obsahy trojuholn\u00edkov AFE, AHF, ABG a BGH.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"69\/69z8a1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\">(Eva Semer\u00e1dov\u00e1)<\/div>\n<p><strong>Z8-I-6<\/strong><br \/>\nV Koc\u00farkove pou\u017e\u00edvaj\u00fa mince iba s dvoma hodnotami, ktor\u00e9 s\u00fa vyjadren\u00e9 v koc\u00farkovsk\u00fdch korun\u00e1ch kladn\u00fdmi cel\u00fdmi \u010d\u00edslami. Pomocou dostato\u010dn\u00e9ho mno\u017estva tak\u00fdch minc\u00ed je mo\u017en\u00e9 zaplati\u0165 ak\u00fako\u013evek celo\u010d\u00edseln\u00fa sumu v\u00e4\u010d\u0161iu ako 53 koc\u00farkovsk\u00fdch kor\u00fan, a to presne a bez vyd\u00e1vania. Sumu 53 koc\u00farkovsk\u00fdch kor\u00fan v\u0161ak bez vyd\u00e1vania zaplati\u0165 nemo\u017eno. Zistite, ktor\u00e9 hodnoty mohli by\u0165 na koc\u00farkovsk\u00fdch minciach. Ur\u010dte aspo\u0148 dve rie\u0161enia.<\/p>\n<div class=\"pri\">(Al\u017ebeta Bohinikov\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=\"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><input name=\"butt82z\" type=\"button\" value=\"Zadanie\" \/><input name=\"butt82r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z8-II-1<\/strong><\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z8-II-2<\/strong><\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z8-II-3<\/strong><\/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=\"http:\/\/skmo.sk\/dokument.php?id=3363\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt91z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=3398\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt91r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-I-1<\/strong><br \/>\nOndro, Ma\u0165o a Kubo sa vracaj\u00fa zo zbierania orechov, dokopy ich maj\u00fa 120. Ma\u0165o sa s\u0165a\u017euje, \u017ee Ondro m\u00e1 ako v\u017edy najviac. Otec prik\u00e1\u017ee Ondrovi, aby prisypal zo svojho Ma\u0165ovi tak, aby mu po\u010det orechov zdvojn\u00e1sobil. Teraz sa s\u0165a\u017euje Kubo, \u017ee najviac m\u00e1 Ma\u0165o. Na otcov pr\u00edkaz prisype Ma\u0165o Kubovi tak, \u017ee mu po\u010det orechov zdvojn\u00e1sob\u00ed. Na to sa hnev\u00e1 Ondro, \u017ee najmenej zo v\u0161etk\u00fdch m\u00e1 teraz on. Kubo teda prisype Ondrovi tak, \u017ee mu po\u010det orechov zdvojn\u00e1sob\u00ed. Teraz maj\u00fa v\u0161etci rovnako a kone\u010dne je k\u013eud. Ko\u013eko orechov mal p\u00f4vodne ka\u017ed\u00fd z chlapcov?<\/p>\n<div class=\"pri\">(Marta Volfov\u00e1)<\/div>\n<p><strong>Z9-I-2<\/strong><br \/>\nV trojuholn\u00edku ABC le\u017e\u00ed bod P v tretine \u00fase\u010dky AB bli\u017e\u0161ie bodu A, bod R je v tretine \u00fase\u010dky PB bli\u017e\u0161ie bodu P a bod Q le\u017e\u00ed na \u00fase\u010dke BC tak, \u017ee uhly PCB a RQB s\u00fa zhodn\u00e9. Ur\u010dte pomer obsahov trojuholn\u00edkov ABC a PQC.<\/p>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\u00e1)<\/div>\n<p><strong>Z9-I-3<\/strong><br \/>\nPre ktor\u00e9 cel\u00e9 \u010d\u00edsla x je podiel<img decoding=\"async\" src=\"69\/69z9a1.png\" align=\"middle\" border=\"0\" \/>cel\u00fdm \u010d\u00edslom? N\u00e1jdite v\u0161etky rie\u0161enia.