{"id":1520,"date":"2021-09-22T19:46:09","date_gmt":"2021-09-22T19:46:09","guid":{"rendered":"http:\/\/matematika.besaba.com\/?p=1520"},"modified":"2022-11-12T21:17:58","modified_gmt":"2022-11-12T21:17:58","slug":"sutazne-ulohy-kategorii-z5-z9-9","status":"publish","type":"post","link":"https:\/\/matematika.besaba.com\/?p=1520","title":{"rendered":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9"},"content":{"rendered":"

<\/a>71. ro\u010dn\u00edk matematickej olympi\u00e1dy 2021-2022
\nS\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9<\/p>\n

aktualizovan\u00e9 22.2.2022 18:00<\/span><\/p>\n\n

<\/p>\n\n\n\n\n\n\n
Z5<\/span><\/td>\nZ6<\/span><\/td>\nZ7<\/span><\/td>\nZ8<\/span><\/td>\nZ9<\/span><\/td>\n<\/tr>\n
dom\u00e1ce<\/a><\/span><\/td>\ndom\u00e1ce<\/a><\/span><\/td>\ndom\u00e1ce<\/a><\/span><\/td>\ndom\u00e1ce<\/a><\/span><\/td>\ndom\u00e1ce<\/a><\/span><\/td>\n<\/tr>\n
okresn\u00e9<\/a><\/span><\/td>\nokresn\u00e9<\/a><\/span><\/td>\nokresn\u00e9<\/a><\/span><\/td>\nokresn\u00e9<\/a><\/span><\/td>\nokresn\u00e9<\/a><\/span><\/td>\n<\/tr>\n
<\/td>\nkrajsk\u00e9<\/a><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

<\/a><\/a><\/form>\n

Z5-I-1<\/strong>
\nDo kruhovy\u0301ch pol\u00edc\u030cok dopln\u030cte prirodzene\u0301 c\u030c\u00edsla od 1 do 20 tak, aby kaz\u030cde\u0301 c\u030c\u00edslo bolo pouz\u030cite\u0301 pra\u0301ve raz a su\u0301c\u030casne platili vs\u030cetky uvedene\u0301 vz\u0165ahy.<\/p>\n

\"\"<\/div>\n
(Miroslava Farkas Smitkova\u0301)<\/div>\n

Z5-I-2<\/strong>
\nTrpasl\u0131\u0301ci natierali kocky zelenou a bielou farbou tak, z\u030ce kaz\u030cd\u00e1 stena bola cel\u00e1 ofarben\u00e1 jednou z ty\u0301chto dvoch farieb. Po chv\u0131\u0301li si vs\u030cimli, z\u030ce niektore\u0301 ofarbene\u0301 kocky vyzeraju\u0301 po vhodnom pootoc\u030cen\u00ed u\u0301plne rovnako, a zac\u030cali ich pod\u013ea toho triedi\u0165 do skup\u0131\u0301n (v rovnakej skupine su\u0301 rovnako ofarbene\u0301 kocky). Najviac ko\u013eko skup\u0131\u0301n mohli takto dosta\u0165?<\/p>\n

(Eva Semer\u00e1dov\u00e1)<\/div>\n

Z5-I-3<\/strong>
\nAdam prepoc\u030c\u00edtaval svoju zbierku du\u0301hovy\u0301ch gu\u013eo\u0302c\u030cok. Zistil, z\u030ce ich mo\u0302z\u030ce rozdeli\u0165 do rovnako poc\u030cetny\u0301ch ko\u0302pok, a to viacery\u0301mi spo\u0302sobmi. Keby ich rozdelil do troch ko\u0302pok, bolo by v kaz\u030cdej ko\u0302pke o osem gu\u013eo\u0302c\u030cok viac, nez\u030c by bolo v kaz\u030cdej ko\u0302pke pri delen\u00ed do s\u030ctyroch ko\u0302pok. Ko\u013eko mal Adam du\u0301hovy\u0301ch gu\u013eo\u0302c\u030cok?<\/p>\n

(Eva Semera\u0301dova\u0301)<\/div>\n

Z5-I-4<\/strong>
\nJaro vystrihol z rohu s\u030cachovnice nasleduju\u0301ci u\u0301tvar pozosta\u0301vaju\u0301ci z pa\u0308tna\u0301stich pol\u00ed:<\/p>\n

