Marinada '25 - Školjka

Vrijeme: 11:06

Četverodimenzionalna komba | 4D Combinatorics #5

Karlo je zatim počeo razmišljati kako bi cijelom ovom dubokoumnom lancu htio dati poučan završetak i podučiti čitatelja kako rješavati olimpijske zadatke koristeći 4D prostor. Puno stvari u životu može biti simulirano u 4D (n-D) prostoru. Naprimjer, četveroznamenkasti pin može biti prikazan kao $10 \times 10 \times 10 \times 10$ hiperkocka (uzevši u obzir da uubacujemo znamenke od $0$ do $9$ ). Naravno, kompleksni kombinatorni zadatci se uglavnom olakšaju kad se vizualiziraju (misli IMO 2024 P3). Jedan problem koji showcasea ovu ideju (i u duhu je ovog lanca) je sljedeći:

Uzmimo sve djeljitelje nekog $n$ i poredajmo ih u niz tako da vrijedi da je barem jedan od $\frac{p_i}{p_{i+1}}$ ili $\frac{p_{i+1}}{p_i}$ prost broj. Dokaži da je za bilo koji $n$ moguće tako poredati djelitelje (naprimjer, niz $2,3,6$ ne radi, ali $3,6,2$ ) radi.

Kad si dokazao ovo (a stvarno je cool dokaz ako koristiš n dimenzija), kao odgovor na ovo pitanje napiši "ok".

Karlo then began thinking about how he could give a meaningful conclusion to this deep chain of thoughts and teach the reader how to solve olympiad-style problems using 4D space. Many things in life can be simulated in 4D (or n-dimensional) space. For example, a four-digit PIN can be represented as a $10\times 10\times 10\times 10$ hypercube (taking into account that we insert digits from 0 to 9). Naturally, complex combinatorial problems are often easier to tackle when visualized (think IMO 2024 P3).

One problem that showcases this idea (and is in the spirit of this chain) is the following:

Consider all the divisors of a positive integer $n$ , and arrange them in a sequence such that for each consecutive pair $p_i, p_{i+1}$ , at least one of the ratios $\frac{p_i}{p_{i+1}} \quad \text{or} \quad \frac{p_{i+1}}{p_i}$ is a prime number. Prove that it is always possible to arrange the divisors of any $n$ in such a sequence.

Example. The sequence $2,3,6$ does not work, but $3,6,2$ does.

As the answer to this question, input "ok".

[lang=hr] Karlo je zatim počeo razmišljati kako bi cijelom ovom dubokoumnom lancu htio dati poučan završetak i podučiti čitatelja kako rješavati olimpijske zadatke koristeći 4D prostor. Puno stvari u životu može biti simulirano u 4D (n-D) prostoru. Naprimjer, četveroznamenkasti pin može biti prikazan kao $10 \times 10 \times 10 \times 10$ hiperkocka (uzevši u obzir da uubacujemo znamenke od $0$ do $9$). Naravno, kompleksni kombinatorni zadatci se uglavnom olakšaju kad se vizualiziraju (misli IMO 2024 P3). Jedan problem koji showcasea ovu ideju (i u duhu je ovog lanca) je sljedeći: Uzmimo sve djeljitelje nekog $n$ i poredajmo ih u niz tako da vrijedi da je barem jedan od $ \frac{p_i}{p_{i+1}} $ ili $\frac{p_{i+1}}{p_i}$ prost broj. Dokaži da je za bilo koji $n$ moguće tako poredati djelitelje (naprimjer, niz $2,3,6$ ne radi, ali $3,6,2$) radi. Kad si dokazao ovo (a stvarno je cool dokaz ako koristiš n dimenzija), kao odgovor na ovo pitanje napiši "ok". [/lang] [lang=en] Karlo then began thinking about how he could give a meaningful conclusion to this deep chain of thoughts and teach the reader how to solve olympiad-style problems using 4D space. Many things in life can be simulated in 4D (or n-dimensional) space. For example, a four-digit PIN can be represented as a $10\times 10\times 10\times 10$ hypercube (taking into account that we insert digits from 0 to 9). Naturally, complex combinatorial problems are often easier to tackle when visualized (think IMO 2024 P3). One problem that showcases this idea (and is in the spirit of this chain) is the following: Consider all the divisors of a positive integer $n$, and arrange them in a sequence such that for each consecutive pair $p_i, p_{i+1}$, at least one of the ratios \[ \frac{p_i}{p_{i+1}} \quad \text{or} \quad \frac{p_{i+1}}{p_i} \] is a prime number. Prove that it is always possible to arrange the divisors of any $n$ in such a sequence. \textbf{Example.} The sequence $2,3,6$ does not work, but $3,6,2$ does. As the answer to this question, input "ok". [/lang]