2019 Taiwan TST Round 1

Part of TST Round 1

Subcontests

(4)

1

Mathematical Reflections

Given a triangle

\triangle ABC

. Denote its incenter and orthocenter by

I, H

, respectively. If there is a point

K

AH+AK = BH+BK = CH+CK

H, I, K

are collinear.
Proposed by Evan Chen

1

So Many Arrows

Find all positive integers

n

with the following property: It is possible to fill a

n \times n

chessboard with one of arrows

\uparrow, \downarrow, \leftarrow, \rightarrow

such that
1. Start from any grid, if we follows the arrows, then we will eventually go back to the start point.
2. For every row, except the first and the last, the number of

\uparrow

and the number of

\downarrow

are the same.
3. For every column, except the first and the last, the number of

\leftarrow

and the number of

\rightarrow

3

Extremely Hard Inequality :>

a_{1} \ge a_{2} \ge \dots \ge a_{107} > 0

\sum\limits_{k=1}^{107}{a_{k}} \ge M

b_{107} \ge b_{106} \ge \dots \ge b_{1} > 0

\sum\limits_{k=1}^{107}{b_{k}} \le M

. Prove that for any

m \in \{1,2, \dots, 107\}

, the arithmetic mean of the following numbers

\frac{a_{1}}{b_{1}}, \frac{a_{2}}{b_{2}}, \dots, \frac{a_{m}}{b_{m}}

is greater than or equal to

\frac{M}{N}

The Only One Can Beat 9 Points, is 9-Point

Given a triangle

\triangle{ABC}

with orthocenter

H

. On its circumcenter, choose an arbitrary point

P

A,B,C

M

be the mid-point of

HP

. Now, we find three points

D,E,F

BC, CA, AB

, respectively, such that

AP \parallel HD, BP \parallel HE, CP \parallel HF

D, E, F, M

Functional equation 013

Find all functions

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

f\left(xf\left(y\right)-f\left(x\right)-y\right) = yf\left(x\right)-f\left(y\right)-x

x,y \in \mathbb{R}

4