\"\"<\/div>\n

Na\u0301sledne odstrihol nieko\u013eko \u010fals\u030c\u00edch pol\u00ed, a to tak, z\u030ce vy\u0301sledny\u0301 u\u0301tvar neobsahoval diery a nerozpadal sa, mal rovnaky\u0301 poc\u030cet c\u030ciernych a bielych pol\u00ed a mal najva\u0308c\u030cs\u030c\u00ed moz\u030cny\u0301 obsah. Navys\u030ce zistil, z\u030ce zo vs\u030cetky\u0301ch moz\u030cny\u0301ch u\u0301tvarov s ty\u0301mito vlastnos\u0165ami mal ten jeho najva\u0308c\u030cs\u030c\u0131\u0301 moz\u030cny\u0301 obvod. Ktore\u0301 polia Jaro dodatoc\u030cne odstrihol? Urc\u030cte vs\u030cetky moz\u030cnosti.<\/p>\n

(Michaela Petrova\u0301)<\/div>\n

Z5-I-5<\/strong>
\nNa papieri bol narysovany\u0301 s\u030ctvorec \ud835\udc34\ud835\udc35\ud835\udc36\ud835\udc37 so stranou 4 cm. Pavol zostrojil vrcholy obd\u013az\u030cnika, ktory\u0301 mal trikra\u0301t va\u0308c\u030cs\u030c\u00ed obsah nez\u030c s\u030ctvorec \ud835\udc34\ud835\udc35\ud835\udc36\ud835\udc37. Pritom rysoval iba kruz\u030cnice, pretoz\u030ce prav\u0131\u0301tko nenas\u030ciel. Ako mohol Pavol postupova\u0165? Pop\u0131\u0301s\u030cte aspon\u030c jednu kons\u030ctrukciu.<\/p>\n

(Karel Pazourek)<\/div>\n

Z5-I-6<\/strong>
\nNa parkovisku sta\u0301li aut\u00e1 a bicykle. Ak by pris\u030clo jedno \u010fals\u030cie auto, bolo by ich rovnako ve\u013ea ako bicyklov. Ak by pris\u030clo pa\u0308\u0165 \u010fals\u030c\u00edch bicyklov, mali by vs\u030cetky bicykle rovnaky\u0301 poc\u030cet kolies ako vs\u030cetky aut\u00e1. Ko\u013eko sta\u0301lo na parkovisku a\u0301ut a ko\u013eko bicyklov?<\/p>\n

(Monika Dillingerova\u0301)<\/div>\n\n\n\n
\n
\n<\/td>\n		<\/a>\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/a><\/a><\/form>\n

Z5-II-1<\/strong>
\nV obchode maj\u00fa jeden druh l\u00edzaniek a jeden druh nanukov. Cena ako l\u00edzaniek, tak nanukov je uveden\u00e1 v cel\u00fdch gro\u0161och. Barborka k\u00fapila tri l\u00edzanky. Eli\u0161ka k\u00fapila \u0161tyri l\u00edzanky a nieko\u013eko nanukov \u2013 vieme len, \u017ee to bolo viac ako jeden a menej ako desa\u0165 nanukov. Janko k\u00fapil jednu l\u00edzanku a jeden nanuk. Barborka platila 24 gro\u0161ov a Eli\u0161ka 109 gro\u0161ov. Ko\u013eko gro\u0161ov platil Janko? <\/p>\n

(Libu\u0161e Hozov\u00e1)<\/div>\n

Z5-II-2<\/strong>
\nX\u00e9nia mala obd\u013a\u017enik s rozmermi 24 cm a 30 cm. Rozdelila ho tromi \u00fase\u010dkami na nieko\u013eko rovnak\u00fdch obd\u013a\u017enikov\u00fdch dielov. Ak\u00e9 mohli by\u0165 rozmery t\u00fdchto dielov? Ur\u010dte \u0161tyri mo\u017enosti.<\/p>\n

