Math

Puzzle Play music “Pascual: España Cañí Pasodoble” to set the vibe. In a bullring, a brave matador is engaged in a splendid fight with a furious bull. The matador is agile, meaning that he can change his velocity immediately (even at accelerations up to 100g), but he has a maximum velocity. In contrast, the bull has no upper limit on its velocity, but it has a maximum acceleration due to its massive build. The bull wants to gore the matador while the matador wants to dodge the bull’s charge. Can the bull knock down the matador in a finite time? Or can the matador always dodge the bull’s charge for as long as he wants? Note: we model both the matador and the bull as a point without volume. ...

Notation $[n]$ denotes $\{1,2,\cdots,n\}$. $[m:n]$ denotes $\{m,m+1,\cdots,n\}$ if $m\geq n$. It’s an empty set otherwise. $a_{m:n}$ denotes $a_m,a_{m+1},\cdots,a_n$. $\{\{1,1,2^{(3)}\}\}$ denotes the multiset with two 1s and three 2s in it. The Problem Given several positive integers $a_{1:n}$, what’s the maximum value that can be achieved by four basic operations, additions, subtractions, multiplications and divisions? Every number is used exactly once. (Provider: 雷伊布是真的狗 aka 雷狗) For example, given 1,1,2,3, a possible result is $4/3=(1+1)\times 2/3$ but the largest result is $12=(1+1)\times 2\times3$. ...