{"id":119,"date":"2010-07-28T18:55:13","date_gmt":"2010-07-28T18:55:13","guid":{"rendered":"http:\/\/matematika.okamzite.eu\/?p=119"},"modified":"2010-07-28T18:55:13","modified_gmt":"2010-07-28T18:55:13","slug":"kategorie-z4-z9-2","status":"publish","type":"post","link":"https:\/\/matematika.besaba.com\/?p=119","title":{"rendered":"Kateg\u00f3rie Z4-Z9"},"content":{"rendered":"

<\/a><\/p>\n

58. ro\u010dn\u00edk matematickej olympi\u00e1dy 2008-2009<\/p>\n

<\/p>\n\n\n\n\n\n\n
Z4<\/span><\/td>\nZ5<\/span><\/td>\nZ6<\/span><\/td>\nZ7<\/span><\/td>\nZ8<\/span><\/td>\nZ9<\/span><\/td>\n<\/tr>\n
1. kolo<\/a><\/span><\/td>\n1. kolo<\/a><\/span><\/td>\n1. kolo<\/a><\/span><\/td>\n1. kolo<\/a><\/span><\/td>\n1. kolo<\/a><\/span><\/td>\n1. kolo<\/a><\/span><\/td>\n<\/tr>\n
2. kolo<\/a><\/span><\/td>\n2. kolo<\/a><\/span><\/td>\n2. kolo<\/a><\/span><\/td>\n2. kolo<\/a><\/span><\/td>\n2. kolo<\/a><\/span><\/td>\n2. kolo<\/a><\/span><\/td>\n<\/tr>\n
<\/td>\n3. kolo<\/a><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

<\/a><\/p>\n

Z4-I-1<\/strong><\/p>\n

Na stole so \u0161tvorcovou doskou o strane 1\u00a0m bola „trochu nakrivo“<\/em> umiestnen\u00e1 kruhov\u00e1 de\u010dka. Od najbli\u017e\u0161ej strany dosky stola bol jej kraj vzdialen\u00fd 10\u00a0cm, od susednej strany potom 20\u00a0cm a od najvzdialenej\u0161ej strany 40\u00a0cm.<\/p>\n

a)<\/strong> Ako \u010faleko bol okraj de\u010dky od \u0161tvrtej strany dosky stola?
\n b)<\/strong> Ak\u00fd polomer mala de\u010dka?<\/p>\n

(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n


<\/p>\n

Z4-I-2<\/strong><\/p>\n

Jo\u017eo Nudilsa sa zab\u00e1val t\u00fdm, \u017ee p\u00edsal za sebou postupne prirodzen\u00e9 \u010d\u00edsla. Za\u010dal jednotkou: 1234567891011…<\/strong> Po \u010dase ho to prestalo bavi\u0165 a kriticky sa pozrel na svoj v\u00fdtvor. Zistil, \u017ee v postupnosti \u010d\u00edslic, ktor\u00e9 nap\u00edsal, sa vyskytuj\u00fa iba raz tri p\u00e4\u0165ky priamo za sebou.<\/p>\n

a)<\/strong> Najmenej ko\u013eko za sebou id\u00facich prirodzen\u00fdch \u010d\u00edsel nap\u00edsal Jo\u017eo?
\n b)<\/strong> Najmenej ko\u013eko \u010d\u00edslic nap\u00edsal Jo\u017eo?<\/p>\n

(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n


<\/p>\n

Z4-I-3<\/strong><\/p>\n

B\u00fdvam v Tom\u00e1\u0161ovciach, ale pracujem v Rimavskej Sobote. Autobus, ktor\u00fdm do pr\u00e1ce cestujem, m\u00e1 nasleduj\u00face zast\u00e1vky (v uvedenom porad\u00ed): Tom\u00e1\u0161ovce, B\u00e1tka, Rokytn\u00edk, B\u00e1tka, Bakta, Vinica, Rimavsk\u00e1 Sobota. Z B\u00e1tky do Rimavskej Soboty cez Baktu a Vinicu je to po ceste 11\u00a0km, z Rokytn\u00edka cez B\u00e1tku a Baktu do Vinice 12\u00a0km, z B\u00e1tky cez Baktu do Vinice 9\u00a0km. Z Tom\u00e1\u0161oviec do B\u00e1tky je to rovnako \u010faleko ako z Vinice do Rimavskej Soboty.<\/p>\n

a)<\/strong> Ko\u013eko km prejde autobus z Tom\u00e1\u0161oviec do Rimavskej Soboty touto trasou?
\n b)<\/strong> Ko\u013eko km by to bolo z Tom\u00e1\u0161oviec do Rimavskej Soboty, keby autobus nezach\u00e1dzal do Rokytn\u00edka?<\/p>\n

(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n


<\/p>\n

Z4-I-4<\/strong>
\n Dopl\u0148 do pr\u00e1zdnych pol\u00ed\u010dok prirodzen\u00e9 \u010d\u00edsla od 1 do 16 (ka\u017ed\u00e9 \u010d\u00edslo m\u00f4\u017ee\u0161 pou\u017ei\u0165 len raz) tak, aby platili matematick\u00e9 vz\u0165ahy.<\/p>\n

\"\"<\/p>\n

(M. Smitkov\u00e1)<\/div>\n


<\/p>\n

Z4-I-5<\/strong>
\n Pa\u013eko s Radkou si kupuj\u00fa spolu cukr\u00edky. Pri poslednom n\u00e1kupe platil Pa\u013eko 92\u00a0Sk za 5\u00a0balen\u00ed z dvoch druhov cukr\u00edkov. S\u00e1m si vzal z ka\u017ed\u00e9ho druhu po jednom balen\u00ed a Radka dostala jedno balenie gumen\u00fdch a dve balenia \u010dokol\u00e1dov\u00fdch cukr\u00edkov. Jej n\u00e1kup bol tak o\u00a020\u00a0 Sk drah\u0161\u00ed ako Pa\u013ekov.<\/p>\n

a)<\/strong> Ko\u013eko kor\u00fan m\u00e1 za n\u00e1kup da\u0165 Radka Pa\u013ekovi?
\n b)<\/strong> Ko\u013eko stoj\u00ed jedno balenie gumen\u00fdch cukr\u00edkov?<\/p>\n

(M. Dillingerov\u00e1)<\/div>\n


<\/p>\n

Z4-I-6<\/strong>
\n Danko si zo \u0161tvor\u010dekovej siete vystrihol \u00fatvar ako na obr\u00e1zku:<\/p>\n

\"\"<\/p>\n

Odstrihni dva \u0161tvor\u010deky siete tak, aby sa v\u00fdsledn\u00fd \u00fatvar nerozpadol a aby mal \u010do najv\u00e4\u010d\u0161\u00ed obvod. N\u00e1jdi dve rie\u0161enia.<\/p>\n

(M. Dillingerov\u00e1)<\/div>\n<\/p>\n\n\n
\n
\n<\/td>\n		<\/a>\"\"<\/a><\/td>\n<\/tr>\n<\/table>\n

Z4-II-1<\/strong>
\n Mirko si zo \u0161tvorcovej siete s vp\u00edsan\u00fdmi \u010d\u00edslami vystrihol \u00fatvar na obr\u00e1zku:<\/p>\n

\"\"<\/p>\n

Odstrihni dva \u0161tvor\u010deky \u00fatvaru tak, aby sa v\u00fdsledn\u00fd \u00fatvar nerozpadol, aby po odstrihnut\u00ed oboch \u0161tvor\u010dekov mal rovnak\u00fd obvod ako p\u00f4vodne a aby s\u00fa\u010det vp\u00edsan\u00fdch \u010d\u00edsel bol najmen\u0161\u00ed mo\u017en\u00fd.<\/p>\n

