2020 Taiwan TST Round 1

Part of TST Round 1

Subcontests

(4)

1

I swear this is the last Taiwan TST I'm gonna post

N>2^{5000}

be a positive integer. Prove that if

1\leq a_1<\cdots<a_k<100

are distinct positive integers then the number

\prod_{i=1}^{k}\left(N^{a_i}+a_i\right)

k

distinct prime factors.
Note. Results with

2^{5000}

replaced by some other constant

N_0

will be awarded points depending on the value of

N_0

.
Proposed by Evan Chen

2

Bisecting Segment

H

be the orthocenter of a scalene triangle

ABC

AH

intersects with the circumcircle

\Omega

ABC

P

BH, CH

AC,AB

E

F

, respectively. Let

PE, PF

\Omega

Q,R

, respectively. Point

Y

\Omega

AY,QR

EF

are concurrent. Prove that

PY

EF

A simple functional equation

\mathbb{R}

be the set of all real numbers. Find all functions

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

such that for any

x,y\in \mathbb{R}

f(x+f(y))+f(xy)=yf(x)+f(y)+f(f(x)).

1

Equilateral triangle

O

be the center of the equilateral triangle

ABC

. Pick two points

P_1

P_2

B

O

C

\odot(BOC)

so that on this circle

B

P_1

P_2

O

C

are placed in this order. Extensions of

BP_1

CP_1

intersects respectively with side

CA

AB

R

S

AP_1

RS

intersects at point

Q_1

. Analogously point

Q_2

is defined. Let

\odot(OP_1Q_1)

\odot(OP_2Q_2)

meet again at point

U

O

2\,\angle Q_2UQ_1 + \angle Q_2OQ_1 = 360^\circ

\odot(XYZ)

denotes the circumcircle of triangle

XYZ

1

Easy inequality

a

b

c

d

be real numbers satisfying \begin{align*} (a + c)(b + d) = \sqrt{2}(ac - 2bd - 1). \end{align*} Show that \begin{align*} (ab - 1)^2 + (bc - 1)^2 + (cd - 1)^2 + (da - 1)^2 + (ac - 1)^2 + (2bd + 1)^2 \ge 4. \end{align*}