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Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2001 Moldova Team Selection Test
2001 Moldova Team Selection Test
Part of
Moldova Team Selection Test
Subcontests
(12)
12
1
Hide problems
$2k$ divides $n(n+1)$ and $2k\leq n+1$
Let
n
n{}
n
(
n
≥
1
)
(n\geq 1)
(
n
≥
1
)
be an integer and a set
A
=
{
1
,
2
,
…
,
n
}
A=\{1,2,\ldots,n\}
A
=
{
1
,
2
,
…
,
n
}
. The set
A
A{}
A
is
k
−
p
a
r
t
i
t
i
o
n
a
b
l
e
k-partitionable
k
−
p
a
r
t
i
t
i
o
nab
l
e
if it can be partitioned in
k
k{}
k
disjoint sets with the same sum of elements. Show that
A
A{}
A
is
k
−
p
a
r
t
i
t
i
o
n
a
b
l
e
k-partitionable
k
−
p
a
r
t
i
t
i
o
nab
l
e
if and only if
2
k
2k
2
k
divides
n
(
n
+
1
)
n(n+1)
n
(
n
+
1
)
and
2
k
≤
n
+
1
2k\leq n+1
2
k
≤
n
+
1
.
11
1
Hide problems
A clock with hands of the same length has stopped at a certain time
A clock with hands of the same length has stopped at a certain time between
00
:
00
00:00
00
:
00
and
12
:
00
12:00
12
:
00
. Is it possible to determine the correct time when the clock stopped, no matter when it stopped, if it has: a) two hands, showing the hour and the minute? b) three hands, showing the hour, the minute and the second?
9
1
Hide problems
show that $|z|<1$
If
z
∈
C
z\in\mathbb{C}
z
∈
C
is a solution of the equation
x
n
+
a
1
x
n
−
1
+
a
2
x
n
−
2
+
…
+
a
n
=
0
x^n+a_1x^{n-1}+a_2x^{n-2}+\ldots+a_n=0
x
n
+
a
1
x
n
−
1
+
a
2
x
n
−
2
+
…
+
a
n
=
0
with real coefficients
0
<
a
n
≤
a
n
−
1
≤
…
≤
a
1
<
1
0<a_n\leq a_{n-1}\leq\ldots\leq a_1<1
0
<
a
n
≤
a
n
−
1
≤
…
≤
a
1
<
1
, show that
∣
z
∣
<
1
|z|<1
∣
z
∣
<
1
.
8
1
Hide problems
Prove that there exists an m-prefered permutation if and only if $km\leq m(k-1)$
A group of
n
n{}
n
(
n
>
1
)
(n>1)
(
n
>
1
)
people each visited
k
k{}
k
(
k
>
1
)
(k>1)
(
k
>
1
)
citites. Each person makes a list of these
k
k
k
cities in the order they want to visit them. A permutation
(
a
1
,
a
2
,
…
,
a
k
)
(a_1,a_2,\ldots,a_k)
(
a
1
,
a
2
,
…
,
a
k
)
is called
m
−
p
r
e
f
e
r
e
d
m-prefered
m
−
p
re
f
ere
d
(
m
∈
N
)
(m\in\mathbb{N})
(
m
∈
N
)
, if for every
i
=
1
,
2
,
…
,
k
i=1,2,\ldots,k
i
=
1
,
2
,
…
,
k
there are at least
m
m
m
people that would prefer to visit the city
a
i
a_i
a
i
before the city
a
i
+
1
a_{i+1}
a
i
+
1
,
(
a
k
+
1
=
a
1
)
(a_{k+1}=a_1)
(
a
k
+
1
=
a
1
)
. Prove that there exists an m-prefered permutation if and only if
k
m
≤
n
(
k
−
1
)
km\leq n(k-1)
km
≤
n
(
k
−
1
)
.
