2019 Taiwan TST Round 3

Part of TST Round 3

Subcontests

(4)

1

Last Problem :D

Given a triangle

\triangle{ABC}

with circumcircle

\Omega

. Denote its incenter and

A

I, J

, respectively. Let

T

be the reflection of

J

BC

P

is the intersection of

BC

AT

. If the circumcircle of

\triangle{AIP}

BC

X \neq P

and there is a point

Y \neq A

\Omega

IA = IY

\odot\left(IXY\right)

tangents to the line

AI

1

Permutations of Puddings

n

kinds of puddings. There are

a_{i}

puddings which are

i

-th type and those

S = a_{1}+\cdots+a_{n}

puddings are distinct. Now, for a given arrangement of puddings:

p_{1}, \dots, p_{n}

c_{i}

\#\{1 \le j \le S-1 \ \mid \ p_{j}, p_{j+1} \text{ are the same type.}\}

S

is composite, then the sum of

\prod_{i=1}^{n}{c_{i}}

over all possible arrangements is a multiple of

S

2

JustSleep

p = 8k+1

for some integer

k

r

be the remainder when

\binom{4k}{k}

p

\sqrt{r}

is not an integer.
Proposed by Evan Chen

Simple Graph is not Simple

Given a simple graph with

4038

vertices. Assume we arbitrarily choose

2019

vertices as a group (the other

2019

is another group, of course), there are always

k

edges that connect two groups. Find all possible value of

k

2

Sum of Numbers in Rectangle isnt Prime

For each lattice point on the Cartesian Coordinate Plane, one puts a positive integer at the lattice such that for any given rectangle with sides parallel to coordinate axes, the sum of the number inside the rectangle is not a prime. Find the minimal possible value of the maximum of all numbers.

Perpendicularity in the configuration consisting of Four Circumcenters

\triangle ABC

P.

O

D

E

F

be the circumcenter of

\triangle ABC

\triangle BPC

\triangle CPA

\triangle APB,

respectively and let

T

be the intersection of

BC

EF.

Prove that the reflection of

O

EF

lies on the perpendicular from

D

PT.

Proposed by Telv Cohl