(M. Petrov\u00e1, M. Dillingerov\u00e1)<\/div>\n


<\/p>\n

Z4-II-2<\/strong>
\n Jeden detsk\u00fd l\u00edstok na plav\u00e1re\u0148 stoj\u00ed 1\u00a0\u20ac. Jeden dospel\u00fd l\u00edstok stoj\u00ed 2\u00a0\u20ac. Teta Eva a ujo Adam i\u0161li na plav\u00e1re\u0148 s de\u0165mi. V\u0161etky l\u00edstky pri pokladni zaplatil ujo Adam. Teta Eva mu potom dala za seba a v\u0161etky diev\u010dat\u00e1 5\u00a0\u20ac. Ko\u013eko platil ujo Adam pri pokladni ak chlapcov bolo dvakr\u00e1t to\u013eko ako diev\u010dat?<\/p>\n

(M. Dillingerov\u00e1)<\/div>\n


<\/p>\n

Z4-II-3<\/strong>
\n Pani Jedin\u00e1\u010dkovej sa narodili troji\u010dky. Prv\u00e1 pri\u0161la na svet naj\u0165a\u017e\u0161ia Katka, po nej Lenka a posledn\u00e1 naj\u013eah\u0161ia Marienka. Keby Katka v\u00e1\u017eila pri naroden\u00ed o\u00a02\u00a0310\u00a0gramov viac, v\u00e1\u017eila by to\u013eko, \u010do Lenka s Marienkou spolu. Keby v\u00e1\u017eila o\u00a04\u00a0660\u00a0g viac, v\u00e1\u017eila by to\u013eko, \u010do v\u0161etky tri spolu. Zistite p\u00f4rodn\u00fa hmotnos\u0165 jednotliv\u00fdch diev\u010datiek v gramoch, ak viete, \u017ee sa ud\u00e1va s presnos\u0165ou na desiatky gramov.<\/p>\n

(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n<\/p>\n\n\n
\n
\n<\/td>\n		<\/a>\"\"<\/a><\/td>\n<\/tr>\n<\/table>\n

Z5-I-1<\/strong>
\n U\u010dite\u013eka Kadrno\u017ekov\u00e1 kupovala v pokladni zoologickej z\u00e1hrady vstupenky pre svojich \u017eiakov a pre seba. Vstupenka pre dospel\u00e9ho bola drah\u0161ia ako pre \u0161kol\u00e1ka, ale nie viac ako dvakr\u00e1t. U\u010dite\u013eka Kadrno\u017ekov\u00e1 zaplatila 994 Sk. U\u010dite\u013e Hniezdo mal so sebou o troch \u017eiakov viac ako u\u010dite\u013eka Kadrno\u017ekov\u00e1, a za svojich \u017eiakov a za seba zaplatil 1120 Sk.<\/p>\n

a)<\/strong> Ko\u013eko \u017eiakov mal so sebou u\u010dite\u013e Hniezdo?
\n b)<\/strong> Ko\u013eko st\u00e1la vstupenka pre dospel\u00e9ho?<\/p>\n

(L. \u0160im\u016fnek)<\/div>\n


<\/p>\n

Z5-I-2<\/strong>
\n Fero Nudilsa sa zab\u00e1val t\u00fdm, \u017ee p\u00edsal za sebou id\u00face prirodzen\u00e9 \u010d\u00edsla. Za\u010dal jednotkou: 1234567891011…<\/em><\/strong> Po \u010dase ho to prestalo bavi\u0165, dokon\u010dil pr\u00e1ve rozp\u00edsan\u00e9 \u010d\u00edslo a kriticky sa pozrel na svoj v\u00fdtvor. Zistil, \u017ee v postupnosti \u010d\u00edslic, ktor\u00e9 nap\u00edsal, sa vyskytuje p\u00e4\u0165 jednotiek za sebou.<\/p>\n

a)<\/strong> Najmenej ko\u013eko za sebou id\u00facich prirodzen\u00fdch \u010d\u00edsel nap\u00edsal Fero?
\n b)<\/strong> Najmenej ko\u013eko \u010d\u00edslic nap\u00edsal Fero?<\/p>\n

(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n


<\/p>\n

Z5-I-3<\/strong>
\n Najvy\u0161\u0161ia zn\u00e1ma sopka na zemeguli je Mauna Kea na Havajsk\u00fdch ostrovoch. Jej v\u00fd\u0161ka od \u00fap\u00e4tia po vrchol je dokonca o\u00a0358 metrov v\u00e4\u010d\u0161ia, ako je nadmorsk\u00e1 v\u00fd\u0161ka najvy\u0161\u0161ej hory sveta, Mont Everestu. Nedv\u00edha sa v\u0161ak z pevniny, ale z dna Tich\u00e9ho oce\u00e1nu, z 5000 metrovej h\u013abky. Keby morsk\u00e1 hladina v tejto oblasti klesla o\u00a0397 metrov, bola by ponoren\u00e1 \u010das\u0165 Mauna Key presne rovnako vysok\u00e1, ako \u010das\u0165, ktor\u00e1 by vy\u010dnievala nad hladinu.<\/p>\n

a)<\/strong> Ak\u00fa nadmorsk\u00fa v\u00fd\u0161ku m\u00e1 vrchol sopky?
\n b)<\/strong> Ko\u013eko meria Mauna Kea od \u00fap\u00e4tia po vrchol?
\n c)<\/strong> Ak\u00fa nadmorsk\u00fa v\u00fd\u0161ku m\u00e1 Mont Everest?<\/p>\n

(\u00dadaje o nadmorsk\u00fdch v\u00fd\u0161kach uv\u00e1dzan\u00e9 v r\u00f4znych literat\u00farach sa m\u00f4\u017eu l\u00ed\u0161i\u0165. Je to sp\u00f4soben\u00e9 jednak nepresnos\u0165ami niektor\u00fdch meran\u00ed, jednak pohybmi zemskej k\u00f4ry \u2013 tieto v\u00fd\u0161ky sa skuto\u010dne menia! Pri rie\u0161en\u00ed \u00falohy preto vych\u00e1dzaj len z \u00fadajov uveden\u00fdch v \u00falohe.)<\/em><\/p>\n

(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n


<\/p>\n

Z5-I-4<\/strong>
\n Klasick\u00e1 hracia kocka sa kot\u00fa\u013eala nazna\u010den\u00fdm smerom po pl\u00e1ne na obr\u00e1zku. Pri jej pohybe na ka\u017edom pol\u00ed\u010dku ostali otla\u010den\u00e9 bodky zo steny, ktorou sa pl\u00e1nu dot\u00fdkala. S\u00fa\u010det v\u0161etk\u00fdch bodiek otla\u010den\u00fdch na pl\u00e1ne bol 23. Ko\u013eko bodiek bolo otla\u010den\u00fdch na zafarbenom pol\u00ed\u010dku?<\/p>\n

\"\"<\/div>\n

(Klasick\u00e1 hracia kocka m\u00e1 na sten\u00e1ch bodky 1, 2, …, 6 umiestnen\u00e9 tak, \u017ee s\u00fa\u010det po\u010dtu bodiek na proti\u013eahl\u00fdch sten\u00e1ch je 7. Pl\u00e1n pozost\u00e1va zo \u0161tvorcov, ktor\u00e9 s\u00fa rovnako ve\u013ek\u00e9 ako steny kocky.)<\/em><\/p>\n

(M. Dillingerov\u00e1)<\/div>\n


<\/p>\n

Z5-I-5<\/strong>
\n Digit\u00e1lne hodiny ukazuj\u00fa hodiny a min\u00faty, napr\u00edklad 14:37<\/strong>. Ak\u00fa dobu (v min\u00fatach) svieti za 24 hod\u00edn na t\u00fdchto hodin\u00e1ch aspo\u0148 jedna p\u00e4\u0165ka?<\/p>\n

(M. Volfov\u00e1)<\/div>\n


<\/p>\n

Z5-I-6<\/strong>
\n Danko si zo \u0161tvor\u010dekovej siete vystrihol \u00fatvar ako na obr\u00e1zku:<\/p>\n

