Marinada '24 - Školjka

Vrijeme: 09:15

Sokrat i Platon | Socrates and Plato #3

Nakon što je Sokrat Platonu ispričao kako se dosjetio rješenja zadnjega zadatka, njih dvojica začuju misteriozan smijeh iz treće kabine - netko ih je cijelo vrijeme prisluškivao.

"Tko si ti" upitaju u isti glas.

"Dečki, pojma vi nemate o filozofiji, a bome niti o minimumima niti o maksimumima!" odgovori im glas iz kabine.

"No, možda se varam, evo dat ću vam ovaj mali zadatak koji sam sad smislio da vas testiram...Nek je $A=\{1,2,\ldots,99\} ,$ i nek su $X,Y$ dva (ne nužno različita) neprazna podskupa od $A.$ Nek $\textup{max} X$ i $\textup{min} Y$ označavaju najveći element skupa $X$ te najmanji element skupa $Y,$ redom. Odredite broj uređenih parova $(X,Y)$ za koje vrijedi $\textup{max} X>\textup{min} Y.$ "

Platon i Sokrat nakon dugog razmišljanja shvate kako nemaju pojma o čemu ovaj priča. "Predajemo se", kažu, "Barem nam reci tko si".

"Ja sam najveći filozof svih vremena, znate" odgovorio im je.

"Gracin. Krešimir Gracin." kaže dok Platon i Sokrat ostaju bez teksta.

Kako niti Platon niti sokrat ne znaju odgovor na Krešimirovo pitanje, znate li vi?

After Socrates explained to Plato how he figured out the solution to the last problem, the two of them heard a mysterious laugh from the third booth—someone had been eavesdropping on them the whole time.

"Who are you?" they asked in unison.

"Guys, you have no idea about philosophy, and certainly not about minimums or maximums!" replied the voice from the booth.

"Well, maybe I'm mistaken. Here, I'll give you this little problem I just thought of to test you... Let $A = \{1, 2, \ldots, 99\}$ , and let $X, Y$ be two (not necessarily different) nonempty subsets of $A$ . Let $\textup{max} X$ and $\textup{min} Y$ denote the largest element of set $X$ and the smallest element of set $Y$ , respectively. Determine the number of ordered pairs $(X, Y)$ such that $\textup{max} X > \textup{min} Y$ ."

Plato and Socrates, after a long moment of thought, realized they had no idea what this was all about. "We give up," they said, "At least tell us who you are."

"I am the greatest philosopher of all time, you know," the voice replied.

"Krešimir Gracin," he said, leaving Plato and Socrates speechless.

Since neither Plato nor Socrates knows the answer to Krešimir's question, do you?

[lang=hr] Nakon što je Sokrat Platonu ispričao kako se dosjetio rješenja zadnjega zadatka, njih dvojica začuju misteriozan smijeh iz treće kabine - netko ih je cijelo vrijeme prisluškivao. "Tko si ti" upitaju u isti glas. "Dečki, pojma vi nemate o filozofiji, a bome niti o minimumima niti o maksimumima!" odgovori im glas iz kabine. "No, možda se varam, evo dat ću vam ovaj mali zadatak koji sam sad smislio da vas testiram...Nek je $A=\{1,2,\ldots,99\} ,$ i nek su $X,Y$ dva (ne nužno različita) neprazna podskupa od $A.$ Nek $\textup{max} X$ i $\textup{min} Y$ označavaju najveći element skupa $X$ te najmanji element skupa $Y,$ redom. Odredite broj uređenih parova $(X,Y)$ za koje vrijedi $\textup{max} X>\textup{min} Y.$" Platon i Sokrat nakon dugog razmišljanja shvate kako nemaju pojma o čemu ovaj priča. "Predajemo se", kažu, "Barem nam reci tko si". "Ja sam najveći filozof svih vremena, znate" odgovorio im je. "Gracin. Krešimir Gracin." kaže dok Platon i Sokrat ostaju bez teksta. Kako niti Platon niti sokrat ne znaju odgovor na Krešimirovo pitanje, znate li vi? [/lang] [lang=en] After Socrates explained to Plato how he figured out the solution to the last problem, the two of them heard a mysterious laugh from the third booth—someone had been eavesdropping on them the whole time. "Who are you?" they asked in unison. "Guys, you have no idea about philosophy, and certainly not about minimums or maximums!" replied the voice from the booth. "Well, maybe I'm mistaken. Here, I'll give you this little problem I just thought of to test you... Let $ A = \{1, 2, \ldots, 99\} $, and let $ X, Y $ be two (not necessarily different) nonempty subsets of $ A $. Let $ \textup{max} X $ and $ \textup{min} Y $ denote the largest element of set $ X $ and the smallest element of set $ Y $, respectively. Determine the number of ordered pairs $ (X, Y) $ such that $ \textup{max} X > \textup{min} Y $." Plato and Socrates, after a long moment of thought, realized they had no idea what this was all about. "We give up," they said, "At least tell us who you are." "I am the greatest philosopher of all time, you know," the voice replied. "Krešimir Gracin," he said, leaving Plato and Socrates speechless. Since neither Plato nor Socrates knows the answer to Krešimir's question, do you? [/lang]