{"id":1777,"date":"2026-01-05T13:45:43","date_gmt":"2026-01-05T12:45:43","guid":{"rendered":"https:\/\/matematika.besaba.com\/?p=1777"},"modified":"2026-07-17T20:29:31","modified_gmt":"2026-07-17T19:29:31","slug":"sutazne-ulohy-kategorii-z5-z9-13","status":"publish","type":"post","link":"https:\/\/matematika.besaba.com\/?p=1777","title":{"rendered":"S\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3rii Z5 \u2013 Z9"},"content":{"rendered":"

<\/a>75. ro\u010dn\u00edk matematickej olympi\u00e1dy 2025-2026
\nS\u00fa\u0165a\u017en\u00e9 \u00falohy kateg\u00f3ri\u00ed Z5-Z9<\/p>\n

aktualizovan\u00e9 05.01.2026 14:30<\/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><\/a><\/form>\n

Z5-I-1<\/strong>
\nZvl\u00e1\u0161tna kalkula\u010dka m\u00e1 iba dve funk\u010dn\u00e9 tla\u010didl\u00e1. Po stla\u010den\u00ed prv\u00e9ho tla\u010didla sa k \u010d\u00edslu na displeji pripo\u010d\u00edta \u010d\u00edslo 1, po stla\u010den\u00ed druh\u00e9ho tla\u010didla sa \u010d\u00edslo na displeji vyn\u00e1sob\u00ed \u010d\u00edslom 2. Na displeji po ka\u017edom stla\u010den\u00ed tla\u010didla svieti spr\u00e1vny v\u00fdsledok.
\nN\u00e1jdite dva r\u00f4zne sp\u00f4soby, ako pomocou 6 stla\u010den\u00ed tla\u010didiel dosta\u0165 na displeji z \u010d\u00edsla 1 \u010d\u00edslo 15.<\/p>\n

(Iveta Jan\u010digov\u00e1)<\/div>\n

Z5-I-2<\/strong>
\nK z\u00e1mku patr\u00ed park v tvare \u0161tvorca so stranou d\u013a\u017eky 240 metrov. N\u00e1\u010drt parku je na obr\u00e1zku. Po stran\u00e1ch \u0161tvorca ved\u00fa cesty, v jeho vrcholoch stoja ag\u00e1t, buk, c\u00e9der a dub. Park kri\u017euj\u00fa dva \u010fal\u0161ie chodn\u00edky rovnobe\u017en\u00e9 so strana mi \u0161tvorcov\u00e9ho parku \u2013 jeden vedie od javora k studni\u010dke, druh\u00fd vedie od lipy k orechu.<\/p>\n

\"\"<\/div>\n

Princezn\u00e1 pri svojich prech\u00e1dzkach parkom chodila len po cest\u00e1ch a chodn\u00edkoch, nikdy sa nevracala, ani zbyto\u010dne neodbo\u010dovala. Zistila, \u017ee:
\n\u2022 prech\u00e1dzka od buka k studni\u010dke okolo javora, ag\u00e1ta, orecha a duba je dvakr\u00e1t dlh\u0161ia ako okolo lipy a c\u00e9dra,
\n\u2022 prech\u00e1dzka od buka k studni\u010dke okolo lipy a c\u00e9dra je rovnako dlh\u00e1 ako prech\u00e1dzka od duba k lipe okolo studni\u010dky a c\u00e9dra.
\nAk\u00e1 dlh\u00e1 je priama cesta od ag\u00e1ta k orechu?<\/p>\n

(Maria\u0301n Macko)<\/div>\n

Z5-I-3<\/strong>
\nDanka a Janka ka\u017ed\u00e1 pre seba nazbierali jahody. Keby mala Janka o polovicu viac jah\u00f4d, ne\u017e nazbierala, mala by ich rovnako ako Danka. Keby mala Janka dvakr\u00e1t viac jah\u00f4d, ne\u017e nazbierala, mala by ich o 48 viac ako Danka.
\nKo\u013eko jah\u00f4d nazbierala Janka a ko\u013eko Danka?<\/p>\n

