MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2000 Moldova Team Selection Test
2000 Moldova Team Selection Test
Part of
Moldova Team Selection Test
Subcontests
(9)
11
1
Hide problems
$$\sum_{A\in M} (-1)^{n-|A|}\cdot f(A)=f(S)-\max\{f(A)|A\in M, A\neq S\},$$
Let
S
S
S
be a finite set with
n
n{}
n
(
n
>
1
)
(n>1)
(
n
>
1
)
elements,
M
M{}
M
the set of all subsets of
S
S{}
S
and a function
f
:
M
→
R
f:M\rightarrow\mathbb{R}
f
:
M
→
R
, that verifies the relation
f
(
A
∩
B
)
=
min
{
f
(
A
)
,
f
(
B
)
}
,
∀
A
,
B
∈
M
f(A\cap B)=\min\{f(A),f(B)\}, \forall A,B\in M
f
(
A
∩
B
)
=
min
{
f
(
A
)
,
f
(
B
)}
,
∀
A
,
B
∈
M
. Show that
∑
A
∈
M
(
−
1
)
n
−
∣
A
∣
⋅
f
(
A
)
=
f
(
S
)
−
max
{
f
(
A
)
∣
A
∈
M
,
A
≠
S
}
,
\sum_{A\in M} (-1)^{n-|A|}\cdot f(A)=f(S)-\max\{f(A)|A\in M, A\neq S\},
A
∈
M
∑
(
−
1
)
n
−
∣
A
∣
⋅
f
(
A
)
=
f
(
S
)
−
max
{
f
(
A
)
∣
A
∈
M
,
A
=
S
}
,
where
∣
A
∣
|A|
∣
A
∣
is the number of elements of subset
A
A{}
A
.
10
1
Hide problems
there is a point $M$ inside it such that the half lines $(A_iM, i=1,2,\ldots,n$
Convex polygon
A
1
A
2
…
A
n
A_1A_2\ldots A_n
A
1
A
2
…
A
n
is called
b
a
l
a
n
c
e
d
balanced
ba
l
an
ce
d
if there is a point
M
M{}
M
inside it such that the half lines
(
A
i
M
,
(
i
=
1
,
2
,
…
,
n
)
(A_iM, (i=1,2,\ldots,n)
(
A
i
M
,
(
i
=
1
,
2
,
…
,
n
)
intersect disctinct sides of the polygon. a) Show that if
n
>
3
n>3
n
>
3
is even, then every polygon with
n
n{}
n
sides is not balanced. b) Do polygons with an odd number of sides that are not balanced exist?
6
1
Hide problems
the squares that have $BC$ and $AD$ as diagonals also have a common vertex
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral. Two squares are constructed such that
A
B
AB{}
A
B
and
C
D
CD{}
C
D
are their diagonals. Show that if these squares have a common vertex inside
A
B
C
D
ABCD
A
BC
D
, then the squares that have
B
C
BC{}
BC
and
A
D
AD{}
A
D
as diagonals also have a common vertex inside
A
B
C
D
ABCD
A
BC
D
.
5
1
Hide problems
there is a term in the Fibonacci sequence that is divided by $r$
Let
(
F
n
)
n
∈
N
(F_n)_{n\in\mathbb{N}}
(
F
n
)
n
∈
N
be the Fibonacci sequence difined as
F
0
=
F
1
=
1
,
F
n
+
2
=
F
n
+
1
+
F
n
,
∀
n
∈
N
F_0=F_1=1, F_{n+2}=F_{n+1}+F_n, \forall n\in\mathbb{N}
F
0
=
F
1
=
1
,
F
n
+
2
=
F
n
+
1
+
F
n
,
∀
n
∈
N
. Show that for every nonnegative integer
r
r
r
there is a term in the Fibonacci sequence that is divided by
r
r
r
.