\"\"<\/p>\n

Odstrihni dva \u0161tvor\u010deky siete tak, aby sa v\u00fdsledn\u00fd \u00fatvar nerozpadol a aby mal \u010do najv\u00e4\u010d\u0161\u00ed obvod. N\u00e1jdi dve rie\u0161enia.<\/p>\n

(M. Dillingerov\u00e1)<\/div>\n


<\/p>\n\n\n
\n
\n<\/td>\n		<\/a>\"\"<\/a><\/td>\n<\/tr>\n<\/table>\n

Z5-II-1<\/strong>
\n Mirko si zo \u0161tvorcovej siete s vp\u00edsan\u00fdmi \u010d\u00edslami vystrihol \u00fatvar na obr\u00e1zku:<\/p>\n

\"\"<\/p>\n

Odstrihni dva \u0161tvor\u010deky \u00fatvaru tak, aby sa v\u00fdsledn\u00fd \u00fatvar nerozpadol, aby po odstrihnut\u00ed oboch \u0161tvor\u010dekov mal rovnak\u00fd obvod ako p\u00f4vodne a aby s\u00fa\u010det vp\u00edsan\u00fdch \u010d\u00edsel bol najmen\u0161\u00ed mo\u017en\u00fd.<\/p>\n

(M. Petrov\u00e1, M. Dillingerov\u00e1)<\/div>\n


<\/p>\n

Z5-II-2<\/strong>
\n P\u00e4tn\u00e1s\u0165 rovnak\u00fdch na sebe polo\u017een\u00fdch listov papiera som naraz prelo\u017eil napoly. Z\u00edskal som tak „zo\u0161it“, ktor\u00e9ho str\u00e1nky som o\u010d\u00edsloval po porad\u00ed \u010d\u00edslami 1 a\u017e 60. Ktor\u00e9 \u010fal\u0161ie tri \u010d\u00edsla s\u00fa nap\u00edsan\u00e9 na tom istom liste papiera ako \u010d\u00edslo 25?<\/p>\n

(L. \u0160im\u016fnek)<\/div>\n


<\/p>\n

Z5-II-3<\/strong>
\n Fero V\u0161\u00edmav\u00fd si opakoval mal\u00fa n\u00e1sobilku jednotliv\u00fdch \u010d\u00edsel t\u00fdm, \u017ee vypisoval jej v\u00fdsledky za sebou bez medzier a \u010diarok. Napr\u00edklad u n\u00e1sobilky \u010d\u00edsla 2 by mal nap\u00edsan\u00e9 2468101214161820. Do jedn\u00e9ho riadku takto zap\u00edsal n\u00e1sobilku 3, za \u0148ou ihne\u010f n\u00e1sobilku 5 a nakoniec 9. Potom si tento riadok prezrel a zistil, \u017ee sa v \u0148om objavuj\u00fa zrkadlov\u00e9 \u010d\u00edsla. (Zrkadlov\u00e9 \u010d\u00edslo m\u00e1 aspo\u0148 3 \u010d\u00edslice a \u010d\u00edta sa zozadu rovnako ako spredu, napr\u00edklad: 272, 3553, 98089.) Nap\u00ed\u0161 3 najmen\u0161ie zrkadlov\u00e9 \u010d\u00edsla a jedno najv\u00e4\u010d\u0161ie zrkadlov\u00e9 \u010d\u00edslo z Ferovho riadka.<\/p>\n

(L. Hozov\u00e1)<\/div>\n


<\/p>\n\n\n
\n
\n<\/td>\n		<\/a>\"\"<\/a><\/td>\n<\/tr>\n<\/table>\n

Z6-I-1<\/strong>
\n Na obr\u00e1zku je \u0161tvorcov\u00e1 sie\u0165, ktorej \u0161tvorce maj\u00fa stranu d\u013a\u017eky 1\u00a0cm. V sieti je zakreslen\u00fd obrazec vyfarben\u00fd \u0161edou. Libor m\u00e1 narysova\u0165 priamku, ktor\u00e1 je rovnobe\u017en\u00e1 s priamku MO<\/em> a rozde\u013euje \u0161ed\u00fd obrazec na dve \u010dasti s rovnak\u00fdm obsahom. V akej vzdialenosti od priamky MO<\/em> povedie Libor t\u00fato rovnobe\u017eku?<\/p>\n

\"\"<\/p>\n

(L. \u0160im\u016fnek)<\/div>\n


<\/p>\n

Z6-I-2<\/strong>
\n Do pr\u00e1zdnych pol\u00ed vp\u00ed\u0161 \u010d\u00edsla 2, 4, 6, 8, 12, 14 a 21 tak, aby ka\u017ed\u00e9 tri \u010d\u00edsla nap\u00edsan\u00e9 na jednej \u00fase\u010dke d\u00e1vali v\u017edy rovnak\u00fd s\u00fa\u010din. Nap\u00ed\u0161 svoj postup.<\/p>\n

\"\"<\/div>\n
(L. \u0160im\u016fnek)<\/div>\n


<\/p>\n

Z6-I-3<\/strong>
\n B\u00e9-banka vyd\u00e1va bankomatov\u00e9 karty so \u0161tvormiestnym PIN k\u00f3dom, ktor\u00fd neobsahuje \u010d\u00edslicu\u00a00. P\u00e1n Skler\u00f3za sa b\u00e1l, \u017ee zabudne PIN k\u00f3d svojej karty, preto si ho nap\u00edsal priamo na kartu. Aby to v\u0161ak pr\u00edpadn\u00fd zlodej nemal tak\u00e9 \u013eahk\u00e9, nap\u00edsal si ho tam r\u00edmskymi \u010d\u00edslicami: IIIVIIIXIV. Svoj n\u00e1pad prezradil najlep\u0161iemu priate\u013eovi, p\u00e1novi Odkukalovi. Tomu sa n\u00e1pad tak zap\u00e1\u010dil, \u017ee spravil so svoj\u00edm PIN k\u00f3dom to ist\u00e9 a na kartu si spr\u00e1vne zap\u00edsal: IVIIIVI. Na svoje ve\u013ek\u00e9 prekvapenie v\u0161ak z r\u00edmskeho z\u00e1pisu nevedel svoj PIN k\u00f3d presne ur\u010di\u0165!<\/p>\n

a)<\/strong> Ak\u00fd PIN k\u00f3d m\u00e1 karta p\u00e1na Skler\u00f3zu?
\n b)<\/strong> Ak\u00fd PIN k\u00f3d m\u00f4\u017ee ma\u0165 karta p\u00e1na Odkukala?<\/p>\n

(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n


<\/p>\n

Z6-I-4<\/strong>
\n Na\u010drtni v\u0161etky mo\u017en\u00e9 tvarovo r\u00f4zne \u0161tvoruholn\u00edky, ktor\u00e9 maj\u00fa vrcholy vo vrcholoch dan\u00e9ho pravideln\u00e9ho \u0161es\u0165uholn\u00edka. Ak\u00e9 by boli ich obsahy, keby \u0161es\u0165uholn\u00edk mal obsah 156 cm2<\/sup>?<\/p>\n

(M. Volfov\u00e1)<\/div>\n


<\/p>\n

Z6-I-5<\/strong>
\n Pani Ku\u010derov\u00e1 bola na sedemdennej dovolenke a Katka jej po cel\u00fd ten \u010das ven\u010dila psa a k\u0155mila kr\u00e1liky. Dostala za to ve\u013ek\u00fa tortu a 700\u00a0Sk. Po \u010fal\u0161ej dovolenke pod\u013ea rovnak\u00fdch pravidiel dostala Katka za \u0161tyri dni ven\u010denia a k\u0155menia rovnak\u00fa tortu a 340\u00a0Sk. Ak\u00fa cenu mala torta?<\/p>\n

