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Problems
Contests
National and Regional Contests
Taiwan Contests
TST Round 2
2005 Taiwan TST Round 2
2005 Taiwan TST Round 2
Part of
TST Round 2
Subcontests
(4)
3
1
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ellipse
In the interior of an ellipse with major axis 2 and minor axis 1, there are more than 6 segments with total length larger than 15. Prove that there exists a line passing through all of the segments.
4
1
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quadrilateral problem
A quadrilateral
P
Q
R
S
PQRS
PQRS
has an inscribed circle, the points of tangencies with sides
P
Q
PQ
PQ
,
Q
R
QR
QR
,
R
S
RS
RS
,
S
P
SP
SP
being
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
, respectively. Let the midpoints of
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
,
D
A
DA
D
A
be
E
E
E
,
F
F
F
,
G
G
G
,
H
H
H
, respectively. Prove that the angle between segments
P
R
PR
PR
and
Q
S
QS
QS
is equal to the angle between segments
E
G
EG
EG
and
F
H
FH
F
H
.
2
3
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difficult inequality
Find all positive integers
n
≥
3
n \ge 3
n
≥
3
such that there exists a positive constant
M
n
M_n
M
n
satisfying the following inequality for any
n
n
n
positive reals
a
1
,
a
2
,
…
,
a
n
a_1, a_2,\dots\>,a_n
a
1
,
a
2
,
…
,
a
n
:
a
1
+
a
2
+
⋯
+
a
n
a
1
a
2
⋯
a
n
n
≤
M
n
(
a
2
a
1
+
a
3
a
2
+
⋯
+
a
n
a
n
−
1
+
a
1
a
n
)
.
\displaystyle \frac{a_1+a_2+\cdots\>+a_n}{\sqrt[n]{a_1a_2\cdots\>a_n}} \le M_n \biggl( \frac{a_2}{a_1} + \frac{a_3}{a_2} +\cdots\>+ \frac{a_n}{a_{n-1}} + \frac {a_1}{a_n} \biggr).
n
a
1
a
2
⋯
a
n
a
1
+
a
2
+
⋯
+
a
n
≤
M
n
(
a
1
a
2
+
a
2
a
3
+
⋯
+
a
n
−
1
a
n
+
a
n
a
1
)
.
Moreover, find the minimum value of
M
n
M_n
M
n
for such
n
n
n
. The difficulty is finding
M
n
M_n
M
n
...
Interesting algorithm...
Starting from a positive integer
n
n
n
, we can replace the current number with a multiple of the current number or by deleting one or more zeroes from the decimal representation of the current number. Prove that for all values of
n
n
n
, it is possible to obtain a single-digit number by applying the above algorithm a finite number of times. There is a nice solution to this...
cyclic quadrilaterals
In
△
A
B
C
\triangle ABC
△
A
BC
,
A
D
AD
A
D
is the bisector of
∠
A
\angle A
∠
A
, and
E
E
E
,
F
F
F
are the feet of the perpendiculars from
D
D
D
to
A
C
AC
A
C
and
A
B
AB
A
B
, respectively.
H
H
H
is the intersection of
B
E
BE
BE
and
C
F
CF
CF
, and
G
G
G
,
I
I
I
are the feet of the perpendiculars from
D
D
D
to
B
E
BE
BE
and
C
F
CF
CF
, respectively. Prove that both
A
F
E
H
AFEH
A
FE
H
and
A
E
I
H
AEIH
A
E
I
H
are cyclic quadrilaterals.
1
5
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