<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z9-I-4<\/strong><br \/>\nMat\u00fa\u0161 dopadol pad\u00e1kom na ostrov ob\u00fdvan\u00fd dvoma druhmi domorodcov: Poctivcami, ktor\u00ed v\u017edy hovoria pravdu, a Klam\u00e1rmi, ktor\u00ed v\u017edy klam\u00fa. Pred dopadom zahliadol v dia\u013eke pr\u00edstav, ku ktor\u00e9mu sa hodlal dosta\u0165. Na prvom r\u00e1zcest\u00ed stretol Mat\u00fa\u0161 jedn\u00e9ho domorodca a ob\u010fale\u010d videl druh\u00e9ho. Po\u017eiadal prv\u00e9ho, aby sa sp\u00fdtal toho druh\u00e9ho, \u010di je Klam\u00e1r, alebo Poctivec. Prv\u00fd domorodec Mat\u00fa\u0161ovi vyhovel, i\u0161iel sa sp\u00fdta\u00bb a ke\u010f sa vr\u00e1til, ozn\u00e1mil Mat\u00fa\u0161ovi, \u017ee druh\u00fd domorodec tvrd\u00ed, \u017ee je Klam\u00e1r. Potom sa Mat\u00fa\u0161 prv\u00e9ho domorodca sp\u00fdtal, ktor\u00e1 cesta vedie k pr\u00edstavu. Ten mu jednu cestu uk\u00e1zal a \u010falej si Mat\u00fa\u0161a nev\u0161\u00edmal. M\u00e1, alebo nem\u00e1 Mat\u00fa\u0161 domorodcovi veri\u0165? Vedie, alebo nevedie t\u00e1to cesta k pr\u00edstavu?<\/p>\n<div class=\"pri\">(Marta Volfov\u00e1)<\/div>\n<p><strong>Z9-I-5<\/strong><br \/>\nMajka sk\u00famala viaccifern\u00e9 \u010d\u00edsla, v ktor\u00fdch sa po jednej striedaj\u00fa nep\u00e1rne a p\u00e1rne cifry. Tie, ktor\u00e9 za\u010d\u00ednaj\u00fa nep\u00e1rnou cifrou, nazvala komick\u00e9 a tie, ktor\u00e9 za\u010d\u00ednaj\u00fa p\u00e1rnou cifrou, nazvala vesel\u00e9. (Napr. \u010d\u00edslo 32387 je komick\u00e9, \u010d\u00edslo 4529 je vesel\u00e9.) Majka vytvorila jedno trojcifern\u00e9 komick\u00e9 a jedno trojcifern\u00e9 vesel\u00e9 \u010d\u00edslo, pri\u010dom \u0161es\u0165 pou\u017eit\u00fdch cifier bolo navz\u00e1jom r\u00f4znych a nebola medzi nimi 0. S\u00fa\u010det t\u00fdchto dvoch \u010d\u00edsel bol 1617. S\u00fa\u010din t\u00fdchto dvoch \u010d\u00edsel kon\u010dil dvoj\u010d\u00edsl\u00edm 40. Ur\u010dte Majkine \u010d\u00edsla a dopo\u010d\u00edtajte ich s\u00fa\u010din<\/p>\n<div class=\"pri\">(Monika Dillingerov\u00e1)<\/div>\n<p><strong>Z9-I-6<\/strong><br \/>\nKrist\u00edna zvolila ist\u00e9 nep\u00e1rne prirodzen\u00e9 \u010d\u00edslo delite\u013en\u00e9 tromi. Jakub s D\u00e1vidom potom sk\u00famali trojuholn\u00edky, ktor\u00e9 maj\u00fa obvod v milimetroch rovn\u00fd Krist\u00ednou zvolen\u00e9mu \u010d\u00edslu a ktor\u00fdch strany maj\u00fa d\u013a\u017eky v milimetroch vyjadren\u00e9 navz\u00e1jom r\u00f4znymi cel\u00fdmi \u010d\u00edslami. Jakub na\u0161iel tak\u00fd trojuholn\u00edk, v ktorom najdlh\u0161ia zo str\u00e1n m\u00e1 najv\u00e4\u010d\u0161iu mo\u017en\u00fa d\u013a\u017eku, a t\u00fato hodnotu zap\u00edsal na tabu\u013eu. D\u00e1vid na\u0161iel tak\u00fd trojuholn\u00edk, v ktorom najkrat\u0161ia zo str\u00e1n m\u00e1 najv\u00e4\u010d\u0161iu mo\u017en\u00fa d\u013a\u017eku, a t\u00fato hodnotu tie\u017e zap\u00edsal na tabu\u013eu. Krist\u00edna obe d\u013a\u017eky na tabuli spr\u00e1vne s\u010d\u00edtala a vy\u0161lo jej 1 681 mm. Ur\u010dte, ktor\u00e9 \u010d\u00edslo Krist\u00edna zvolila. <\/p>\n<div class=\"pri\">(Lucie R\u016f\u017ei\u010dkov\u00e1)<\/div>\n<table style=\"width: 100%; height: 25px;\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td align=\"left\" width=\"85%\">\n<hr \/>\n<\/td>\n<td class=\"h\" width=\"12%\"><a name=\"z92\"><\/a><a href=\"#top\"><img decoding=\"async\" src=\"\/img\/top.gif\" alt=\"\" align=\"right;\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<form><a href=\"http:\/\/skmo.sk\/dokument.php?id=3395\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt92z\" type=\"button\" value=\"Zadanie\" \/><\/a><a href=\"http:\/\/skmo.sk\/dokument.php?id=3393\" target=\"_blank\" rel=\"noopener noreferrer\"><input name=\"butt92r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/a><\/form>\n<p><strong>Z9-II-1<\/strong><br \/>\nPat a Mat mali ka\u017ed\u00fd svoje ob\u013e\u00faben\u00e9 prirodzen\u00e9 \u010d\u00edslo, ale ka\u017ed\u00fd in\u00e9. Obe \u010d\u00edsla postupne s\u010d\u00edtali, od\u010d\u00edtali (men\u0161ie od v\u00e4\u010d\u0161ieho), vyn\u00e1sobili a vydelili (v\u00e4\u010d\u0161ie men\u0161\u00edm). Ke\u010f takto z\u00edskan\u00e9 v\u00fdsledky s\u010d\u00edtali, vy\u0161lo im 98. Ktor\u00e9 ob\u013e\u00faben\u00e9 \u010d\u00edsla mali Pat a Mat?