(Karel Pazourek)<\/div>\n

Z5-II-3<\/strong>
\nVojto sa sna\u017e\u00ed ulo\u017ei\u0165 svojich 20 hra\u010diek do \u0161kat\u00fa\u013e tak, aby v ka\u017edej \u0161katuli bola aspo\u0148 jedna hra\u010dka a v \u017eiadnych dvoch \u0161katuliach nebol rovnak\u00fd po\u010det hra\u010diek.<\/p>\n

    a)<\/strong>  Op\u00ed\u0161te, ako m\u00f4\u017ee hra\u010dky ulo\u017ei\u0165 do piatich \u0161kat\u00fa\u013e.
\n    b)<\/strong>  M\u00f4\u017ee hra\u010dky ulo\u017ei\u0165 do \u0161iestich \u0161kat\u00fa\u013e?<\/p>\n

(Josef Tkadlec)<\/div>\n\n\n\n
\n
\n<\/td>\n		<\/a>\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/a><\/form>\n

Z6-I-1<\/strong>
\nMo\u0302j jediny\u0301 syn sa narodil, ke\u010f som mala 37 rokov. To bolo pra\u0301ve 32 rokov po smrti dedka, ktory\u0301 zomrel, ke\u010f mal 64 rokov. Dedko bol o 12 rokov stars\u030c\u00ed nez\u030c babka, brali sa v roku 1947, pra\u0301ve ke\u010f mala babka 18 rokov. V ktorom roku sa narodil mo\u0302j syn?<\/p>\n

(Miroslava Farkas Smitkova\u0301)<\/div>\n

Z6-I-2<\/strong>
\nPeter mal obd\u013a\u017enik \u0161\u00edrky 2 cm a nezn\u00e1mej d\u013a\u017eky. Radka mala obd\u013a\u017enik \u0161\u00edrky 2 cm, ktor\u00e9ho d\u013a\u017eka bola rovn\u00e1 obvodu Petrovho obd\u013a\u017enika. Ke\u010f k sebe obd\u013a\u017eniky prilo\u017eili ich \u0161\u00edrkami, z\u00edskali novy\u0301 obd\u013a\u017enik s obvodom 63 cm. Ur\u010dte obsah Petrovho obd\u013a\u017enika.<\/p>\n

(Karel Pazourek)<\/div>\n

Z6-I-3<\/strong>
\nMis\u030cka sku\u0301ma c\u030c\u00edsla, ktore\u0301 sa daju\u0301 vyjadri\u0165 ako su\u0301c\u030cet aspon\u030c dvoch po sebe idu\u0301cich prirodzeny\u0301ch c\u030c\u00edsel. Obzvla\u0301s\u030c\u0165 ju zauj\u0131\u0301maju\u0301 c\u030c\u00edsla, ktore\u0301 sa takto daju\u0301 vyjadri\u0165 viacery\u0301mi spo\u0302sobmi (napr. 18 = 5 + 6 + 7 = 3 + 4 + 5 + 6). \u010c\u00edsla, ktore\u0301 moz\u030cno takto vyjadri\u0165 aspon\u030c tromi spo\u0302sobmi, nazvala ve\u013ekolep\u00e9<\/strong><\/em>. Na\u0301jdite aspon\u030c tri Mis\u030ckine ve\u013ekolepe\u0301<\/em> c\u030c\u00edsla.<\/p>\n

(Veronika Huc\u00edkova\u0301)<\/div>\n

Z6-I-4<\/strong>
\nKubo si nap\u00edsal s\u030ctvormiestne c\u030c\u00edslo, ktore\u0301ho dve c\u030c\u00edslice boli p\u00e1rne a dve nep\u00e1rne. Ak by v tomto c\u030c\u00edsle vys\u030ckrtol obe p\u00e1rne c\u030c\u00edslice, dostal by c\u030c\u00edslo s\u030ctyrikra\u0301t mens\u030cie, nez\u030c keby v n\u030com vys\u030ckrtol obe nep\u00e1rne c\u030c\u00edslice. Ake\u0301 najva\u0308c\u030cs\u030cie c\u030c\u00edslo s ty\u0301mito vlastnos\u0165ami si mohol Kubo nap\u00edsa\u0165?<\/p>\n

(Michaela Petrova\u0301)<\/div>\n

Z6-I-5<\/strong>
\nMojm\u0131\u0301r rozstrihal pravidelny\u0301 s\u030ces\u0165uholn\u0131\u0301k na 12 zhodny\u0301ch dielov. Z ty\u0301chto dielov (nie nutne zo vs\u030cetky\u0301ch) skladal rozlic\u030cne\u0301 pravouhle\u0301 trojuholn\u0131\u0301ky. Ako mohli Mojm\u0131\u0301rove zloz\u030cene\u0301 trojuholn\u0131\u0301ky vyzera\u0165? Na\u0301jdite aspon\u030c s\u030ctyri moz\u030cnosti.<\/p>\n

