MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
1997 Moldova Team Selection Test
1997 Moldova Team Selection Test
Part of
Moldova Team Selection Test
Subcontests
(11)
12
1
Hide problems
Let $A$ be a set of $n$ positive integers.
For every nonempty set of real numbers
S
S{}
S
denote
σ
(
S
)
\sigma(S)
σ
(
S
)
the sum of its elements. Let
A
A{}
A
be a set of
n
n{}
n
positive integers. Show that the set of all sums
σ
\sigma{}
σ
of all nonempty sets of
A
A{}
A
can be partitioned in
n
n{}
n
groups such that the ratio between the greatest number and the smallest number from each group is less than
2
2
2
.
11
1
Hide problems
Show that $P(n)\in\mathbb{Z}, \forall n\in\mathbb{N}$
Let
P
(
X
)
P(X)
P
(
X
)
be a polynomial with real coefficients such that
{
P
(
n
)
}
≤
1
n
,
∀
n
∈
N
\{P(n)\}\leq\frac{1}{n}, \forall n\in\mathbb{N}
{
P
(
n
)}
≤
n
1
,
∀
n
∈
N
, where
{
a
}
\{a\}
{
a
}
is the fractional part of the number
a
a
a
. Show that
P
(
n
)
∈
Z
,
∀
n
∈
N
P(n)\in\mathbb{Z}, \forall n\in\mathbb{N}
P
(
n
)
∈
Z
,
∀
n
∈
N
.
10
1
Hide problems
Let there be a regular hexagon with sidelength $1$
Let there be a regular hexagon with sidelength
1
1
1
. Find the greatest integer
n
≥
2
n\geq2
n
≥
2
for which there exist
n
n{}
n
points inside or on the sides of the hexagon such that the distance between every two points is no less than
2
\sqrt{2}
2
.
8
1
Hide problems
there are $\frac{n(n+1)}{2}$ distinct remainders
Find all integers
n
>
1
n>1
n
>
1
for which there exist positive integers
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
such that when divided by
a
i
+
a
j
,
1
≤
i
≤
j
≤
n
a_i+a_j, 1\leq i\leq j\leq n
a
i
+
a
j
,
1
≤
i
≤
j
≤
n
there are
n
(
n
+
1
)
2
\frac{n(n+1)}{2}
2
n
(
n
+
1
)
distinct remainders.
6
1
Hide problems
Prove that $0\leq a_{n}-a_{n+1}<\frac{2}{n^2}, \forall n\in\mathbb{N}.$
Let
(
a
n
)
n
∈
N
(a_n)_{n\in\mathbb{N}}
(
a
n
)
n
∈
N
be a sequence of positive numbers such that
a
n
−
2
a
n
+
1
+
a
n
+
2
≥
0
and
∑
j
=
1
n
a
j
≤
1
,
∀
n
∈
N
.
a_n-2a_{n+1}+a_{n+2}\geq 0 \text{ and } \sum_{j=1}^{n} a_j \leq 1, \forall n\in\mathbb{N}.
a
n
−
2
a
n
+
1
+
a
n
+
2
≥
0
and
j
=
1
∑
n
a
j
≤
1
,
∀
n
∈
N
.
Prove that
0
≤
a
n
−
a
n
+
1
<
2
n
2
,
∀
n
∈
N
.
0\leq a_{n}-a_{n+1}<\frac{2}{n^2}, \forall n\in\mathbb{N}.
0
≤
a
n
−
a
n
+
1
<
n
2
2
,
∀
n
∈
N
.
4
1
Hide problems
Let $A=\{1,2,\ldots,1997\}$ be a set
Let
A
=
{
1
,
2
,
…
,
1997
}
A=\{1,2,\ldots,1997\}
A
=
{
1
,
2
,
…
,
1997
}
be a set. Find the samllest integer
k
>
1
k>1
k
>
1
such that in each subset
M
M{}
M
of
A
A{}
A
, which cointain
k
k{}
k
elements, there is a multiple of the smallest element from
M
M{}
M
, different from itself.
1
1
Hide problems
the term $x_n$ is the greatest odd integer of $x_{n-1}+x_{n-2}$
Let
a
a
a
and
b
b
b
be two odd positive integers. Define the sequence
(
x
n
)
n
∈
N
(x_n)_{n\in\mathbb{N}}
(
x
n
)
n
∈
N
as such:
x
1
=
a
,
x
2
=
b
,
x_1=a, x_2=b,
x
1
=
a
,
x
2
=
b
,
for every
n
≥
3
n\geq3
n
≥
3
the term
x
n
x_n{}
x
n
is the greatest odd integer of
x
n
−
1
+
x
n
−
2
x_{n-1}+x_{n-2}
x
n
−
1
+
x
n
−
2
. Show that starting with a term, all the following terms are constant.
7
1
Hide problems
collinear with H wanted, tangents through A at circle of diameter BC
Let
A
B
C
ABC
A
BC
be a triangle with orthocenter
H
H
H
. Let the circle
ω
\omega
ω
have
B
C
BC
BC
as the diameter. Draw tangents
A
P
AP
A
P
,
A
Q
AQ
A
Q
to the circle
ω
\omega
ω
at the point
P
,
Q
P, Q
P
,
Q
respectively. Prove that
P
,
H
,
Q
P,H,Q
P
,
H
,
Q
lie on the same line .
5
1
Hide problems
Polynomial
Let
P
(
x
)
∈
Z
[
x
]
P(x)\in\mathbb{Z}[x]
P
(
x
)
∈
Z
[
x
]
with deg
P
=
2015
P=2015
P
=
2015
. Let
Q
(
x
)
=
(
P
(
x
)
)
2
−
9
Q(x)=(P(x))^2-9
Q
(
x
)
=
(
P
(
x
)
)
2
−
9
. Prove that: the number of distinct roots of
Q
(
x
)
Q(x)
Q
(
x
)
can not bigger than
2015
2015
2015
2
1
Hide problems
in a convex pentagon every diagonal is parallel to one side
In a convex pentagon every diagonal is parallel to one side. Show that the ratios between the lengths of diagonals and the sides parallel to them are equal and find their value.
9
1
Hide problems
3 Nov FE problem
Find all
t
∈
Z
t\in \mathbb Z
t
∈
Z
such that: exists a function
f
:
Z
+
→
Z
f:\mathbb Z^+\to \mathbb Z
f
:
Z
+
→
Z
such that:
f
(
1997
)
=
1998
f(1997)=1998
f
(
1997
)
=
1998
∀
x
,
y
∈
Z
+
,
gcd
(
x
,
y
)
=
d
:
f
(
x
y
)
=
f
(
x
)
+
f
(
y
)
+
t
f
(
d
)
:
P
(
x
,
y
)
\forall x,y\in \mathbb Z^+ , \text{gcd}(x,y)=d : f(xy)=f(x)+f(y)+tf(d):P(x,y)
∀
x
,
y
∈
Z
+
,
gcd
(
x
,
y
)
=
d
:
f
(
x
y
)
=
f
(
x
)
+
f
(
y
)
+
t
f
(
d
)
:
P
(
x
,
y
)