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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
2000 Vietnam Team Selection Test
2000 Vietnam Team Selection Test
Part of
Vietnam Team Selection Test
Subcontests
(3)
3
2
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game on polynomials
Two players alternately replace the stars in the expression
∗
x
2000
+
∗
x
1999
+
.
.
.
+
∗
x
+
1
*x^{2000}+*x^{1999}+...+*x+1
∗
x
2000
+
∗
x
1999
+
...
+
∗
x
+
1
by real numbers. The player who makes the last move loses if the resulting polynomial has a real root
t
t
t
with
∣
t
∣
<
1
|t| < 1
∣
t
∣
<
1
, and wins otherwise. Give a winning strategy for one of the players.
A collection of $2000$ congruent circles
A collection of
2000
2000
2000
congruent circles is given on the plane such that no two circles are tangent and each circle meets at least two other circles. Let
N
N
N
be the number of points that belong to at least two of the circles. Find the smallest possible value of
N
N
N
.
2
2
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$x_n = [an]$ for all $n$
Let
k
k
k
be a given positive integer. Define
x
1
=
1
x_{1}= 1
x
1
=
1
and, for each
n
>
1
n > 1
n
>
1
, set
x
n
+
1
x_{n+1}
x
n
+
1
to be the smallest positive integer not belonging to the set
{
x
i
,
x
i
+
i
k
∣
i
=
1
,
.
.
.
,
n
}
\{x_{i}, x_{i}+ik | i = 1, . . . , n\}
{
x
i
,
x
i
+
ik
∣
i
=
1
,
...
,
n
}
. Prove that there is a real number
a
a
a
such that
x
n
=
[
a
n
]
x_{n}= [an]
x
n
=
[
an
]
for all
n
∈
N
n \in\mathbb{ N}
n
∈
N
.
ineq function
Let
a
>
1
a > 1
a
>
1
and
r
>
1
r > 1
r
>
1
be real numbers. (a) Prove that if
f
:
R
+
→
R
+
f : \mathbb{R}^{+}\to\mathbb{ R}^{+}
f
:
R
+
→
R
+
is a function satisfying the conditions (i)
f
(
x
)
2
≤
a
x
r
f
(
x
a
)
f (x)^{2}\leq ax^{r}f (\frac{x}{a})
f
(
x
)
2
≤
a
x
r
f
(
a
x
)
for all
x
>
0
x > 0
x
>
0
, (ii)
f
(
x
)
<
2
2000
f (x) < 2^{2000}
f
(
x
)
<
2
2000
for all
x
<
1
2
2000
x < \frac{1}{2^{2000}}
x
<
2
2000
1
, then
f
(
x
)
≤
x
r
a
1
−
r
f (x) \leq x^{r}a^{1-r}
f
(
x
)
≤
x
r
a
1
−
r
for all
x
>
0
x > 0
x
>
0
. (b) Construct a function
f
:
R
+
→
R
+
f : \mathbb{R}^{+}\to\mathbb{ R}^{+}
f
:
R
+
→
R
+
satisfying condition (i) such that for all
x
>
0
,
f
(
x
)
>
x
r
a
1
−
r
x > 0, f (x) > x^{r}a^{1-r}
x
>
0
,
f
(
x
)
>
x
r
a
1
−
r
.
1
2
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prove 5 point lie on a circle
Two circles
C
1
C_{1}
C
1
and
C
2
C_{2}
C
2
intersect at points
P
P
P
and
Q
Q
Q
. Their common tangent, closer to
P
P
P
than to
Q
Q
Q
, touches
C
1
C_{1}
C
1
at
A
A
A
and
C
2
C_{2}
C
2
at
B
B
B
. The tangents to
C
1
C_{1}
C
1
and
C
2
C_{2}
C
2
at
P
P
P
meet the other circle at points
E
≠
P
E \not = P
E
=
P
and
F
≠
P
F \not = P
F
=
P
, respectively. Let
H
H
H
and
K
K
K
be the points on the rays
A
F
AF
A
F
and
B
E
BE
BE
respectively such that
A
H
=
A
P
AH = AP
A
H
=
A
P
and
B
K
=
B
P
BK = BP
B
K
=
BP
. Prove that
A
,
H
,
Q
,
K
,
B
A,H,Q,K,B
A
,
H
,
Q
,
K
,
B
lie on a circle.
stubborn numbers
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be pairwise coprime natural numbers. A positive integer
n
n
n
is said to be stubborn if it cannot be written in the form
n
=
b
c
x
+
c
a
y
+
a
b
z
n = bcx+cay+abz
n
=
b
c
x
+
c
a
y
+
ab
z
, for some
x
,
y
,
z
∈
N
.
x, y, z \in\mathbb{ N}.
x
,
y
,
z
∈
N
.
Determine the number of stubborn numbers.