MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
2001 Vietnam Team Selection Test
2001 Vietnam Team Selection Test
Part of
Vietnam Team Selection Test
Subcontests
(3)
3
1
Hide problems
we can find one pair of a boy and a girl
Some club has 42 members. It’s known that among 31 arbitrary club members, we can find one pair of a boy and a girl that they know each other. Show that from club members we can choose 12 pairs of knowing each other boys and girls.
2
2
Hide problems
H be a point symmetric to B under PT
In the plane let two circles be given which intersect at two points
A
,
B
A, B
A
,
B
; Let
P
T
PT
PT
be one of the two common tangent line of these circles (
P
,
T
P, T
P
,
T
are points of tangency). Tangents at
P
P
P
and
T
T
T
of the circumcircle of triangle
A
P
T
APT
A
PT
meet each other at
S
S
S
. Let
H
H
H
be a point symmetric to
B
B
B
under
P
T
PT
PT
. Show that
A
,
S
,
H
A, S, H
A
,
S
,
H
are collinear.
painted orthogonal coordinate system
Let an integer
n
>
1
n > 1
n
>
1
be given. In the space with orthogonal coordinate system
O
x
y
z
Oxyz
O
x
yz
we denote by
T
T
T
the set of all points
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
with
x
,
y
,
z
x, y, z
x
,
y
,
z
are integers, satisfying the condition:
1
≤
x
,
y
,
z
≤
n
1 \leq x, y, z \leq n
1
≤
x
,
y
,
z
≤
n
. We paint all the points of
T
T
T
in such a way that: if the point
A
(
x
0
,
y
0
,
z
0
)
A(x_0, y_0, z_0)
A
(
x
0
,
y
0
,
z
0
)
is painted then points
B
(
x
1
,
y
1
,
z
1
)
B(x_1, y_1, z_1)
B
(
x
1
,
y
1
,
z
1
)
for which
x
1
≤
x
0
,
y
1
≤
y
0
x_1 \leq x_0, y_1 \leq y_0
x
1
≤
x
0
,
y
1
≤
y
0
and
z
1
≤
z
0
z_1 \leq z_0
z
1
≤
z
0
could not be painted. Find the maximal number of points that we can paint in such a way the above mentioned condition is satisfied.
1
2
Hide problems
a_n= a_{n-1} + a_{[n/3]}
Let a sequence of integers
{
a
n
}
\{a_n\}
{
a
n
}
,
n
∈
N
n \in \mathbb{N}
n
∈
N
be given, defined by
a
0
=
1
,
a
n
=
a
n
−
1
+
a
[
n
/
3
]
a_0 = 1, a_n= a_{n-1} + a_{[n/3]}
a
0
=
1
,
a
n
=
a
n
−
1
+
a
[
n
/3
]
for all
n
∈
N
∗
n \in \mathbb{N}^{*}
n
∈
N
∗
. Show that for all primes
p
≤
13
p \leq 13
p
≤
13
, there are infinitely many integer numbers
k
k
k
such that
a
k
a_k
a
k
is divided by
p
p
p
. (Here
[
x
]
[x]
[
x
]
denotes the integral part of real number
x
x
x
).
minimum of 1/a + 1/b + 1/c
Let’s consider the real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfying the condition
21
⋅
a
⋅
b
+
2
⋅
b
⋅
c
+
8
⋅
c
⋅
a
≤
12.
21 \cdot a \cdot b + 2 \cdot b \cdot c + 8 \cdot c \cdot a \leq 12.
21
⋅
a
⋅
b
+
2
⋅
b
⋅
c
+
8
⋅
c
⋅
a
≤
12.
Find the minimal value of the expression
P
(
a
,
b
,
c
)
=
1
a
+
1
b
+
1
c
.
P(a, b, c) = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.
P
(
a
,
b
,
c
)
=
a
1
+
b
1
+
c
1
.