(M. Volfov\u00e1)<\/div>\n


<\/p>\n

Z6-I-6<\/strong>
\n Na ka\u017ed\u00fa stenu kocky sme nap\u00edsali in\u00e9 prvo\u010d\u00edslo men\u0161ie ako\u00a020 tak, aby s\u00fa\u010dty dvoch \u010d\u00edsel na proti\u013eahl\u00fdch sten\u00e1ch boli v\u017edy rovnak\u00e9. Kocku sme polo\u017eili na prv\u00e9 pol\u00ed\u010dko pl\u00e1nu na obr\u00e1zku najmen\u0161\u00edm \u010d\u00edslom nadol a potom sme ju kot\u00fa\u013eali nazna\u010den\u00fdm smerom po pl\u00e1ne. Pri ka\u017edom dotyku kocky s pl\u00e1nom sme na pol\u00ed\u010dko pl\u00e1nu nap\u00edsali \u010d\u00edslo, ktor\u00fdm sa ho kocka dotkla. Ktor\u00fdm \u010d\u00edslom sa kocka dotkla zafarben\u00e9ho pol\u00ed\u010dka pl\u00e1nu, ak s\u00fa\u010det v\u0161etk\u00fdch nap\u00edsan\u00fdch \u010d\u00edsel bol najmen\u0161\u00ed mo\u017en\u00fd?<\/p>\n

\"\"<\/p>\n

(Pl\u00e1n pozost\u00e1va zo \u0161tvorcov, ktor\u00e9 s\u00fa rovnako ve\u013ek\u00e9 ako steny kocky.)<\/em><\/p>\n

(M. Dillingerov\u00e1)<\/div>\n


<\/p>\n\n\n
\n
\n<\/td>\n		<\/a>\"\"<\/a><\/td>\n<\/tr>\n<\/table>\n

Z6-II-1<\/strong>
\n Katka chce obdarova\u0165 svoje kamar\u00e1tky a rozm\u00fd\u0161\u013ea: keby som ka\u017edej k\u00fapila sponku za 2,80 \u20ac, ostalo by mi e\u0161te 2,90 \u20ac, ale keby to bol medved\u00edk za 4,20 \u20ac, tak by mi e\u0161te 1,30 \u20ac ch\u00fdbalo. Ko\u013eko m\u00e1 Katka kamar\u00e1tok a ko\u013eko pe\u0148az\u00ed na dar\u010deky?<\/p>\n

(M. Volfov\u00e1)<\/div>\n


<\/p>\n

Z6-II-2<\/strong><\/p>\n

Na kocku sme na ka\u017ed\u00fa stenu nap\u00edsali prvo\u010d\u00edslo men\u0161ie ako\u00a020. Potom sme zistili, \u017ee s\u00fa\u010det ka\u017ed\u00fdch dvoch \u010d\u00edsel le\u017eiacich na proti\u013eahl\u00fdch sten\u00e1ch je v\u017edy\u00a0rovnak\u00fd. Kocku sme polo\u017eili na prv\u00e9 pol\u00ed\u010dko pl\u00e1nu na obr\u00e1zku stenou s\u00a0najmen\u0161\u00edm \u010d\u00edslom nadol. Potom sme ju kot\u00fa\u013eali nazna\u010den\u00fdm smerom po pl\u00e1ne. Pri ka\u017edom dotyku kocky s\u00a0pl\u00e1nom sme na pol\u00ed\u010dko pl\u00e1nu nap\u00edsali \u010d\u00edslo, ktor\u00fdm sa ho kocka dotkla. Ktor\u00fdm \u010d\u00edslom sa kocka mohla dotkn\u00fa\u0165 zafarben\u00e9ho pol\u00ed\u010dka pl\u00e1nu, ak s\u00fa\u010det v\u0161etk\u00fdch nap\u00edsan\u00fdch \u010d\u00edsel bol<\/p>\n

1.<\/strong> najmen\u0161\u00ed mo\u017en\u00fd,
\n 2.<\/strong> najv\u00e4\u010d\u0161\u00ed mo\u017en\u00fd?<\/p>\n

\"\"<\/div>\n

(Pl\u00e1n pozost\u00e1va zo \u0161tvorcov, ktor\u00e9 s\u00fa rovnako ve\u013ek\u00e9 ako steny kocky.)<\/em><\/p>\n

(M. Dillingerov\u00e1, L. Hozov\u00e1)<\/div>\n


<\/p>\n

Z6-II-3<\/strong>
\n Traja z\u00e1hradn\u00edci mali ve\u013ek\u00fa \u00farodu mrkvy a tak sk\u00fasili mrkvu od\u0161\u0165avova\u0165. Po od\u0161\u0165aven\u00ed v\u0161etku z\u00edskan\u00fa \u0161\u0165avu naliali do 9 poh\u00e1rov. V\u0161etky boli pln\u00e9, ka\u017ed\u00fd v\u0161ak mal in\u00fd objem: 1 dl, 2 dl, 3 dl, …, 9 dl. Chceli sa spravodlivo podeli\u0165 tak, aby ka\u017ed\u00fd dostal rovnak\u00fd po\u010det poh\u00e1rov i rovnako ve\u013ea \u0161\u0165avy. Ako to mohli urobi\u0165?<\/p>\n

(M. Volfov\u00e1)<\/div>\n


<\/p>\n\n\n
\n
\n<\/td>\n		<\/a>\"\"<\/a><\/td>\n<\/tr>\n<\/table>\n

Z7-I-1<\/strong><\/p>\n

Na ka\u017ed\u00fa stenu kocky sme nap\u00edsali in\u00e9 prvo\u010d\u00edslo men\u0161ie ako 20 tak, aby s\u00fa\u010det dvoch \u010d\u00edsel na proti\u013eahl\u00fdch sten\u00e1ch bol v\u017edy rovnak\u00fd. Kocku sme polo\u017eili na prv\u00e9 pol\u00ed\u010dko pl\u00e1nu na obr\u00e1zku najv\u00e4\u010d\u0161\u00edm \u010d\u00edslom nadol a potom sme ju kot\u00fa\u013eali nazna\u010den\u00fdm smerom po pl\u00e1ne. Pri ka\u017edom dotyku kocky s pl\u00e1nom sme na pol\u00ed\u010dko pl\u00e1nu nap\u00edsali \u010d\u00edslo, ktor\u00fdm sa ho kocka dotkla. Ktor\u00fdm \u010d\u00edslom sa kocka dotkla zafarben\u00e9ho pol\u00ed\u010dka, ak s\u00fa\u010det v\u0161etk\u00fdch nap\u00edsan\u00fdch \u010d\u00edsel bol najv\u00e4\u010d\u0161\u00ed mo\u017en\u00fd?<\/p>\n

\"\"<\/p>\n

(Pl\u00e1n pozost\u00e1va zo \u0161tvorcov, ktor\u00e9 s\u00fa rovnako ve\u013ek\u00e9 ako steny kocky.)<\/em><\/p>\n

(M. Dillingerov\u00e1)<\/div>\n


<\/p>\n

Z7-I-2<\/strong><\/p>\n

Na obr\u00e1zku je \u0161tvorcov\u00e1 sie\u0165, ktorej \u0161tvorce maj\u00fa stranu d\u013a\u017eky 1\u00a0cm. V sieti je zakreslen\u00fd obrazec vyfarben\u00fd \u0161edou. Libor m\u00e1 narysova\u0165 priamku, ktor\u00e1 je rovnobe\u017en\u00e1 s priamku MO<\/em> a rozde\u013euje \u0161ed\u00fd obrazec na dve \u010dasti s rovnak\u00fdm obsahom. V akej vzdialenosti od priamky MO<\/em> povedie Libor t\u00fato rovnobe\u017eku?<\/p>\n