<\/p>\n<div class=\"pri\">(Libu\u0161e Hozov\u00e1)<\/div>\n<p><strong>Z9-II-2<\/strong><br \/>\nObr\u00e1zok predstavuje poh\u013ead zhora na trojvrstvov\u00fa pyram\u00eddu tvoren\u00fa 14 zhodn\u00fdmi kockami. Ka\u017edej kocke prisl\u00facha jedno prirodzen\u00e9 \u010d\u00edslo, a to tak, \u017ee \u010d\u00edsla zodpovedaj\u00face kock\u00e1m v spodnej vrstve s\u00fa navz\u00e1jom r\u00f4zne a \u010d\u00edslo na ka\u017edej \u010fal\u0161ej kocke je s\u00fa\u010dtom \u010d\u00edsel zo \u0161tyroch susediacich kociek z ni\u017e\u0161ej vrstvy. Ur\u010dte najmen\u0161ie \u010d\u00edslo delite\u013en\u00e9 \u0161tyrmi, ktor\u00e9 m\u00f4\u017ee prisl\u00facha\u0165 najvrchnej\u0161ej kocke.<\/p>\n<div class=\"pc\"><img decoding=\"async\" src=\"69\/69z9b1.png\" alt=\"\" border=\"0\" \/><\/div>\n<div class=\"pri\"> (Al\u017ebeta Bohinikov\u00e1)<\/div>\n<p><strong>Z9-II-3<\/strong><br \/>\nUva\u017eujme \u0161tvorcifern\u00e9 prirodzen\u00e9 \u010d\u00edslo s nasleduj\u00facou vlastnos\u0165ou: ak prehod\u00edme jeho prv\u00e9 dvoj\u010d\u00edslie s druh\u00fdm, dostaneme \u0161tvorcifern\u00e9 \u010d\u00edslo o 99 men\u0161ie. Ko\u013eko je tak\u00fdch \u010d\u00edsel celkom a ko\u013eko z nich je delite\u013en\u00fdch 9?<\/p>\n<div class=\"pri\"> (Karel Pazourek)<\/div>\n<p><strong>Z9-II-4<\/strong><br \/>\nDo v\u0161eobecn\u00e9ho trojuholn\u00edka <em>ABC<\/em> narysujte bod <em>D<\/em> tak, aby obsah trojuholn\u00edka <em>ABD<\/em> bol rovn\u00fd polovici obsahu trojuholn\u00edka <em>ABC<\/em> a obsah trojuholn\u00edka <em>BCD<\/em> bol rovn\u00fd \u0161estine obsahu trojuholn\u00edka <em>ABC<\/em>.<\/p>\n<p><em>(Rie\u0161enie m\u00e1 by\u0165 v\u0161eobecne platn\u00e9, teda nez\u00e1visl\u00e9 na zvolenom trojuholn\u00edku, jeho \u0161peci\u00e1lnych vlastnostiach \u010di rozmeroch. Kon\u0161trukcia nem\u00f4\u017ee by\u0165 zalo\u017een\u00e1 na meran\u00ed a po\u010d\u00edtan\u00ed. Zvo\u013ete si trojuholn\u00edk, ktor\u00fd nie je rovnoramenn\u00fd ani pravouhl\u00fd.)<\/em><\/p>\n<div class=\"pri\">\u00a0<\/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><input name=\"butt93z\" type=\"button\" value=\"Zadanie\" \/><input name=\"butt93r\" type=\"button\" value=\"Rie\u0161enie\" \/><\/form>\n<p><strong>Z9-III-1<\/strong><\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z9-III-2<\/strong><\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z9-III-3<\/strong><\/p>\n<div class=\"pri\">()<\/div>\n<p><strong>Z9-III-4<\/strong><\/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 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>69. ro\u010dn\u00edk matematickej olympi\u00e1dy 2019-2020 S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9 aktualizovan\u00e9 31.1.2020 13:00<\/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":[28],"tags":[],"class_list":["post-1365","post","type-post","status-publish","format-standard","hentry","category-69-rocnik"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.6 - 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