MathDB

The winners of first AGMC in 2019 gifts the person in charge of the organiser, which is a polyhedron formed by

60

congruent triangles. From the photo, we can see that this polyhedron formed by

60

quadrilateral spaces. (Note: You can find the photo in 3.4 of https://files.alicdn.com/tpsservice/18c5c7b31a7074edc58abb48175ae4c3.pdf?spm=a1zmmc.index.0.0.51c0719dNAbw3C&file=18c5c7b31a7074edc58abb48175ae4c3.pdf) A quadrilateral space is the plane figures that we fold the figures following the diagonal on a

n

sides on a plane (i.e. form an appropriate dihedral angle in where the chosen diagonal is). "Two figure spaces are congruent" means they can coincide completely by isometric transform in

\mathbb{R}^3

. A polyhedron is the bounded space region, whose boundary is formed by the common edge of finite polygon. (a) We know that

2021=43\times 47

. Does there exist a polyhedron, whose surface can be formed by

43

congruent

47

-gon? (b) Prove your answer in (a) with logical explanation.

Consider a computer network consisting of servers and bi-directional communication channels among them. Unfortunately, not all channels operate. Each direction of each channel fails with probability

p

and operates otherwise. (All of these stochastic events are mutually independent, and

0 \le p \le 1

.) There is a root serve, denoted by

r

. We call the network operational, if all serves can reach

r

using only operating channels. Note that we do not require

r

to be able to reach any servers.
Show that the probability of the network to be operational does not depend on the choice of

r

. (In other words, for any two distinct root servers

r_1

and

r_2

, the operational probability is the same.)

2

Problems(2)