\"\"<\/p>\n

(L. \u0160im\u016fnek)<\/div>\n


<\/p>\n

Z7-I-3<\/strong><\/p>\n

Turisti pl\u00e1novali dlh\u00fa t\u00faru na tri dni tak, \u017ee ka\u017ed\u00fd de\u0148 prejd\u00fa tretinu celej trasy. To v\u0161ak dodr\u017eali iba prv\u00fd de\u0148. Druh\u00fd de\u0148 pre\u0161li iba tretinu zvy\u0161ku cesty, a tret\u00ed de\u0148, unaven\u00ed, len \u0161tvrtinu zvy\u0161ku. Posledn\u00fdch 24\u00a0km do cie\u013ea ich doviezlo ter\u00e9nne auto. Ak\u00e1 dlh\u00e1 mala by\u0165 cel\u00e1 t\u00fara a ko\u013eko kilometrov pre\u0161li (pe\u0161o) prv\u00fd, druh\u00fd a tret\u00ed de\u0148?<\/p>\n

(M. Volfov\u00e1)<\/div>\n


<\/p>\n

Z7-I-4<\/strong><\/p>\n

P\u00e1n Hor\u00e1k je o\u00a03\u00a0roky star\u0161\u00ed ako jeho \u017eena a ich prvoroden\u00fd syn je o\u00a04\u00a0roky star\u0161\u00ed ako ich druhoroden\u00fd. V\u0161etci \u0161tyria oslavuj\u00fa narodeniny v ten ist\u00fd de\u0148 a teraz maj\u00fa spolu 81\u00a0rokov. Pred 5\u00a0rokmi mali spolu 62\u00a0rokov. Ur\u010di dne\u0161n\u00fd vek rodi\u010dov aj oboch synov.<\/p>\n

(M. Volfov\u00e1)<\/div>\n


<\/p>\n

Z7-I-5<\/strong><\/p>\n

Zuzka nap\u00edsala p\u00e4\u0165cifern\u00e9 \u010d\u00edslo. Ke\u010f prip\u00edsala jednotku na koniec tohto \u010d\u00edsla, dostala \u010d\u00edslo, ktor\u00e9 je trikr\u00e1t v\u00e4\u010d\u0161ie ako \u010d\u00edslo, ktor\u00e9 by dostala, keby nap\u00edsala jednotku pred p\u00f4vodn\u00e9 \u010d\u00edslo. Ktor\u00e9 p\u00e4\u0165cifern\u00e9 \u010d\u00edslo Zuzka nap\u00edsala?<\/p>\n

(L. Hozov\u00e1)<\/div>\n


<\/p>\n

Z7-I-6<\/strong>
\n Dan\u00fd je obd\u013a\u017enik ABCD<\/em>. Bodom A<\/em> vedieme priamku, ktor\u00e1 pretne \u00fase\u010dku CD<\/em> v bode X<\/em> tak, \u017ee pre obsahy vzniknut\u00fdch \u00fatvarov plat\u00ed SAXD<\/em><\/sub> :\u00a0SABCX<\/em><\/sub> =\u00a01:2<\/strong>. Bodom X<\/em> vedieme priamku, ktor\u00e1 pretne \u00fase\u010dku AB<\/em> v bode Y<\/em> tak, \u017ee plat\u00ed SAXY<\/em><\/sub> :\u00a0SYBCX<\/em><\/sub> =\u00a01:2<\/strong>. Bodom Y<\/em> op\u00e4\u0165 vedieme priamku, ktor\u00e1 pretne \u00fase\u010dku XC<\/em> v bode Z<\/em> tak, \u017ee plat\u00ed SXYZ<\/em><\/sub> :\u00a0SYBCZ<\/em><\/sub> =\u00a01:2<\/strong>. Vypo\u010d\u00edtaj pomer obsahov SAXD<\/em><\/sub> :\u00a0SAXZY<\/em><\/sub><\/strong>.<\/p>\n

(M. Dillingerov\u00e1)<\/div>\n


<\/p>\n\n\n
\n
\n<\/td>\n		<\/a>\"\"<\/a><\/td>\n<\/tr>\n<\/table>\n

Z7-II-1<\/strong>
\n Rado \u010d\u00edta zauj\u00edmav\u00fa kni\u017eku. V\u010dera pre\u010d\u00edtal 15 str\u00e1n, dnes \u010fal\u0161\u00edch 12 str\u00e1n. S \u00fadivom si uvedomil, \u017ee s\u00fa\u010det \u010d\u00edsel str\u00e1n, ktor\u00e9 pre\u010d\u00edtal v\u010dera, je rovnak\u00fd ako s\u00fa\u010det \u010d\u00edsel str\u00e1n, ktor\u00e9 pre\u010d\u00edtal dnes. Zisti \u010d\u00edslo na str\u00e1nke, ktorou za\u010dne najbli\u017e\u0161ie \u010d\u00edtanie?
\n(Rado pri \u010d\u00edtan\u00ed \u017eiadne str\u00e1nky nepreskakuje ani ne\u010d\u00edta \u017eiadnu str\u00e1nku dva a viackr\u00e1t. Ka\u017edodenn\u00e9 \u010d\u00edtanie nikdy neskon\u010d\u00ed roz\u010d\u00edtanou str\u00e1nkou.)<\/em><\/p>\n

(M. Petrov\u00e1)<\/div>\n<\/p>\n

Z7-II-2<\/strong>
\n Tajn\u00fd agent sa sna\u017e\u00ed rozl\u00fa\u0161ti\u0165 pr\u00edstupov\u00fd k\u00f3d. Zatia\u013e z\u00edskal tieto inform\u00e1cie:
\n \u2022 je to \u0161tvorcifern\u00e9 \u010d\u00edslo,
\n \u2022 nie je delite\u013en\u00e9 siedmimi,
\n \u2022 \u010d\u00edslica na mieste desiatok je s\u00fa\u010dtom \u010d\u00edslice na mieste jednotiek a \u010d\u00edslice na mieste stoviek,
\n \u2022 \u010d\u00edslo vytvoren\u00e9 z prv\u00fdch dvoch \u010d\u00edslic k\u00f3du (v tomto porad\u00ed) je p\u00e4tn\u00e1s\u0165n\u00e1sobkom poslednej \u010d\u00edslice k\u00f3du,
\n \u2022 prv\u00e1 a posledn\u00e1 \u010d\u00edslica k\u00f3du (v tomto porad\u00ed) tvoria prvo\u010d\u00edslo.<\/p>\n

Sta\u010dia mu tieto inform\u00e1cie na rozl\u00fa\u0161tenie k\u00f3du? Svoj z\u00e1ver zd\u00f4vodni.<\/p>\n

(M. Petrov\u00e1)<\/div>\n<\/p>\n

Z7-II-3<\/strong>
\n Na obr\u00e1zku je \u0161tvorec ABCD<\/em> so stranou d\u013a\u017eky 10 cm a men\u0161\u00ed \u0161tvorec EFGH<\/em>, ktor\u00e9ho vrcholy E, F, G, H<\/em> le\u017eia na uhloprie\u010dkach AC, BD<\/em> \u0161tvorca ABCD<\/em>. Plocha, ktor\u00e1 le\u017e\u00ed vn\u00fatri \u0161tvorca ABCD<\/em> a pritom mimo \u0161tvorca EFGH<\/em>, je \u0161edo vyzna\u010den\u00e1. Priamka p<\/em>, ktor\u00e1 je rovnobe\u017en\u00e1 s AB<\/em>, od AB<\/em> vzdialen\u00e1 6,5 cm a prech\u00e1dza \u0161tvorcom EFGH<\/em>, rozde\u013euje \u0161ed\u00fa plochu na dve \u010dasti. Obsah jednej tejto \u010dasti je o 13,8 cm2<\/sup> v\u00e4\u010d\u0161\u00ed ne\u017e obsah druhej. Vypo\u010d\u00edtajte d\u013a\u017eku strany EF<\/em>.<\/p>\n