(Monika Dillingerov\u00e1)<\/div>\n

Z5-I-4<\/strong>
\nAne\u017eka spr\u00e1vne vyn\u00e1sobila ur\u010dit\u00e9 prirodzen\u00e9 \u010d\u00edslo \u010d\u00edslom 7 a v\u00fdsledn\u00e9 p\u00e4\u0165cifern\u00e9 \u010d\u00edslo nap\u00edsala na papier. Papag\u00e1j kus papiera vy\u010fobal, a tak sa prv\u00e1 cifra v\u00fdsledku stala ne\u010ditate\u013enou. Na zvy\u0161ku papiera zostalo nap\u00edsan\u00e9 2887.
\nAk\u00e1 mohla by\u0165 prv\u00e1 cifra Ane\u017ekinho v\u00fdsledku? N\u00e1jdite v\u0161etky mo\u017enosti.<\/p>\n

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

Z5-I-5<\/strong>
\nVeky siedmich kamar\u00e1tov s\u00fa 8, 9, 10, 11, 11, 13 a 14 rokov. Traja kamar\u00e1ti s\u00fa pr\u00e1ve v kine, dvaja s\u00fa na futbale a dvaja doma. S\u00fa\u010det vekov t\u00fdch v kine je 30 rokov a s\u00fa\u010det vekov t\u00fdch na futbale je 24 rokov. Ka\u017ed\u00fd z kamar\u00e1tov na futbale m\u00e1 viac rokov ako Ondrej, ktor\u00fd zostal doma.
\nKo\u013eko rokov m\u00f4\u017ee ma\u0165 Ondrej? N\u00e1jdite v\u0161etky mo\u017enosti.<\/p>\n

(Mari\u00e1n Macko)<\/div>\n

Z5-I-6<\/strong>
\n\u0160tyri diev\u010dat\u00e1 dl\u00e1\u017edili \u0161tvorcov\u00fdmi dla\u017edicami terasu v rohu dvora. Viola pou\u017e\u00edvala dla\u017edice so stranou 1 dm, Ru\u017eenka so stranou 2 dm, Bianka so stranou 3 dm a Karmen so stranou 4 dm. Prv\u00e9 z diev\u010dat polo\u017eilo jednu zo svojich dla\u017ed\u00edc do rohu. Druh\u00e9 polo\u017eilo svoje dla\u017edice pozd\u013a\u017e vo\u013en\u00fdch str\u00e1n predch\u00e1dzaj\u00facej dla\u017edice a pridalo jednu navy\u0161e tak, aby vznikol \u0161tvorec (napr\u00edklad ako na obr\u00e1zku).<\/p>\n

\"\"<\/div>\n

Podobn\u00fdm sp\u00f4sobom polo\u017eilo svoje dla\u017edice tretie a nakoniec aj \u0161tvrt\u00e9 diev\u010da. Takto pokryli \u0161tvorcov\u00fa terasu bez medzier a prekryvov.
\nV akom porad\u00ed mohli diev\u010dat\u00e1 kl\u00e1s\u0165 dla\u017edice a ko\u013eko dla\u017ed\u00edc celkovo pou\u017eili? N\u00e1jdite v\u0161etky mo\u017enosti.<\/p>\n

(Mari\u00e1n Macko)<\/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>
\nNa obr\u00e1zku je trin\u00e1s\u0165 geometrick\u00fdch \u00fatvarov, z ktor\u00fdch niektor\u00e9 s\u00fa rovnak\u00e9. Do ka\u017ed\u00e9ho u\u0301tvaru vp\u00ed\u0161te jedno z \u010d\u00edsel 1, 2, \u2026, 9 tak, aby v rovnak\u00fdch \u00fatvaroch boli rovnak\u00e9 \u010d\u00edsla a v r\u00f4znych r\u00f4zne.
Ak\u00fd najv\u00e4\u010d\u0161\u00ed a ak\u00fd najmen\u0161\u00ed m\u00f4\u017ee by\u0165 s\u00fa\u010det \u010d\u00edsel v \u00fatvaroch?<\/p>\n

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

Z5-II-2<\/strong>
\nVojtech a Pavol sa v\u00e1\u017eili spolu so svoj\u00edm psom Brokom. Zistili, \u017ee Brok v\u00e1\u017ei 13 kg, Pavol s Brokom v\u00e1\u017eia rovnako ako Vojtech a Vojtech spolu s Brokom v\u00e1\u017ei dvakr\u00e1t to\u013eko ako Pavol.
Ur\u010dte, ko\u013eko v\u00e1\u017ei Vojtech a ko\u013eko Pavol.<\/p>\n

