MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
2004 Vietnam Team Selection Test
2004 Vietnam Team Selection Test
Part of
Vietnam Team Selection Test
Subcontests
(3)
3
2
Hide problems
there are two circles intersecting each at two points
In the plane, there are two circles
Γ
1
,
Γ
2
\Gamma_1, \Gamma_2
Γ
1
,
Γ
2
intersecting each other at two points
A
A
A
and
B
B
B
. Tangents of
Γ
1
\Gamma_1
Γ
1
at
A
A
A
and
B
B
B
meet each other at
K
K
K
. Let us consider an arbitrary point
M
M
M
(which is different of
A
A
A
and
B
B
B
) on
Γ
1
\Gamma_1
Γ
1
. The line
M
A
MA
M
A
meets
Γ
2
\Gamma_2
Γ
2
again at
P
P
P
. The line
M
K
MK
M
K
meets
Γ
1
\Gamma_1
Γ
1
again at
C
C
C
. The line
C
A
CA
C
A
meets
Γ
2
\Gamma_2
Γ
2
again at
Q
Q
Q
. Show that the midpoint of
P
Q
PQ
PQ
lies on the line
M
C
MC
MC
and the line
P
Q
PQ
PQ
passes through a fixed point when
M
M
M
moves on
Γ
1
\Gamma_1
Γ
1
. [Moderator edit: This problem was also discussed on http://www.mathlinks.ro/Forum/viewtopic.php?t=21414 .]
greatest element from S
Let
S
S
S
be the set of positive integers in which the greatest and smallest elements are relatively prime. For natural
n
n
n
, let
S
n
S_n
S
n
denote the set of natural numbers which can be represented as sum of at most
n
n
n
elements (not necessarily different) from
S
S
S
. Let
a
a
a
be greatest element from
S
S
S
. Prove that there are positive integer
k
k
k
and integers
b
b
b
such that
∣
S
n
∣
=
a
⋅
n
+
b
|S_n| = a \cdot n + b
∣
S
n
∣
=
a
⋅
n
+
b
for all
n
>
k
n > k
n
>
k
.
2
2
Hide problems
function from VietNam
Find all real values of
α
\alpha
α
, for which there exists one and only one function
f
:
R
↦
R
f: \mathbb{R} \mapsto \mathbb{R}
f
:
R
↦
R
and satisfying the equation
f
(
x
2
+
y
+
f
(
y
)
)
=
(
f
(
x
)
)
2
+
α
⋅
y
f(x^2 + y + f(y)) = (f(x))^2 + \alpha \cdot y
f
(
x
2
+
y
+
f
(
y
))
=
(
f
(
x
)
)
2
+
α
⋅
y
for all
x
,
y
∈
R
x, y \in \mathbb{R}
x
,
y
∈
R
.
convex hexagon
Let us consider a convex hexagon ABCDEF. Let
A
1
,
B
1
,
C
1
,
D
1
,
E
1
,
F
1
A_1, B_1,C_1, D_1, E_1, F_1
A
1
,
B
1
,
C
1
,
D
1
,
E
1
,
F
1
be midpoints of the sides
A
B
,
B
C
,
C
D
,
D
E
,
E
F
,
F
A
AB, BC, CD, DE, EF,FA
A
B
,
BC
,
C
D
,
D
E
,
EF
,
F
A
respectively. Denote by
p
p
p
and
p
1
p_1
p
1
, respectively, the perimeter of the hexagon
A
B
C
D
E
F
A B C D E F
A
BC
D
EF
and hexagon
A
1
B
1
C
1
D
1
E
1
F
1
A_1B_1C_1D_1E_1F_1
A
1
B
1
C
1
D
1
E
1
F
1
. Suppose that all inner angles of hexagon
A
1
B
1
C
1
D
1
E
1
F
1
A_1B_1C_1D_1E_1F_1
A
1
B
1
C
1
D
1
E
1
F
1
are equal. Prove that
p
≥
2
⋅
3
3
⋅
p
1
.
p \geq \frac{2 \cdot \sqrt{3}}{3} \cdot p_1 .
p
≥
3
2
⋅
3
⋅
p
1
.
When does equality hold ?
1
2
Hide problems
distinct elements with gcd greater than 1
Let us consider a set
S
=
{
a
1
<
a
2
<
…
<
a
2004
}
S = \{ a_1 < a_2 < \ldots < a_{2004}\}
S
=
{
a
1
<
a
2
<
…
<
a
2004
}
, satisfying the following properties:
f
(
a
i
)
<
2003
f(a_i) < 2003
f
(
a
i
)
<
2003
and f(a_i) = f(a_j) \forall i, j from
{
1
,
2
,
…
,
2004
}
\{1, 2,\ldots , 2004\}
{
1
,
2
,
…
,
2004
}
, where
f
(
a
i
)
f(a_i)
f
(
a
i
)
denotes number of elements which are relatively prime with
a
i
a_i
a
i
. Find the least positive integer
k
k
k
for which in every
k
k
k
-subset of
S
S
S
, having the above mentioned properties there are two distinct elements with greatest common divisor greater than 1.
sequence from VietNam
Let
{
x
n
}
\left\{x_n\right\}
{
x
n
}
, with n \equal{} 1, 2, 3, \ldots, be a sequence defined by x_1 \equal{} 603, x_2 \equal{} 102 and x_{n \plus{} 2} \equal{} x_{n \plus{} 1} \plus{} x_n \plus{} 2\sqrt {x_{n \plus{} 1} \cdot x_n \minus{} 2}
∀
n
≥
1
\forall n \geq 1
∀
n
≥
1
. Show that: (1) The number
x
n
x_n
x
n
is a positive integer for every
n
≥
1
n \geq 1
n
≥
1
. (2) There are infinitely many positive integers
n
n
n
for which the decimal representation of
x
n
x_n
x
n
ends with 2003. (3) There exists no positive integer
n
n
n
for which the decimal representation of
x
n
x_n
x
n
ends with 2004.