2024 Taiwan TST Round 1

Part of TST Round 1

Subcontests

(3)

1

partitioning 1 to p-1 into several a+b=c (mod p)

Given a prime number

p

, a set is said to be

p

-good if the set contains exactly three elements

a, b, c

a + b \equiv c \pmod{p}

. Find all prime number

p

\{ 1, 2, \cdots, p-1 \}

can be partitioned into several

p

-good sets.
Proposed by capoouo

1

Geometry in Taiwan TST

For the quadrilateral

ABCD

AC

BD

E

AB

CD

F

AD

BC

G

. Additionally, let

W, X, Y

Z

be the points of symmetry to

E

with respect to

AB, BC, CD,

DA

respectively. Prove that one of the intersection points of

\odot(FWY)

\odot(GXZ)

lies on the line

FG

.
Proposed by chengbilly

2

Spy Family Game

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

Family of sets

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