MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
2012 Vietnam Team Selection Test
2012 Vietnam Team Selection Test
Part of
Vietnam Team Selection Test
Subcontests
(3)
2
2
Hide problems
Minimum number of fountains to spray each square
Consider a
m
×
n
m\times n
m
×
n
rectangular grid with
m
m
m
rows and
n
n
n
columns. There are water fountains on some of the squares. A water fountain can spray water onto any of it's adjacent squares, or a square in the same column such that there is exactly one square between them. Find the minimum number of fountains such that each square can be sprayed in the case that a)
m
=
4
m=4
m
=
4
; b)
m
=
3
m=3
m
=
3
.
Best constant c for the positive reals a_i
Prove that
c
=
10
24
c=10\sqrt{24}
c
=
10
24
is the largest constant such that if there exist positive numbers
a
1
,
a
2
,
…
,
a
17
a_1,a_2,\ldots ,a_{17}
a
1
,
a
2
,
…
,
a
17
satisfying:
∑
i
=
1
17
a
i
2
=
24
,
∑
i
=
1
17
a
i
3
+
∑
i
=
1
17
a
i
<
c
\sum_{i=1}^{17}a_i^2=24,\ \sum_{i=1}^{17}a_i^3+\sum_{i=1}^{17}a_i<c
i
=
1
∑
17
a
i
2
=
24
,
i
=
1
∑
17
a
i
3
+
i
=
1
∑
17
a
i
<
c
then for every
i
,
j
,
k
i,j,k
i
,
j
,
k
such that
1
≤
1
<
j
<
k
≤
17
1\le 1<j<k\le 17
1
≤
1
<
j
<
k
≤
17
, we have that
x
i
,
x
j
,
x
k
x_i,x_j,x_k
x
i
,
x
j
,
x
k
are sides of a triangle.
3
2
Hide problems
When ax+by+cz+d is divisible by p
Let
p
≥
17
p\ge 17
p
≥
17
be a prime. Prove that
t
=
3
t=3
t
=
3
is the largest positive integer which satisfies the following condition: For any integers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
such that
a
b
c
abc
ab
c
is not divisible by
p
p
p
and
(
a
+
b
+
c
)
(a+b+c)
(
a
+
b
+
c
)
is divisible by
p
p
p
, there exists integers
x
,
y
,
z
x,y,z
x
,
y
,
z
belonging to the set
{
0
,
1
,
2
,
…
,
⌊
p
t
⌋
−
1
}
\{0,1,2,\ldots , \left\lfloor \frac{p}{t} \right\rfloor - 1\}
{
0
,
1
,
2
,
…
,
⌊
t
p
⌋
−
1
}
such that
a
x
+
b
y
+
c
z
+
d
ax+by+cz+d
a
x
+
b
y
+
cz
+
d
is divisible by
p
p
p
.
42 students split into groups
There are
42
42
42
students taking part in the Team Selection Test. It is known that every student knows exactly
20
20
20
other students. Show that we can divide the students into
2
2
2
groups or
21
21
21
groups such that the number of students in each group is equal and every two students in the same group know each other.
1
2
Hide problems
Sequence involving 2011 gives pefect square
Consider the sequence
(
x
n
)
n
≥
1
(x_n)_{n\ge 1}
(
x
n
)
n
≥
1
where
x
1
=
1
,
x
2
=
2011
x_1=1,x_2=2011
x
1
=
1
,
x
2
=
2011
and
x
n
+
2
=
4022
x
n
+
1
−
x
n
x_{n+2}=4022x_{n+1}-x_n
x
n
+
2
=
4022
x
n
+
1
−
x
n
for all
n
∈
N
n\in\mathbb{N}
n
∈
N
. Prove that
x
2012
+
1
2012
\frac{x_{2012}+1}{2012}
2012
x
2012
+
1
is a perfect square.
Tangents to cirumcircle meet at the fixed point T
Consider a circle
(
O
)
(O)
(
O
)
and two fixed points
B
,
C
B,C
B
,
C
on
(
O
)
(O)
(
O
)
such that
B
C
BC
BC
is not the diameter of
(
O
)
(O)
(
O
)
.
A
A
A
is an arbitrary point on
(
O
)
(O)
(
O
)
, distinct from
B
,
C
B,C
B
,
C
. Let
D
,
J
,
K
D,J,K
D
,
J
,
K
be the midpoints of
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
, respectively,
E
,
M
,
N
E,M,N
E
,
M
,
N
be the feet of perpendiculars from
A
A
A
to
B
C
BC
BC
,
B
B
B
to
D
J
DJ
D
J
,
C
C
C
to
D
K
DK
DK
, respectively. The two tangents at
M
,
N
M,N
M
,
N
to the circumcircle of triangle
E
M
N
EMN
EMN
meet at
T
T
T
. Prove that
T
T
T
is a fixed point (as
A
A
A
moves on
(
O
)
(O)
(
O
)
).