2023 Taiwan TST Round 2

Part of TST Round 2

Subcontests

(6)

1

Line drawing game

There is an equilateral triangle

ABC

on the plane. Three straight lines pass through

A

B

C

, respectively, such that the intersections of these lines form an equilateral triangle inside

ABC

. On each turn, Ming chooses a two-line intersection inside

ABC

, and draws the straight line determined by the intersection and one of

A

B

C

of his choice. Find the maximum possible number of three-line intersections within

ABC

after 300 turns.
Proposed by usjl

1

3k+1cities with 4k roads

n

k

n > 2023k^3

. Kingdom Kitty has

n

cities, with at most one road between each pair of cities. It is known that the total number of roads in the kingdom is at least

2n^{3/2}

. Prove that we can choose

3k + 1

cities such that the total number of roads with both ends being a chosen city is at least

4k

1

A delicious geometry in Taiwan TST

\Omega

be the circumcircle of an acute triangle

ABC

D

E

F

are the midpoints of the inferior arcs

BC

CA

AB

, respectively, on

\Omega

G

be the antipode of

D

\Omega

X

be the intersection of lines

GE

AB

Y

the intersection of lines

FG

CA

. Let the circumcenters of triangles

BEX

CFY

S

T

, respectively. Prove that

D

S

T

are collinear. Proposed by kyou46 and Li4.

1

True or False?

Is there a scalene triangle

ABC

similar to triangle

IHO

I

H

O

are the incenter, orthocenter, and circumcenter, respectively, of triangle

ABC

?
Proposed by Li4 and usjl.

2

Joints type inequality

For each positive integer

k

1

, find the largest real number

t

such that the following hold: Given

n

distinct points

a^{(1)}=(a^{(1)}_1,\ldots, a^{(1)}_k)

\ldots

a^{(n)}=(a^{(n)}_1,\ldots, a^{(n)}_k)

\mathbb{R}^k

, we define the score of the tuple

a^{(i)}

\prod_{j=1}^{k}\#\{1\leq i'\leq n\textup{ such that }\pi_j(a^{(i')})=\pi_j(a^{(i)})\}

\#S

is the number of elements in set

S

\pi_j

is the projection

\mathbb{R}^k\to \mathbb{R}^{k-1}

j

-th coordinate. Then the

t

-th power mean of the scores of all

a^{(i)}

n

t

-th power mean of positive real numbers

x_1,\ldots,x_n

\left(\frac{x_1^t+\cdots+x_n^t}{n}\right)^{1/t}

t\neq 0

\sqrt[n]{x_1\cdots x_n}

t=0

.
Proposed by Cheng-Ying Chang and usjl

Function equation in Taiwan TST

Find all functions

f : \mathbb{R} \to \mathbb{R}

f\left(xy+f(y)\right)f(x)=x^2f(y)+f(xy)

x,y \in \mathbb{R}

Proposed by chengbilly

2

Sequence of polynomials

f_n

be a polynomial with real coefficients for all

n \in \mathbb{Z}

. Suppose that f_n(k) = f_{n+k}(k) n, k \in \mathbb{Z}. (a) Does

f_n = f_m

necessarily hold for all

m,n \in \mathbb{Z}

? (b) If furthermore

f_n

is a polynomial with integer coefficients for all

n \in\mathbb{Z}

f_n = f_m

necessarily hold for all

m, n \in\mathbb{Z}

?
Proposed by usjl

Polynomials evaluated to denominator

Find all polynomials

P

with real coefficients satisfying that there exist infinitely many pairs

(m, n)

of coprime positives integer such that

P(\frac{m}{n})=\frac{1}{n}

.
Proposed by usjl