4
1
Hide problems
Show that if $x\in S, y\in S, x\neq y,$ then $\frac{x+y}{2}\notin S$
Let
S
S{}
S
be the set of nonnegative integers, which cointain only digits
0
0
0
and
1
1
1
in base
4
4
4
numeral system. a) Show that if
x
∈
S
,
y
∈
S
,
x
≠
y
,
x\in S, y\in S, x\neq y,
x
∈
S
,
y
∈
S
,
x
=
y
,
then
x
+
y
2
∉
S
\frac{x+y}{2}\notin S
2
x
+
y
∈
/
S
. b) Let
T
T
T
be a set of nonnegative integers such that
S
⊂
T
,
T
≠
S
S\subset T, T\neq S
S
⊂
T
,
T
=
S
. Show that there exist
x
∈
T
,
y
∈
T
,
x
≠
y
,
x\in T, y\in T, x\neq y,
x
∈
T
,
y
∈
T
,
x
=
y
,
such that
x
+
y
2
∈
T
\frac{x+y}{2} \in T
2
x
+
y
∈
T
.
2
1
Hide problems
Find the greatest possible value of $\frac{\min\{S(BPB_1),S(CPC_1)\}}{S(ABC)}$
In triangle
A
B
C
ABC
A
BC
points
B
1
B_1
B
1
and
C
1
C_1
C
1
are on
A
B
AB
A
B
and
A
C
AC
A
C
respectively and
P
P{}
P
is a point on the segment
B
1
C
1
B_1C_1
B
1
C
1
. Find the greatest possible value of
min
{
S
(
B
P
B
1
)
,
S
(
C
P
C
1
)
}
S
(
A
B
C
)
\frac{\min\{S(BPB_1),S(CPC_1)\}}{S(ABC)}
S
(
A
BC
)
m
i
n
{
S
(
BP
B
1
)
,
S
(
CP
C
1
)}
, where
S
(
X
Y
Z
)
S(XYZ)
S
(
X
Y
Z
)
is the area o the triangle
A
B
C
ABC
A
BC
.
9
1
Hide problems
Sequence
The sequence
x
n
x_{n}
x
n
is de fined by:
x
0
=
1
,
x
1
=
0
,
x
2
=
1
,
x
3
=
1
,
x
n
+
3
=
(
n
2
+
n
+
1
)
(
n
+
1
)
n
x
n
+
2
+
(
n
2
+
n
+
1
)
x
n
+
1
−
n
+
1
n
x
n
(
n
=
1
,
2
,
3..
)
x_{0}=1, x_{1}=0, x_{2}=1,x_{3}=1, x_{n+3}=\frac{(n^2+n+1)(n+1)}{n}x_{n+2}+(n^2+n+1)x_{n+1}-\frac{n+1}{n}x_{n} (n=1,2,3..)
x
0
=
1
,
x
1
=
0
,
x
2
=
1
,
x
3
=
1
,
x
n
+
3
=
n
(
n
2
+
n
+
1
)
(
n
+
1
)
x
n
+
2
+
(
n
2
+
n
+
1
)
x
n
+
1
−
n
n
+
1
x
n
(
n
=
1
,
2
,
3..
)
Prove that all members of the sequence are perfect squares.
3
1
Hide problems
Evaluate the sum
For each positive integer
n
n
n
, evaluate the sum \sum_{k\equal{}0}^{2n}(\minus{}1)^{k}\frac{\binom{4n}{2k}}{\binom{2n}{k}}
7
1
Hide problems
6 points in the plane
Suppose that
p
1
,
p
2
,
p
3
,
q
1
,
q
2
,
q
3
p_1,p_2,p_3,q_1,q_2,q_3
p
1
,
p
2
,
p
3
,
q
1
,
q
2
,
q
3
are six points in the plane and that the distance between
p
i
p_i
p
i
and
q
j
q_j
q
j
( i,j \equal{} 1,2,3) is i \plus{} j. Show that the six points are collinear.