C

Part of 2024 Taiwan TST Round 1

Problems(2)

Spy Family Game

Source: 2024 Taiwan TST Round 1 Independent Study 1-C

n \geq 5

be a positive integer. There are

n

stars with values

1

n

, respectively. Anya and Becky play a game. Before the game starts, Anya places the

n

stars in a row in whatever order she wishes. Then, starting from Becky, each player takes the left-most or right-most star in the row. After all the stars have been taken, the player with the highest total value of stars wins; if their total values are the same, then the game ends in a draw. Find all

n

such that Becky has a winning strategy.
Proposed by Ho-Chien Chen

Taiwancombinatorics

Source: 2024 Taiwan TST Round 1 Independent Study 2-C

k

-set is a set with exactly

k

elements. For a

6

A

and any collection

\mathcal{F}

4

-sets, we say that

A

\mathcal{F}

-good if there are exactly three elements

B_1, B_2, B_3

\mathcal{F}

that are subsets of

A

, and they furthermore satisfy

(A \backslash B_1) \cup (A \backslash B_2) \cup (A \backslash B_3) = A.

n \geq 6

so that there exists a collection

\mathcal{F}

4

\{1, 2, \ldots , n\}

such that every

6

A \subseteq \{1, 2, \ldots , n\}

\mathcal{F}

-good.
Proposed by usjl

combinatoricsTaiwan