2014 Taiwan TST Round 2

Part of TST Round 2

Subcontests

(2)

1

Easy for a #6 -- Prove AP and AQ are isogonal

P

be a point inside triangle

ABC

, and suppose lines

AP

BP

CP

meet the circumcircle again at

T

S

R

T \neq A

S \neq B

R \neq C

U

be any point in the interior of

PT

. A line through

U

AB

CR

W

, and the line through

U

AC

BS

V

. Finally, the line through

B

CP

and the line through

C

BP

intersect at point

Q

RS

VW

are parallel, prove that

\angle CAP = \angle BAQ

2

Prove that A, X, O, Y are concyclic

ABC

be a triangle with incenter

I

and circumcenter

O

. A straight line

L

BC

and tangent to the incircle. Suppose

L

IO

X

Y

L

YI

is perpendicular to

IO

A

X

O

Y

are cyclic.
Proposed by Telv Cohl

n-variable product of kth powers [Taiwan 2014 Quizzes]

a_i > 0

i=1,2,\dots,n

a_1 + a_2 + \dots + a_n = 1

. Prove that for any positive integer

k

\left( a_1^k + \frac{1}{a_1^k} \right) \left( a_2^k + \frac{1}{a_2^k} \right) \dots \left( a_n^k + \frac{1}{a_n^k} \right) \ge \left( n^k + \frac{1}{n^k} \right)^n.