(Libus\u030ce Hozova\u0301)<\/div>\n

Z6-I-6<\/strong>
\nPa\u0308tica kamara\u0301tov porovna\u0301vala, ko\u013eko stare\u0301ho z\u030celeza priviezli do zberu. Priemerne to bolo 55 kg, avs\u030cak Ivan priviezol len 43 kg. Ko\u013eko kilogramov v priemere priviezli bez Ivana?<\/p>\n

(Libus\u030ce Hozova\u0301)<\/div>\n\n\n\n
\n
\n<\/td>\n		<\/a>\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/form>\n

Z6-II-1<\/strong><\/p>\n

()<\/div>\n

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

()<\/div>\n

Z6-II-3<\/strong><\/p>\n

()<\/div>\n\n\n\n
\n
\n<\/td>\n		<\/a>\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n<\/a><\/form>\n

Z7-I-1<\/strong>
\nDa\u0301z\u030c\u010fovka s\u030cpira\u0301lova\u0301 raz\u0131\u0301 novy\u0301 tunel: Najprv mieri 10 cm na sever, potom 11 cm na vy\u0301chod, potom 12 cm na juh, 13 cm na za\u0301pad a tak \u010falej (kaz\u030cdy\u0301 u\u0301sek je o 1 cm dlhs\u030c\u0131\u0301 nez\u030c predcha\u0301dzaju\u0301ci, smery opakuje pod\u013ea uvedene\u0301ho vzoru).
\nDa\u0301z\u030c\u010fovka su\u0301radnicova\u0301 mapuje dielo svojej kolegyne: Zac\u030ciatok tunela oznac\u030c\u0131\u0301 su\u0301radnicami (0, 0), prvu\u0301 odboc\u030cku su\u0301radnicami (0, 10), druhu\u0301 odboc\u030cku (11, 10) a tak \u010falej.
\nUrc\u030cte su\u0301radnice konca u\u0301seku, ktory\u0301 ma\u0301 d\u013a\u017eku 100 cm.<\/p>\n

(Iveta Janc\u030cigova)<\/div>\n

Z7-I-2<\/strong>
\nSu\u0301c\u030cin vekov vs\u030cetky\u0301ch det\u0131\u0301 pa\u0301na Na\u0301sobka je 1408. Vek najmlads\u030cieho die\u0165a\u0165a je rovny\u0301 polovici veku najstars\u030cieho die\u0165a\u0165a. Ko\u013eko det\u00ed ma\u0301 pa\u0301n Na\u0301sobok a ko\u013eko maju\u0301 rokov?<\/p>\n

(Libus\u030ce Hozova\u0301)<\/div>\n

Z7-I-3<\/strong>
\nNa strana\u0301ch trojuholn\u00edka ABC<\/em> su\u0301 dan\u00e9 body D, E, F, G<\/em> (pozri obra\u0301zok). Pritom plat\u00ed, z\u030ce s\u030ctvoruholn\u00edk DEFG<\/em> je kosos\u030ctvorec a u\u0301sec\u030cky AD<\/em>, DE<\/em> a EB<\/em> su\u0301 navza\u0301jom zhodne\u0301.<\/p>\n

\"\"<\/div>\n

Urc\u030cte ve\u013ekos\u0165 uhla ACB<\/em>.<\/p>\n

(Iveta Janc\u030cigova)<\/div>\n

Z7-I-4<\/strong>
\nJoz\u030cko vymyslel nasleduju\u0301cu u\u0301lohu:<\/p>\n

M + A + M + R + A + D + M + A + T + E + M + A + T + I + K + U = ?<\/div>\n

Ro\u0302zne p\u00edsmena\u0301 nahradzoval ro\u0302znymi c\u030c\u00edslicami od 1 do 9 a zis\u0165oval, c\u030co vycha\u0301dza.<\/br>
\na)<\/strong> Aky\u0301 najva\u0308c\u030cs\u030c\u00ed vy\u0301sledok mohol Joz\u030cko dosta\u0165?
\nb)<\/strong> Mohol dosta\u0165 vy\u0301sledok 50? Ak a\u0301no, ako?
\nc)<\/strong> Mohol dosta\u0165 vy\u0301sledok 59? Ak a\u0301no, ake\u0301 hodnoty mohol ma\u0165 v takom pr\u00edpade su\u0301c\u030cet \ud835\udc40 + \ud835\udc34 + \ud835\udc40?<\/p>\n