\"\"<\/div>\n
(L. \u0160im\u016fnek)<\/div>\n<\/p>\n\n\n
\n
\n<\/td>\n		<\/a>\"\"<\/a><\/td>\n<\/tr>\n<\/table>\n

Z8-I-1<\/strong><\/p>\n

Mysl\u00edm si nez\u00e1porn\u00e9 \u010d\u00edslo v tvare zlomku s menovate\u013eom 12 a celo\u010d\u00edseln\u00fdm \u010ditate\u013eom. Ke\u010f ho nap\u00ed\u0161em v tvare desatinn\u00e9ho \u010d\u00edsla, bude ma\u0165 pred aj za desatinnou \u010diarkou po jednej platnej \u010d\u00edslici, ktor\u00e9 bud\u00fa obe nenulov\u00e9. \u010c\u00edsel s takouto vlastnos\u0165ou je viac. Ak ich zorad\u00edme od najmen\u0161ieho po najv\u00e4\u010d\u0161ie, to „moje“ bude predposledn\u00e9. Ak\u00e9 \u010d\u00edslo si mysl\u00edm?<\/p>\n

(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n


<\/p>\n

Z8-I-2<\/strong><\/p>\n

Na ka\u017ed\u00fa stenu kocky sme nap\u00edsali in\u00e9 prvo\u010d\u00edslo men\u0161ie ako 20 tak, aby s\u00fa\u010dty dvoch \u010d\u00edsel na proti\u013eahl\u00fdch sten\u00e1ch boli v\u017edy rovnak\u00e9. Kocku sme polo\u017eili na prv\u00e9 pol\u00ed\u010dko pl\u00e1nu na obr\u00e1zku a potom sme ju kot\u00fa\u013eali nazna\u010den\u00fdm smerom po pl\u00e1ne. Pri ka\u017edom dotyku kocky s pl\u00e1nom sme na odpovedaj\u00face pol\u00ed\u010dko pl\u00e1nu nap\u00edsali \u010d\u00edslo, ktor\u00fdm sa ho kocka dotkla. Ktor\u00fdm svoj\u00edm \u010d\u00edslom sa kocka pl\u00e1nu nedotkla, ak s\u00fa\u010det v\u0161etk\u00fdch nap\u00edsan\u00fdch \u010d\u00edsel bol 86?<\/p>\n

\"\"<\/p>\n

(Pl\u00e1n pozost\u00e1va zo \u0161tvorcov, ktor\u00e9 s\u00fa rovnako ve\u013ek\u00e9 ako steny kocky.)<\/em><\/p>\n

(M. Dillingerov\u00e1)<\/div>\n<\/p>\n

Z8-I-3<\/strong><\/p>\n

Grafik v redakcii nov\u00edn dostal dva obr\u00e1zky, aby ich umiestnil k \u010dl\u00e1nku. Prv\u00fd origin\u00e1l bol 13\u00a0cm \u0161irok\u00fd a 9\u00a0cm vysok\u00fd, druh\u00fd mal \u0161\u00edrku 14\u00a0cm a v\u00fd\u0161ku 12\u00a0cm. Grafik sa rozhodol umiestni\u0165 obr\u00e1zky na str\u00e1nku ved\u013ea seba tak, aby sa dot\u00fdkali a aby oba mali rovnak\u00fa v\u00fd\u0161ku. Po vytla\u010den\u00ed nov\u00edn mali obr\u00e1zky spolu \u0161\u00edrku 18,8\u00a0cm. Obr\u00e1zky teda vhodne zmen\u0161il bez toho, aby ich akoko\u013evek orez\u00e1val. Ak\u00e1 bola v\u00fd\u0161ka obr\u00e1zkov vo vytla\u010den\u00fdch novin\u00e1ch?<\/p>\n

(L. \u0160im\u016fnek)<\/div>\n


<\/p>\n

Z8-I-4<\/strong><\/p>\n

Dan\u00e9 s\u00fa tri vz\u00e1jomne r\u00f4zne nenulov\u00e9 \u010d\u00edslice. Na tabu\u013eu sme nap\u00edsali v\u0161etky r\u00f4zne trojcifern\u00e9 \u010d\u00edsla, ktor\u00e9 sa daj\u00fa z uveden\u00fdch \u010d\u00edslic vytvori\u0165, pri\u010dom pre ka\u017ed\u00e9 \u010d\u00edslo sme pou\u017eili v\u0161etky tri \u010d\u00edslice. S\u00fa\u010det \u010d\u00edsel nap\u00edsan\u00fdch na tabuli je 1776. Ktor\u00e9 tri \u010d\u00edslice boli dan\u00e9? N\u00e1jdite v\u0161etky rie\u0161enia.<\/p>\n

(L. \u0160im\u016fnek)<\/div>\n<\/p>\n

Z8-I-5<\/strong><\/p>\n

Na ve\u017ei radnice s\u00fa hodiny, ktor\u00e9 maj\u00fa bl\u00edzko stredu cifern\u00edku dvierka pou\u017e\u00edvan\u00e9 pri \u00fadr\u017ebe. Dvierka sa otv\u00e1raj\u00fa von, \u010do je nepraktick\u00e9 \u2013 napr\u00edklad presne o\u00a012:09 ve\u013ek\u00e1 ru\u010di\u010dka zakryje dvierka, ktor\u00e9 sa preto nedaj\u00fa otvori\u0165. Najsk\u00f4r sa dvierka op\u00e4\u0165 daj\u00fa otvori\u0165 presne o\u00a012:21. Ko\u013eko min\u00fat denne sa dvierka nedaj\u00fa otvori\u0165?
\n (Nezabudnite, \u017ee dvierka m\u00f4\u017ee zakry\u0165 aj mal\u00e1 ru\u010di\u010dka; cel\u00e9 dvierka le\u017eia v kruhu, ktor\u00fd t\u00e1to ru\u010di\u010dka opisuje.)<\/em><\/p>\n

\"\"<\/p>\n

(L. \u0160im\u016fnek)<\/div>\n


<\/p>\n

Z8-I-6<\/strong><\/p>\n

Do kv\u00e1dra ABCDEFGH<\/em> je zakreslen\u00fd hranol s vrcholmi PQRSTUVX<\/em>, ktor\u00e9ho vrcholy s\u00fa stredy hr\u00e1n kv\u00e1dra (pozri n\u00e1\u010drtok). Vypo\u010d\u00edtajte objem a povrch hranola, ak viete, \u017ee: |AB<\/em>|\u00a0=\u00a08\u00a0cm, |BC<\/em>|\u00a0=\u00a06\u00a0cm, |BF<\/em>|\u00a0=\u00a04\u00a0cm.<\/p>\n

\"\"<\/p>\n

(M. Krej\u010dov\u00e1)<\/div>\n


<\/p>\n\n\n
\n
\n<\/td>\n		<\/a>\"\"<\/a><\/td>\n<\/tr>\n<\/table>\n

Z8-II-1<\/strong>
\nPri lese, ktor\u00fd mal tvar rovnoramenn\u00e9ho trojuholn\u00edka, sa u jedn\u00e9ho z jeho vrcholov ut\u00e1borili Ivo a Peter. Uprostred proti\u013eahlej strany bola studni\u010dka. Chlapci sa rozhodli, \u017ee k nej nep\u00f4jdu lesom, ale po jeho obvode. Ka\u017ed\u00fd vy\u0161iel in\u00fdm smerom, ale obaja r\u00fdchlos\u0165ou 4 km\/h. Ivo dorazil k studni\u010dke za 15 min\u00fat, Peter za 12 min\u00fat. Zisti d\u013a\u017eky str\u00e1n trojuholn\u00edka lesa.
\n(D\u013a\u017eky str\u00e1n zaokr\u00fahlite na cel\u00e9 metre.)<\/em><\/p>\n

