Detailed explanation of the first round of CSP-S 2019 improvement group

$\lfloor May$ I no longer fail myself, miracles no longer fail miracles$\rceil$

$\lfloor Rainbow$ after the storm, CSP-J/S 2020, come on! $\rceil$

Multiple choice

6.

Description:
How many different 4-digit numbers are composed of 1, 1, 2, 4, 8, 8?

Solution

Everyone doesn't want to enumerate violently, right? So I used the optimized version of enumeration to get answers faster in the examination room.

There are the following types of numbers:
① $1124$ type: 12 (1124, 1214, 4122, etc.)
②$1128$ type: 12 (1128, 1812, 8121, etc.)
③$.\; 1.\; 1.\; 4.\; 8$ Type: 12
④$1188$ type: 6 pcs
⑤$1248$ type: 24
⑥$1288$ type: 12
⑦$1488$ type: 12
⑧$2488$ type: 12

Add up to $102$ , so I happily choose B~

8.

Description

G is a non-connected undirected graph (no heavy edges and self-loops) with 28 edges in total. How many vertices does the graph have at least?

Solution

Simple structure questions.

We construct a graph like this: first draw a graph containing 8 8Complete graph with$8$ nodes, and then add an isolated point next to it.

Then not only meets the conditions, but also minimizes the number of vertices~

9.

Description

Some numbers can be inverted, such as 0, 1, and 8 upside down or themselves, 6 upside down, or 9, 9 upside down or 6, other numbers upside down do not constitute numbers. Similarly, some multiple digits can be reversed, for example, 106 is reversed to 901. Suppose the license plate of a certain city has only 5 digits, and each digit can be 0-9. How many license plates in this city are inverted and are still the original license plates, and the five digits on the license plate can be divisible by 3?

Solution

Anyway, I enumerate directly and violently in the examination room, there is nothing to say.