((Miroslava Farkas Smitkova\u0301)<\/div>\n

Z7-I-5<\/strong>
\nJano vyrazil do sveta s bat\u00f4\u017ekom buchiet. Na prvom r\u00e1zcest\u00ed stretol Dlh\u00e9ho, \u0160irok\u00e9ho a Bystrozrak\u00e9ho a spravodlivo sa s nimi o svoje buchty rozdelil \u2013 ka\u017ed\u00fd\u0301 dostal \u0161tvrtinu buchiet. Jano zo svojho dielu zjedol dve buchty a \u0161iel \u010falej. Na druhom r\u00e1zcest\u00ed stretol Plav\u010d\u00edka a Vratka aj s nimi sa spravodlivo rozdelil \u2013 ka\u017ed\u00fd\u0301 dostal tretinu zvy\u0161n\u00fdch buchiet. Jano zo svojho dielu zjedol zasa dve buchty a so zvy\u0161n\u00fdmi vyrazil \u010falej. Na tre\u0165om r\u00e1zcest\u00ed stretol Snehulienku. Aj s tou sa spravodlivo rozdelil, tak\u017ee obaja mali polovicu zvy\u0161n\u00fdch buchiet. Ke\u010f Jano zjedol op\u00e4\u0165 svoje dve buchty, bol bat\u00f4\u017etek pr\u00e1zdny, a tak sa vr\u00e1til domov. S ko\u013ek\u00fdmi buchtami vyrazil Jano do sveta?<\/p>\n

(Michaela Petrova\u0301)<\/div>\n

Z7-I-6<\/strong>
\nP\u00e1n Chrt mal vo svojom z\u00e1prahu p\u00e4\u0165 psov \u2013 Al\u00edka, Broka, Muka, Rafa a Pun\u0165a. Ko\u013ek\u00fdmi sp\u00f4sobmi ich mohol zapriahnu\u0165 do radu za sebou tak, aby bol Al\u00edk pred Pun\u0165om?<\/p>\n

(Libus\u030ce Hozova\u0301)<\/div>\n\n\n\n
\n
\n<\/td>\n		<\/a>\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/form>\n

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

()<\/div>\n

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

()<\/div>\n

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

()<\/div>\n\n\n\n
\n
\n<\/td>\n		<\/a>\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/a><\/form>\n

Z8-I-1<\/strong>
\nVierka z troch dan\u00fdch \u010d\u00edslic zostavovala navz\u00e1jom r\u00f4zne trojmiestne \u010d\u00edsla. Ke\u010f v\u0161etky tieto \u010d\u00edsla s\u010d\u00edtala, vy\u0161lo jej 1221. Ak\u00e9 \u010d\u00edslice Vierka pou\u017eila? Ur\u010dte p\u00e4\u0165 mo\u017enost\u00ed.<\/p>\n

(Karel Pazourek)<\/div>\n

Z8-I-2<\/strong>
\nTRN<\/em> a HAM<\/em> s\u00fa zhodne\u0301 rovnostranne\u0301 trojuholn\u00edky. Pritom bod T<\/em> je \u0165a\u017eiskom trojuholn\u00edka HAM<\/em> a bod R<\/em> le\u017e\u00ed na polpriamke TA<\/em>. Ak\u00fd je pomer obsahov \u010dast\u00ed trojuholn\u00edka TRN<\/em>, ktor\u00e9 s\u00fa vn\u00fatri trojuholn\u00edka HAM<\/em>, a t\u00fdch, ktor\u00e9 s\u00fa mimo neho?<\/p>\n