(M. Volfov\u00e1)<\/div>\n<\/p>\n

Z8-II-2<\/strong>
\nEva p\u00edsala za sebou id\u00face prirodzen\u00e9 \u010d\u00edsla: 1234567891011… Ktor\u00e1 \u010d\u00edslica by bola zap\u00edsan\u00e1 na 2 009-tom mieste Evinho \u010d\u00edsla?<\/p>\n

(M. Volfov\u00e1)<\/div>\n<\/p>\n

Z8-II-3<\/strong>
\nTri dan\u00e9 prirodzen\u00e9 \u010d\u00edsla s\u00fa zoraden\u00e9 pod\u013ea ve\u013ekosti. Ur\u010dite tieto \u010d\u00edsla na z\u00e1klade nasleduj\u00facich inform\u00e1ci\u00ed:
\n \u2022 priemer dan\u00fdch troch \u010d\u00edsel je rovn\u00fd prostredn\u00e9mu z nich,
\n \u2022 rozdiel niektor\u00fdch dvoch dan\u00fdch \u010d\u00edsel je 321,
\n \u2022 s\u00fa\u010det niektor\u00fdch dvoch dan\u00fdch \u010d\u00edsel je 777.<\/p>\n

(L. \u0160im\u016fnek)<\/div>\n<\/p>\n\n\n
\n
\n<\/td>\n		<\/a>\"\"<\/a><\/td>\n<\/tr>\n<\/table>\n

Z9-I-1<\/strong>
\nDo troch pr\u00e1zdnych kr\u00fa\u017ekov na obr\u00e1zku patria tak\u00e9 prirodzen\u00e9 \u010d\u00edsla, aby s\u00fa\u010din troch \u010d\u00edsel na ka\u017edej strane trojuholn\u00edka bol rovnak\u00fd. Ktor\u00e9 najmen\u0161ie a ktor\u00e9 najv\u00e4\u010d\u0161ie \u010d\u00edslo m\u00f4\u017ee by\u0165 za tejto podmienky vp\u00edsan\u00e9 v \u0161edo vyfarbenom kr\u00fa\u017eku?<\/p>\n

\"\"<\/p>\n

(L. \u0160im\u016fnek)<\/div>\n<\/p>\n

Z9-I-2<\/strong>
\nAlena, Barbora, Cyril a D\u00e1vid si spolo\u010dne k\u00fapili tandem \u2013 bicykel pre dvoch. Na vych\u00e1dzky na tandeme vyr\u00e1\u017eaj\u00fa v\u017edy v dvojici. Ka\u017ed\u00fd bol s ka\u017ed\u00fdm u\u017e aspo\u0148 raz a nikto in\u00fd sa na tandeme e\u0161te neviezol. Alena bola na vych\u00e1dzke na tandeme jeden\u00e1s\u0165kr\u00e1t, Barbora dvadsa\u0165kr\u00e1t, Cyril iba \u0161tyrikr\u00e1t. Ur\u010dte, ko\u013ekokr\u00e1t minim\u00e1lne a ko\u013ekokr\u00e1t maxim\u00e1lne mohol by\u0165 na vych\u00e1dzke na tandeme D\u00e1vid.<\/p>\n

(L. \u0160im\u016fnek)<\/div>\n<\/p>\n

Z9-I-3<\/strong>
\nUr\u010dte obsah vyfarbenej plochy \u0161tvorcovej siete so stranou 10\u00a0cm, v ktorej je narysovan\u00e1 kru\u017enica s polomerom 20\u00a0cm a so stredom S<\/em> vo vyzna\u010denom mre\u017eovom bode. Body A<\/em>, B<\/em> s\u00fa priese\u010dn\u00edky kru\u017enice so sie\u0165ov\u00fdmi priamkami.<\/p>\n

\"\"<\/p>\n

(L. \u0160im\u016fnek)<\/div>\n<\/p>\n

Z9-I-4<\/strong>
\nDominik si vyrobil „prvo\u010d\u00edseln\u00e9 domino“<\/em> \u2013 ka\u017ed\u00e1 kocka domina mala na sebe jedno dvojcifern\u00e9 prvo\u010d\u00edslo tak, \u017ee na ka\u017edej polovici kocky bola jedna \u010d\u00edslica tohto prvo\u010d\u00edsla. \u017diadne dvojcifern\u00e9 prvo\u010d\u00edslo v domine nech\u00fdbalo, \u017eiadne prvo\u010d\u00edslo nebolo na dvoch kock\u00e1ch. Dominik sa rozhodol, \u017ee v\u0161etky kocky usporiada do kru\u017enice tak, aby kocky le\u017eiace ved\u013ea seba susedili rovnakou \u010d\u00edslicou (pozri obr\u00e1zok). Jeho kamar\u00e1t Filip mu v\u0161ak povedal, \u017ee to nie je mo\u017en\u00e9. Kto mal pravdu? Pre\u010do?<\/p>\n

\"\"<\/p>\n

(M. Petrov\u00e1)<\/div>\n<\/p>\n

Z9-I-5<\/strong>
\nNa stole s kruhovou doskou o priemere 0,6\u00a0m je „nakrivo“<\/em> polo\u017een\u00fd \u0161tvorcov\u00fd obrus so stranou 1\u00a0m \u2013 jeho stred je vzh\u013eadom na stred dosky posunut\u00fd. Jeden c\u00edp obrusu pre\u010dnieva cez hranu dosky stola 0,5\u00a0m, susedn\u00fd c\u00edp 0,3\u00a0m. Zistite d\u013a\u017eku presahu zvy\u0161n\u00fdch dvoch c\u00edpov.<\/p>\n

\"\"<\/p>\n

(S. Bedn\u00e1\u0159ov\u00e1)<\/div>\n<\/p>\n

Z9-I-6<\/strong>
\n\u0160tyria otcovia chceli de\u0165om sponzorova\u0165 ly\u017eiarsky z\u00e1jazd.
\n \u2022 Prv\u00fd s\u013e\u00fabil: d\u00e1m 11 500 Sk,
\n \u2022 druh\u00fd s\u013e\u00fabil: d\u00e1m tretinu toho, \u010do vy ostatn\u00ed spolu,
\n \u2022 tret\u00ed s\u013e\u00fabil: ja d\u00e1m \u0161tvrtinu toho, \u010do vy ostatn\u00ed spolu,
\n \u2022 \u0161tvrt\u00fd s\u013e\u00fabil: ja d\u00e1m p\u00e4tinu toho, \u010do vy ostatn\u00ed spolu.<\/p>\n

Ko\u013eko kor\u00fan s\u013e\u00fabil druh\u00fd, tret\u00ed a \u0161tvrt\u00fd otecko?<\/p>\n

(M. Volfov\u00e1)<\/div>\n<\/p>\n\n\n
\n
\n<\/td>\n		<\/a>\"\"<\/a><\/td>\n<\/tr>\n<\/table>\n

Z9-II-1<\/strong>
\nDo troch pr\u00e1zdnych \u0161tvorcov na obr\u00e1zku patria tak\u00e9 prirodzen\u00e9 \u010d\u00edsla, aby s\u00fa\u010det troch \u010d\u00edsel na ka\u017edej strane trojuholn\u00edka bol rovnak\u00fd. Ko\u013eko r\u00f4znych troj\u00edc prirodzen\u00fdch \u010d\u00edsel je mo\u017en\u00e9 do obr\u00e1zka doplni\u0165?<\/p>\n

