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Most interesting problem

Hey Brilliantinians, here I'm posting the most beautiful problem I've ever seen. The idea is that everyone can share the most interesting/beautiful/awesome problem he/she has ever faced to. The problem needs not to come from Brilliant. Solutions to the problems are also accepted in the comments

Here's mine. Let $$p\geq 3$$ be a prime number. We divide every side of a triangle in $$p$$ equal parts and every point of division is connected to the opposite vertex. Compute the maximum number of pairwise disjoint parts in which the triangle is divided in funciton of $$p$$.

Enjoy!!!

Note by Jordi Bosch
2 years, 3 months ago

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Well, if $$p$$ is an odd prime, then the maximum such number must be $$3{ p }^{ 2 }-3p+1$$

so that for $$p={3,5,7,11,13,19,...}$$, we have $${19,61,127,331,469,817,...}$$

We first note that the triangle is divided into $${ p }^{ 2 }$$ disjoint parts by the first $$2$$ sets of rays from $$2$$ vertices. Then each ray from the last vertex intersects those rays $$2(p-1)$$ times, thereby increasing the number of disjoint parts by $$2(p-1)+1$$, and there are $$p-1$$ such rays from that last vertex. Hence, the total is

$${ p }^{ 2 }+(p-1)(2(p-1)+1)=3{ p }^{ 2 }-3p+1$$

This derivation depends on the fact that no point is shared by $$3$$ such rays, hence the primes only.

This graphic shows the case where $$p=5$$, so one can see how this is done. Affine transforms of this triangle leaves the number of parts unaffected.

· 1 year, 9 months ago