(Eva Semera\u0301dova\u0301)<\/div>\n

Z8-I-3<\/strong>
\nNa novoobjavenej plan\u00e9te \u017eij\u00fa zvierat\u00e1, ktor\u00e9 astronauti pomenovali pod\u013ea po\u010dtu n\u00f4h jednonoz\u030cky, dvojnoz\u030cky, trojnoz\u030cky a tak \u010falej (zvierat\u00e1 bez n\u00f4h tam neboli). Zvierat\u00e1 s nep\u00e1rnym po\u010dtom n\u00f4h maj\u00fa\u0301 dve hlavy, zvierat\u00e1 s p\u00e1rnym po\u010dtom n\u00f4h maj\u00fa jednu hlavu. V istej priehlbine stretli skupinu tak\u00fdch zvierat a napo\u010d\u00edtali na nich 18 hl\u00e1v a 24 n\u00f4h. Ko\u013eko zvierat mohlo by\u0165 v priehlbine? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n

(Toma\u0301s\u030c B\u00e1rta)<\/div>\n

Z8-I-4<\/strong>
\nV danej skupine \u010d\u00edsel je jedno \u010d\u00edslo rovne\u0301 priemeru v\u0161etk\u00fdch, najv\u00e4\u010d\u0161ie \u010d\u00edslo je o 7 v\u00e4\u010d\u0161ie ne\u017e priemer, najmen\u0161ie je o 7 men\u0161ie ne\u017e priemer a v\u00e4\u010d\u0161ina \u010d\u00edsel zo skupiny m\u00e1 podpriemern\u00fa hodnotu. Ak\u00fd najmen\u0161\u00ed po\u010det \u010d\u00edsel m\u00f4\u017ee by\u0165 v skupine?<\/p>\n

(Karel Pazourek)<\/div>\n

Z8-I-5<\/strong>
\nV pravidelnom p\u00e4\u0165uholn\u00edku ABCDE<\/em> je obsiahnut\u00fd rovnostrann\u00fd trojuholn\u00edk ABM<\/em>. Ur\u010dte ve\u013ekos\u0165 uhla BCM<\/em>.<\/p>\n

(Libus\u030ce Hozova\u0301)<\/div>\n

Z8-I-6<\/strong>
\nAlenka dostala list papiera s nasleduj\u00facimi tvrdeniami:<\/br>
\n    A:<\/strong> Najviac jedno z tvrden\u00ed A, B, C, D, E je pravdiv\u00e9.
\n    B:<\/strong>
\n    C:<\/strong> V\u0161etky tvrdenia A, B, C, D, E s\u00fa pravdiv\u00e9.
\n    D:<\/strong>
\n    E:<\/strong> Tvrdenie A je pravdiv\u00e9.<\/br>
\nTvrdenia B<\/em><\/strong> a D<\/em><\/strong> boli nap\u00edsane\u0301 nevidite\u013en\u00fdm atramentom, ktor\u00fd sa d\u00e1 pre\u010d\u00edta\u0165 len pod \u0161peci\u00e1lnou lampou. Aj bez takejto lampy v\u0161ak Alenka dok\u00e1zala rozhodn\u00fa\u0165, \u010di m\u00f4\u017ee t\u00fdmto tvrdeniam d\u00f4verova\u0165. Ur\u010dte i vy, ktor\u00e9 z tvrden\u00ed A, B, C, D, E<\/em> s\u00fa pravdiv\u00e9 a ktor\u00e9 nepravdiv\u00e9.<\/p>\n

(Iveta Janc\u030cigova\u0301)<\/div>\n\n\n\n
\n
\n<\/td>\n		<\/a>\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/form>\n

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

()<\/div>\n

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

()<\/div>\n

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

()<\/div>\n\n\n\n
\n
\n<\/td>\n		<\/a>\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/a><\/a><\/form>\n

Z9-I-1<\/strong>
\nAdam, Boris a Cyril porovn\u00e1vali, ko\u013eko kilogramov ga\u0161tanov nazbierali. Zistili, \u017ee aritmeticky\u0301 priemer toho, \u010do nazbieral Adam s Borisom, je o 10 kg v\u00e4\u010d\u0161\u00ed ne\u017e Cyrilov pr\u00edspevok, a aritmeticky\u0301 priemer toho, \u010do nazbieral Adam s Cyrilom, je o 3 kg men\u0161\u00ed ne\u017e Borisov pr\u00edspevok. Ur\u010dte rozdiel medzi aritmetick\u00fdm priemerom toho, \u010do nazbierali Boris s Cyrilom, a Adamov\u00fdm pr\u00edspevkom.<\/p>\n

