2020 Taiwan TST Round 2

Part of TST Round 2

Subcontests

(6)

1

Exponential Growth

K

consists of at least 3 distinct positive integers. Suppose that

K

can be partitioned into two nonempty subsets

A,B\in K

ab+1

is always a perfect square whenever

a\in A

b\in B

\max_{k\in K}k\geq \left\lfloor (2+\sqrt{3})^{\min\{|A|,|B|\}-1}\right\rfloor+1,

|X|

stands for the cartinality of the set

X

x\in \mathbb{R}

\lfloor x\rfloor

is the greatest integer that does not exceed

x

1

Cool Optimization Problem on Trees.

n

cities in a country, where

n>1

. There are railroads connecting some of the cities so that you can travel between any two cities through a series of railroads (railroads run in both direction.) In addition, in this country, it is impossible to travel from a city, through a series of distinct cities, and return back to the original city. We define the degree of a city as the number of cities directly connected to it by a single segment of railroad. For a city

A

that is directly connected to

x

y

of those cities having a smaller degree than city

A

, the significance of city

A

\frac{y}{x}

.
Find the smallest positive real number

t

so that, for any

n>1

, the sum of the significance of all cities is less than

tn

, no matter how the railroads are paved.
Proposed by houkai

1

Another simple functional equation

\mathbb{R}

denote the set of all real numbers. Determine all functions

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

such that for all real numbers

x

y

f(xy+xf(x))=f(x)\left(f(x)+f(y)\right).

1

Triangle counting

N

acute triangles on the plane. Their vertices are all integer points, their areas are all equal to

2^{2020}

, but no two of them are congruent. Find the maximum possible value of

N

(x,y)

is an integer point if and only if

x

y

are both integers.
Proposed by CSJL

1

Games in Quarantine

Alice and Bob are stuck in quarantine, so they decide to play a game. Bob will write down a polynomial

f(x)

with the following properties:
(a) for any integer

n

f(n)

is an integer; (b) the degree of

f(x)

187

.
Alice knows that

f(x)

satisfies (a) and (b), but she does not know

f(x)

. In every turn, Alice picks a number

k

\{1,2,\ldots,187\}

, and Bob will tell Alice the value of

f(k)

. Find the smallest positive integer

N

so that Alice always knows for sure the parity of

f(0)

N

turns.
Proposed by YaWNeeT

1

Easy Geometry with Simple Property about OI Line

I, O, \omega, \Omega

be the incenter, circumcenter, the incircle, and the circumcircle, respectively, of a scalene triangle

ABC

\omega

is tangent to side

BC

D

S

be the point on the circumcircle

\Omega

AS, OI, BC

are concurrent. Let

H

be the orthocenter of triangle

BIC

T

\Omega

\angle ATI

is a right angle. Prove that the points

D, T, H, S

are concyclic.
Proposed by ltf0501