2017 Taiwan TST Round 1

Part of TST Round 1

Subcontests

(5)

1

Polynomial with power of two

n

be an odd number larger than 1, and

f(x)

is a polynomial with degree

n

f(k)=2^k

k=0,1,\cdots,n

. Prove that there is only finite integer

x

f(x)

is the power of two.

1

Easy geometry from Taiwan TST

BC

EF

are parallel. Let

D

be a point on segment

BC

B

C

I

be the intersection of

BF

CE

. Denote the circumcircle of

\triangle CDE

\triangle BDF

K

L

K

L

are tangent with

EF

E

F

,respectively. Let

A

be the other intersection of circle

K

L

DF

K

intersect again at

Q

DE

L

intersect again at

R

EQ

FR

M

I

A

M

1

Inequality from 2017 Taiwan TST

a,b,c,d>0

\sum_{cyc}\frac{c}{a+2b}+\sum_{cyc}\frac{a+2b}{c}\geq 8(\frac{(a+b+c+d)^2}{ab+ac+ad+bc+bd+cd}-1),

\sum_{cyc}f(a,b,c,d)=f(a,b,c,d)+f(d,a,b,c)+f(c,d,a,b)+f(b,c,d,a)

1

Functional equation 006

Find all injective functions

f:\mathbb{N} \to \mathbb{N}

f^{f\left(a\right)}\left(b\right)f^{f\left(b\right)}\left(a\right)=\left(f\left(a+b\right)\right)^2

a,b \in \mathbb{N}

f^{k}\left(n\right)

\underbrace{f(f(\ldots f}_{k}(n) \ldots ))

3

Easy number theory 2017 Taiwan TST

For postive integers

k,n

f_k(n)=\sum_{m\mid n,m>0}m^k

Find all pairs of positive integer

(a,b)

f_a(n)\mid f_b(n)

for every positive integer

n

Concentric Circles

{\cal C}_1

{\cal C}_2

be concentric circles, with

{\cal C}_2

in the interior of

{\cal C}_1

A

{\cal C}_1

one draws the tangent

AB

{\cal C}_2

B\in {\cal C}_2

C

be the second point of intersection of

AB

{\cal C}_1

D

be the midpoint of

AB

. A line passing through

A

{\cal C}_2

E

F

in such a way that the perpendicular bisectors of

DE

CF

intersect at a point

M

AB

. Find, with proof, the ratio

AM/MC

Polynomial Functional Equation

Find all polynomials

P

with real coefficients which satisfy P(x)P(x+1)=P(x^2-x+3) \forall x \in \mathbb{R}