(Michaela Petrova\u0301)<\/div>\n

Z9-I-2<\/strong>
\nJana si vymyslela 2022-miestne \u010d\u00edslo a jeho cifern\u00fd s\u00fa\u010det po\u0161epkala Petrovi. Peter vypo\u010d\u00edtal cifern\u00fd s\u00fa\u010det \u010d\u00edsla, ktor\u00e9 mu povedala Jana, a v\u00fdsledok po\u0161epkal Zuzke. Zuzka tie\u017e vypo\u010d\u00edtala cifern\u00fd s\u00fa\u010det \u010d\u00edsla, ktor\u00e9 dostala od Petra, a v\u00fdsledok, ktor\u00fdm bolo dvojmiestne \u010d\u00edslo, po\u0161epkala Adamovi. Adam urobil to iste\u0301 s \u010d\u00edslom od Zuzky a vy\u0161iel mu cifern\u00fd s\u00fa\u010det 1. Ktor\u00e9 \u010d\u00edsla mohol po\u0161epka\u0165 Peter Zuzke? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n

(Iveta Janc\u030cigova\u0301)<\/div>\n

Z9-I-3<\/strong>
\nJe dan\u00fd pravideln\u00fd trojbok\u00fd hranol s podstavnou hranou d\u013a\u017eky 3,2 cm a v\u00fd\u0161kou 5 cm. Jeho pl\u00e1\u0161\u0165 omot\u00e1vame \u0161achovnicovou f\u00f3liou, ktor\u00e1 pozost\u00e1va z neprieh\u013eadn\u00fdch a prieh\u013eadn\u00fdch \u0161tvorcov\u00fdch pol\u00ed so stranami d\u013a\u017eky 1 cm. Za\u010diatok f\u00f3lie l\u00edcuje s hranou hranola (pozri obr\u00e1zok) a jej d\u013a\u017eka vysta\u010d\u00ed pr\u00e1ve na dvojn\u00e1sobne\u0301 omotanie cel\u00e9ho pl\u00e1\u0161\u0165a.<\/p>\n

\"\"<\/div>\n

Ko\u013eko percent pl\u00e1\u0161\u0165a hranola bude po omotan\u00ed vidite\u013en\u00fdch cez f\u00f3liu? Hr\u00fabku f\u00f3lie zanedbajte.<\/p>\n

(Karel Pazourek)<\/div>\n

Z9-I-4<\/strong>
\nKonvexn\u00fd \u0161tvoruholn\u00edk ABCD<\/em> so stranou AB<\/em> d\u013a\u017eky 5 cm, so stranou BC<\/em> d\u013a\u017eky 3 cm a s uhlom BCD<\/em> ve\u013ekosti 60\u00b0 je s\u00famern\u00fd pod\u013ea uhloprie\u010dky AC<\/em>. Bod E<\/em> je p\u00e4tou kolmice z vrcholu B<\/em> na stranu AD<\/em> a F<\/em> je p\u00e4tou kolmice z vrcholu D<\/em> na stranu BC<\/em>. Ur\u010dte obvod a obsah \u0161tvoruholn\u00edka DEBF<\/em>.<\/p>\n

(Karel Pazourek)<\/div>\n

Z9-I-5<\/strong>
\nVodn\u00edk Gebu\u013ea nakupoval v obchode, kde ceny v\u0161etk\u00e9ho tovaru boli uvedene\u0301 v cel\u00fdch \u0161upink\u00e1ch. Keby Gebu\u013ea k\u00fapil 2 raky, 3 \u0161keble a 1 \u0161\u0165uku, zaplatil by 49 \u0161upiniek. Ak by prik\u00fapil e\u0161te 5 rakov, 11 \u0161kebl\u0131\u0301 a 1 \u0161\u0165uku, platil by celkovo 154 \u0161upiniek. Ko\u013eko \u0161upiniek by platil za 1 raka, 2 \u0161keble a 3 \u0161\u0165uky? Ur\u010dte v\u0161etky mo\u017enosti.<\/p>\n

