MathDB

1

Rectangle EFGH in incircle, prove that QIM = 90

ABC

be a triangle with incenter

I

, and suppose the incircle is tangent to

CA

and

AB

at

E

and

F

. Denote by

G

and

H

the reflections of

E

and

F

over

I

. Let

Q

be the intersection of

BC

with

GH

, and let

M

be the midpoint of

BC

. Prove that

IQ

and

IM

are perpendicular.

4

1

Midpoint of bisector; prove that B, E, F, C cyclic

Let

ABC

be an acute triangle and let

D

be the foot of the

A

-bisector. Moreover, let

M

be the midpoint of

AD

. The circle

\omega_1

with diameter

AC

meets

BM

at

E

, while the circle

\omega_2

with diameter

AB

meets

CM

at

F

. Assume that

E

and

F

lie inside

ABC

. Prove that

B

,

E

,

F

,

C

are concyclic.

2

3

Inequality on triangle with altitudes [Taiwan 2014 Quizzes]

A triangle has side lengths

a

,

b

,

c

, and the altitudes have lengths

h_a

,

h_b

,

h_c

. Prove that

\left( \frac{a}{h_a} \right)^2 + \left( \frac{b}{h_b} \right)^2 + \left( \frac{c}{h_c} \right)^2 \ge 4.

Arrange 10 sets in a circle

Determine whether there exist ten sets

A_1

,

A_2

,

\dots

,

A_{10}

such that (i) each set is of the form

\{a,b,c\}

, where

a \in \{1,2,3\}

,

b \in \{4,5,6\}

,

c \in \{7,8,9\}

, (ii) no two sets are the same, (iii) if the ten sets are arranged in a circle

(A_1, A_2, \dots, A_{10})

, then any two adjacent sets have no common element, but any two non-adjacent sets intersect. (Note:

A_{10}

is adjacent to

A_1

.)

Nice -- Find all polynomials f such that f(n) divides n!^k

For a fixed integer

k

, determine all polynomials

f(x)

with integer coefficients such that

f(n)

divides

(n!)^k

for every positive integer

n

.

1

4

2014 Taiwan TST Round 1

Subcontests

Rectangle EFGH in incircle, prove that QIM = 90

Midpoint of bisector; prove that B, E, F, C cyclic

Inequality on triangle with altitudes [Taiwan 2014 Quizzes]

Arrange 10 sets in a circle

Nice -- Find all polynomials f such that f(n) divides n!^k