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

Z5-II-3<\/strong>
\nV obd\u013a\u017eniku s rozmermi 808 cm a 707 cm s\u00fa do proti\u013eahl\u00fdch rohov vlo\u017een\u00e9 dva \u0161tvorce so stranami d\u013a\u017eok 505 cm a 404 cm. Ka\u017ed\u00fd zo \u0161tvorcov prilieha dvoma stranami k stran\u00e1m obd\u013a\u017enika. \u0160tvorce sa prekr\u00fdvaj\u00fa v obd\u013a\u017eniku, ktor\u00fd je na obr\u00e1zku vyzna\u010den\u00fd sivou farbou.
Vypo\u010d\u00edtajte d\u013a\u017eky str\u00e1n siv\u00e9ho obd\u013a\u017enika.<\/p>\n

\"\"<\/div>\n
(E. Novotn\u00e1)<\/div>\n\n\n\n
\n
\n<\/td>\n		<\/a>\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/a><\/a><\/a><\/form>\n

Z6-I-1<\/strong>
\nV nasleduj\u00facom s\u010d\u00edtacom algebrograme zodpovedaj\u00fa r\u00f4zne p\u00edsmen\u00e1 r\u00f4znym cifr\u00e1m a rovnak\u00e9 rovnak\u00fdm:<\/p>\n\n\n\n\n\n\n
T<\/td>\nO<\/td>\nN<\/td>\nA<\/td>\n<\/tr>\n
 <\/td>\nO<\/td>\nN<\/td>\nA<\/td>\n<\/tr>\n
 <\/td>\n <\/td>\nN<\/td>\nA<\/td>\n<\/tr>\n
 <\/td>\n <\/td>\n <\/td>\nA<\/td>\n<\/tr>\n
8<\/td>\n6<\/td>\n5<\/td>\n4<\/td>\n<\/tr>\n<\/table>\n
(Iveta Janc\u030cigov\u00e1)<\/div>\n

Z6-I-2<\/strong>
\nOd 1. janu\u00e1ra bol p\u00e1n Nov\u00e1k zamestnan\u00fd v novej firme, pri\u010dom n\u00e1stupn\u00e1 v\u00fd\u0161ka jeho platu bola 1 550 eur mesa\u010dne.
\nPreto\u017ee sa osved\u010dil, od ist\u00e9ho mesiaca v prvom polroku mu zv\u00fd\u0161ili plat o nieko\u013eko cel\u00fdch eur. Tak\u00fdto plat dost\u00e1val mesa\u010dne u\u017e vo v\u0161etk\u00fdch \u010fal\u0161\u00edch mesiacoch. Za cel\u00fd rok si zarobil dokopy 20 000 eur.
\nZa ktor\u00fd mesiac mohol dosta\u0165 prv\u00fdkr\u00e1t zv\u00fd\u0161en\u00fd plat a o ko\u013eko eur? N\u00e1jdite v\u0161etky mo\u017enosti.<\/p>\n

(Mari\u00e1n Macko)<\/div>\n

Z6-I-3<\/strong>
\nObd\u013a\u017enik ABCD<\/em> rozmerov 8 \u00d7 6 je pomocou vodorovn\u00fdch a zvisl\u00fdch \u010diar rozdelen\u00fd na rovnak\u00e9 \u0161tvor\u010deky a bod N<\/em> le\u017e\u00ed v \u0161tvrtine jeho dlh\u0161ej strany AB<\/em> bli\u017e\u0161ie k A<\/em>, tak ako vid\u00edme na obr\u00e1zku.<\/p>\n

\"\"<\/div>\n

Rozde\u013ete obd\u013a\u017enik ABCD<\/em> tromi \u00fase\u010dkami so spolo\u010dn\u00fdm bodom N<\/em> na \u0161tyri \u010dasti s rovnak\u00fdmi obsahmi.<\/p>\n

(Daniela Koval\u010d\u00edkov\u00e1)<\/div>\n

Z6-I-4<\/strong>
\nMajme jedno trojcifern\u00e9 a jedno dvojcifern\u00e9 \u010d\u00edslo. Trojcifern\u00e9 \u010d\u00edslo zaokr\u00fahlime na stovky a dvojcifern\u00e9 zaokr\u00fahlime na desiatky. Rozdiel zaokr\u00fahlen\u00fdch \u010d\u00edsel je 500.
\nAk\u00fd najmen\u0161\u00ed a ak\u00fd najv\u00e4\u010d\u0161\u00ed m\u00f4\u017ee by\u0165 rozdiel p\u00f4vodn\u00fdch nezaokr\u00fahlen\u00fdch \u010d\u00edsel? <\/p>\n