(Karel Pazourek)<\/div>\n

Z9-I-6<\/strong>
\nS\u00fa dane\u0301 dve r\u00f4zne \u010d\u00edsla. Ak od oboch od\u010d\u00edtame \u0161tvrtinu men\u0161ieho z nich, dostaneme \u010d\u00edsla, z ktor\u00fdch jedno bude p\u00e4\u0165kr\u00e1t v\u00e4\u010d\u0161ie ne\u017e druh\u00e9. Ko\u013ekokr\u00e1t je dane\u0301 v\u00e4\u010d\u0161ie \u010d\u00edslo v\u00e4\u010d\u0161ie ne\u017e to men\u0161ie?<\/p>\n

(Libus\u030ce Hozova\u0301)<\/div>\n\n\n\n
\n
\n<\/td>\n		<\/a>\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/a><\/a><\/form>\n

Z9-II-1<\/strong>
\nTrojcifern\u00e9 \u010d\u00edslo m\u00e1 cifern\u00fd s\u00fa\u010det 16. Ak v tomto \u010d\u00edsle zamen\u00edme cifry na miestach stoviek a desiatok, \u010d\u00edslo sa o 360 zmen\u0161\u00ed. Ak v p\u00f4vodnom \u010d\u00edsle zamen\u00edme cifry na miestach desiatok a jednotiek, \u010d\u00edslo sa o 54 zv\u00e4\u010d\u0161\u00ed. N\u00e1jdite toto trojcifern\u00e9 \u010d\u00edslo. <\/p>\n

(Libu\u0161e Hozov\u00e1)<\/div>\n

Z9-II-2<\/strong>
\n\u0160tvoruholn\u00edk ABCD<\/em> je s\u00famern\u00fd pod\u013ea uhloprie\u010dky AC<\/em>. D\u013a\u017eka AC<\/em> je 12 cm, d\u013a\u017eka BC<\/em> je 6 cm a vn\u00fatorn\u00fd uhol pri vrchole B<\/em> je prav\u00fd. Na stran\u00e1ch AB<\/em>, AD<\/em> s\u00fa dan\u00e9 body E<\/em>, F<\/em> tak, \u017ee trojuholn\u00edk ECF<\/em> je rovnostrann\u00fd. Ur\u010dte d\u013a\u017eku \u00fase\u010dky EF<\/em>.<\/p>\n

(Karel Pazourek)<\/div>\n

Z9-II-3<\/strong>
\n\u013dudov\u00edt si pri istom pr\u00edklade na delenie v\u0161imol, \u017ee ke\u010f delenec zdvojn\u00e1sob\u00ed a delite\u013e zv\u00e4\u010d\u0161\u00ed o 12, dostane svoje ob\u013e\u00faben\u00e9 \u010d\u00edslo. To ist\u00e9 \u010d\u00edslo by dostal, aj keby p\u00f4vodn\u00fd delenec zmen\u0161il o 42 a p\u00f4vodn\u00fd delite\u013e zmen\u0161il na polovicu. Ur\u010dte \u013dudov\u00edtovo ob\u013e\u00faben\u00e9 \u010d\u00edslo.<\/p>\n

(Michaela Petrov\u00e1)<\/div>\n

Z9-II-4<\/strong>
\nV konvexnom \u0161tvoruholn\u00edku ABCD<\/em> plat\u00ed, \u017ee E<\/em> je priese\u010dn\u00edkom uhloprie\u010dok, trojuholn\u00edky ADE<\/em>, BCE<\/em>, CDE<\/em> maj\u00fa postupne obsahy 12 cm2<\/sup> , 45 cm2<\/sup> , 18 cm2<\/sup> a d\u013a\u017eka strany AB<\/em> je 7 cm. Ur\u010dte vzdialenos\u0165 bodu D<\/em> od priamky AB<\/em>. <\/p>\n

(Michaela Petrov\u00e1)<\/div>\n\n\n\n
\n
\n<\/td>\n		<\/a>\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/form>\n

Z9-III-1<\/strong><\/p>\n

(Eva Semer\u00e1dov\u00e1)<\/div>\n

Z9-III-2<\/strong><\/p>\n

(Marie Krej\u010dov\u00e1)<\/div>\n

Z9-III-3<\/strong>
\n(Karel Pazourek)<\/p>\n

Z9-III-4<\/strong><\/p>\n

(Eva Semer\u00e1dov\u00e1)<\/div>\n\n\n\n
\n
\n<\/td>\n		\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n