\"\"<\/div>\n
(L. \u0160im\u016fnek)<\/div>\n<\/p>\n

Z9-II-2<\/strong>
\nNo\u010dn\u00fd str\u00e1\u017enik si na skr\u00e1tenie \u010dasu v slu\u017ebe p\u00edsal postupnos\u0165 \u010d\u00edsel. Za\u010dal ist\u00fdm prirodzen\u00fdm \u010d\u00edslom. Ka\u017ed\u00fd \u010fal\u0161\u00ed \u010dlen postupnosti vytvoril tak, \u017ee k predch\u00e1dzaj\u00facemu \u010d\u00edslu pri\u010d\u00edtal ur\u010dit\u00e9 \u010d\u00edslo: k prv\u00e9mu \u010dlenu pri\u010d\u00edtal\u00a01, k druh\u00e9mu\u00a03, k tretiemu\u00a05, k \u0161tvrt\u00e9mu\u00a01, k piatemu\u00a03, k \u0161iestemu\u00a05, k siedmemu\u00a01 a tak \u010falej. Vieme, \u017ee sa v jeho postupnosti nach\u00e1dzaj\u00fa \u010d\u00edsla 40 a 874.<\/p>\n

a)<\/strong> Ktor\u00e9 \u010d\u00edslo nasleduje v postupnosti priamo po \u010d\u00edsle 40 a ktor\u00e9 priamo po \u010d\u00edsle 874?
\n b)<\/strong> V postupnosti n\u00e1jdeme dva priamo po sebe id\u00face \u010dleny, ktor\u00fdch s\u00fa\u010det je 491.<\/p>\n

Ktor\u00e9 dve \u010d\u00edsla to s\u00fa?<\/p>\n

(L. \u0160im\u016fnek)<\/div>\n<\/p>\n

Z9-II-3<\/strong>
\nVojto Vodn\u00edk sa bavil t\u00fdm, \u017ee prelieval vodu medzi tromi n\u00e1dobami. Najprv prelial po jednej tretine vody z druhej n\u00e1doby do prvej a tretej. Potom prelial po jednej \u0161tvrtine vody z prvej n\u00e1doby do druhej a tretej a nakoniec e\u0161te po jednej p\u00e4tine vody z tretej n\u00e1doby do prvej a druhej n\u00e1doby. Nakoniec zostalo v ka\u017edej n\u00e1dobe po 1\u00a0litre vody. Ko\u013eko vody mal Vojto p\u00f4vodne v jednotliv\u00fdch n\u00e1dob\u00e1ch?<\/p>\n

(M. Petrov\u00e1)<\/div>\n<\/p>\n

Z9-II-4<\/strong>
\nV Koc\u00farkove pl\u00e1novali postavi\u0165 ozdobn\u00fd most ponad rieku. Jeho obl\u00fak (vi\u010f obr\u00e1zok) mal by\u0165 \u010das\u0165ou kru\u017enice. Tento obl\u00fak spolu s vozovkou ohrani\u010duje kruhov\u00fd odsek. V p\u00f4vodnom n\u00e1vrhu bola ale v\u00fd\u0161ka obl\u00faku mosta pr\u00edli\u0161 ve\u013ek\u00e1. Postavili teda most, ktor\u00e9ho v\u00fd\u0161ka obl\u00faku bola trikr\u00e1t men\u0161ia, t\u00fdm sa v\u0161ak polomer pr\u00edslu\u0161nej kru\u017enice dvakr\u00e1t zv\u00e4\u010d\u0161il. V akom pomere bola v\u00fd\u0161ka obl\u00faku mosta a polomer pr\u00edslu\u0161nej kru\u017enice v n\u00e1vrhu a v akom u postaven\u00e9ho mostu?<\/p>\n

\"\"<\/div>\n
(M. Petrov\u00e1)<\/div>\n<\/p>\n\n\n
\n
\n<\/td>\n		<\/a>\"\"<\/a><\/td>\n<\/tr>\n<\/table>\n

Z9-III-1<\/strong>
\nNa na\u0161u \u201ezamy\u0161en\u00fa\u201c chalupu sme priviezli my\u0161ilovca koc\u00fara Viliama. V pondelok chytil 1\/2<\/em> v\u0161etk\u00fdch my\u0161\u00ed, v utorok 1\/3<\/em> zvy\u0161ku, v stredu 1\/4<\/em> t\u00fdch, \u010do zostali po utor\u0148aj\u0161om love, a vo \u0161tvrtok u\u017e len 1\/5<\/em> zvy\u0161ku. V piatok sa zost\u00e1vaj\u00face my\u0161i rad\u0161ej ods\u0165ahovali. Ko\u013eko bolo my\u0161\u00ed na chalupe p\u00f4vodne, ak sa v piatok ods\u0165ahovalo o\u00a02 my\u0161i viac ako ich Viliam v utorok chytil?<\/p>\n

(M. Volfov\u00e1, M. Dillingerov\u00e1)<\/div>\n<\/p>\n

Z9-III-2<\/strong>
\nJuraj, Vojto a Oto na s\u00fa\u0165a\u017ei z\u00edskali v\u0161etky 3 medaily, zlat\u00fa, strieborn\u00fa i bronzov\u00fa. Nechceli sa chv\u00e1li\u0165, tak takto \u017eartovali:
\n \u2022 Juraj: \u201eOto z\u00edskal zlat\u00fa!\u201c
\n \u2022 Vojto: \u201eAle nie, Oto z\u00edskal strieborn\u00fa!\u201c
\n \u2022 Oto: \u201eNedostal som ani zlat\u00fa ani strieborn\u00fa!\u201c<\/p>\n

Tr\u00e9ner prezradil, \u017ee nosite\u013e zlatej medaily hovoril pravdu a nosite\u013e bronzovej klamal. Zistite, kto ktor\u00fa medailu z\u00edskal.<\/p>\n

(M. Volfov\u00e1)<\/div>\n<\/p>\n

Z9-III-3<\/strong>
\nV \u0161tvorcovej sieti, ktorej \u0161tvorce maj\u00fa stranu d\u013a\u017eky 10cm, s\u00fa narysovan\u00e9 dve kru\u017enice (pozri obr\u00e1zok). Obe maj\u00fa stred v bode S<\/em> a ka\u017ed\u00e1 prech\u00e1dza \u0161tyrmi mre\u017eov\u00fdmi bodmi. Sivo vyfarben\u00fd obrazec je ohrani\u010den\u00fd \u010das\u0165ami t\u00fdchto kru\u017en\u00edc a jednou sie\u0165ovou priamkou. Ur\u010dite obsah siv\u00e9ho obrazca.<\/p>\n

\"\"<\/div>\n
(L. \u0160im\u016fnek)<\/div>\n<\/p>\n

Z9-III-4<\/strong>
\nAdam s Evou hrali \u0161achy o jablko.
\nAdam vyhral a ute\u0161oval Evu: \u201eTo vie\u0161, ja hr\u00e1vam \u0161achy dlho, dvakr\u00e1t dlh\u0161ie ako ty!\u201c
\nEva sa hnevala: \u201eAle minule si hovoril, \u017ee ich hr\u00e1va\u0161 trikr\u00e1t dlh\u0161ie!\u201c
\nAdam sa divil: \u201eTo \u017ee som hovoril? A kedy to bolo?\u201c
\n\u201ePredvlani!\u201c
\n\u201eNo tak to \u00e1no, hovoril som pravdu \u2013 a dnes tie\u017e.\u201c
\nAko dlho (v rokoch) teda hr\u00e1va Adam \u0161ach?<\/p>\n

(M. Volfov\u00e1)<\/div>\n<\/p>\n\n\n
\n
\n<\/td>\n		\"\"<\/a><\/td>\n<\/tr>\n<\/table>\n\n
<\/div>","protected":false},"excerpt":{"rendered":"

58. ro\u010dn\u00edk matematickej olympi\u00e1dy 2008-2009<\/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":[7],"tags":[],"class_list":["post-119","post","type-post","status-publish","format-standard","hentry","category-58-rocnik"],"yoast_head":"\nKateg\u00f3rie Z4-Z9 - Matematick\u00e1 olympi\u00e1da<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/matematika.besaba.com\/?p=119\" \/>\n<meta property=\"og:locale\" content=\"sk_SK\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Kateg\u00f3rie Z4-Z9 - 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