(Mari\u00e1n Macko)<\/div>\n

Z6-I-5<\/strong>
\nPe\u0165o m\u00e1 ku ka\u017ed\u00e9mu d\u0148u v t\u00fd\u017edni priraden\u00fa farbu: k pondelku modr\u00fa, k utorku zelen\u00fa, k strede bielu, k \u0161tvrtku \u010derven\u00fa, k piatku oran\u017eov\u00fa, k sobote siv\u00fa a k nedeli hned\u00fa. V t\u00fdchto farb\u00e1ch nos\u00ed aj pono\u017eky, a to tak, \u017ee na pravej nohe m\u00e1 pono\u017eku farby d\u0148a, av\u0161ak na \u013eavej nohe nem\u00e1 pono\u017eku farby tohto d\u0148a ani d\u0148a nasleduj\u00faceho. (Napr. v sobotu m\u00e1 na pravej nohe siv\u00fa pono\u017eku a na \u013eavej nem\u00e1 siv\u00fa ani hned\u00fa.)
\nUr\u010dte, ktor\u00fd je de\u0148, ak mal Pe\u0165o na \u013eavej nohe predv\u010derom modr\u00fa pono\u017eku, v\u010dera \u010derven\u00fa a dnes m\u00e1 na nej hned\u00fa. <\/p>\n

(Monika Dillingerov\u00e1)<\/div>\n

Z6-I-6<\/strong>
\nObd\u013a\u017enikov\u00fd park m\u00e1 obvod 228 metrov. Vo vrcholoch obd\u013a\u017enika a na jeho stran\u00e1ch r\u00e1stlo 38 okrasn\u00fdch kr\u00edkov tak, \u017ee vzdialenosti medzi ka\u017ed\u00fdmi dvoma susedn\u00fdmi kr\u00edkmi boli rovnak\u00e9. Na dvoch proti\u013eahl\u00fdch stran\u00e1ch obd\u013a\u017enika zasadil z\u00e1hradn\u00edk medzi ka\u017ed\u00e9 dva kr\u00edky jeden \u010fal\u0161\u00ed. T\u00fdm zv\u00fd\u0161il po\u010det kr\u00edkov po obvode parku na 60.
\nUr\u010dte rozmery parku. <\/p>\n

(Mari\u00e1n Macko)<\/div>\n\n\n\n
\n
\n<\/td>\n		<\/a>\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/a><\/a><\/form>\n

Z6-II-1<\/strong>
\nV nasleduj\u00facom s\u010d\u00edtacom algebrograme nahra\u010fte ka\u017ed\u00fa hviezdi\u010dku nejakou nenulovou cifrou, v\u0161etky p\u00edsmen\u00e1 P tou istou p\u00e1rnou cifrou a v\u0161etky p\u00edsmen\u00e1 N tou istou nep\u00e1rnou cifrou tak, aby bol v\u00fdpo\u010det spr\u00e1vny. N\u00e1jdite v\u0161etky rie\u0161enia.<\/p>\n\n\n\n\n\n
 <\/td>\nP<\/td>\nP<\/td>\nP<\/td>\n<\/tr>\n
 <\/td>\n <\/td>\nN<\/td>\nN<\/td>\n<\/tr>\n
 <\/td>\n <\/td>\n <\/td>\nP<\/td>\n<\/tr>\n
 <\/td>\n*<\/td>\n*<\/td>\n3<\/td>\n<\/tr>\n<\/table>\n
(E. Novotn\u00e1)<\/div>\n

Z6-II-2<\/strong>
\nZ\u00e1hrada v tvare obd\u013a\u017enika je rozdelen\u00e1 na dve rovnak\u00e9 obd\u013a\u017enikov\u00e9 \u010dasti. Obvod celej z\u00e1hrady je 164 metrov a obvod ka\u017edej z dvoch \u010dast\u00ed je 110 metrov.
Ak\u00e9 s\u00fa rozmery z\u00e1hrady?<\/p>\n

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

Z6-II-3<\/strong>
\nSnehulienka a sedem trpasl\u00edkov st\u00e1li v rade za sebou tak, \u017ee:<\/p>\n