7
1
Hide problems
P_1(X)=X-1, P_2(X)=X^2-X-1, P_n(X)=XP_{n-1}(X)-P_{n-2}(X)
Let
(
P
n
(
X
)
)
n
∈
N
(P_n(X))_{n\in\mathbb{N}}
(
P
n
(
X
)
)
n
∈
N
be a sequence of polynomials defined as:
P
1
(
X
)
=
X
−
1
,
P
2
(
X
)
=
X
2
−
X
−
1
,
P
n
(
X
)
=
X
P
n
−
1
(
X
)
−
P
n
−
2
(
X
)
,
∀
n
>
2
P_1(X)=X-1, P_2(X)=X^2-X-1, P_n(X)=XP_{n-1}(X)-P_{n-2}(X), \forall n>2
P
1
(
X
)
=
X
−
1
,
P
2
(
X
)
=
X
2
−
X
−
1
,
P
n
(
X
)
=
X
P
n
−
1
(
X
)
−
P
n
−
2
(
X
)
,
∀
n
>
2
. For every nonnegative integer
n
n{}
n
find all roots of the polynomial
P
n
(
X
)
P_n(X)
P
n
(
X
)
.
3
1
Hide problems
Is the number $|m^2+1-n^2|$ is a perfect square?
Let
m
m
m
and
n
n{}
n
be positive integers of the same parity such that
n
2
−
1
n^2-1
n
2
−
1
divides
∣
m
2
+
1
−
n
2
∣
|m^2+1-n^2|
∣
m
2
+
1
−
n
2
∣
. Is the number
∣
m
2
+
1
−
n
2
∣
|m^2+1-n^2|
∣
m
2
+
1
−
n
2
∣
is a perfect square?
6
1
Hide problems
smallest possible area of a convex pentagon whose vertexes are lattice points
Find the smallest possible area of a convex pentagon whose vertexes are lattice points in a plane.
4
1
Hide problems
Prove that $2\log_3 n \leq f(n) < 5\log_3 n$ for every $n>1$.
For every nonnegative integer
n
n{}
n
let
f
(
n
)
f(n)
f
(
n
)
be the smallest number of digits
1
1
1
which can represent the number
n
n{}
n
using the symbols
"
+
"
,
"
−
"
,
"
×
"
,
"
(
"
,
"
)
"
"+", "-", "\times", "(", ")"
"
+
"
,
"
−
"
,
"
×
"
,
"
(
"
,
"
)
"
. For example,
80
=
(
1
+
1
+
1
+
1
+
1
)
×
(
1
+
1
+
1
+
1
)
×
(
1
+
1
+
1
+
1
)
80=(1+1+1+1+1)\times(1+1+1+1)\times(1+1+1+1)
80
=
(
1
+
1
+
1
+
1
+
1
)
×
(
1
+
1
+
1
+
1
)
×
(
1
+
1
+
1
+
1
)
and
f
(
80
)
≤
13
f(80)\leq 13
f
(
80
)
≤
13
. Prove that
2
log
3
n
≤
f
(
n
)
<
5
log
3
n
2\log_3 n \leq f(n) < 5\log_3 n
2
lo
g
3
n
≤
f
(
n
)
<
5
lo
g
3
n
for every
n
>
1
n>1
n
>
1
.
2
1
Hide problems
Prove that $S\leq\sum_{i=1}^{4}MA_i\cdot MA_i^{&#039;}$
Let
A
i
A_i
A
i
and
A
i
′
A_i^{'}
A
i
′
(
i
=
1
,
2
,
3
,
4
)
(i=1,2,3,4)
(
i
=
1
,
2
,
3
,
4
)
be diametrically opposite vertexes of a rectangular cuboid and
M
M{}
M
a point inside it. Prove that
S
≤
∑
i
=
1
4
M
A
i
⋅
M
A
i
′
S\leq\sum_{i=1}^{4}MA_i\cdot MA_i^{'}
S
≤
∑
i
=
1
4
M
A
i
⋅
M
A
i
′
, where
S
S{}
S
is the total surface area of the rectangular cuboid.
10
1
Hide problems
Delta 2.9
Let
A
B
C
ABC
A
BC
be a triangle and let
D
D
D
and
E
E
E
be points on sides
A
B
AB
A
B
and
A
C
AC
A
C
, respectively, such that
D
E
∥
B
C
DE \parallel BC
D
E
∥
BC
. Let
P
P
P
be any point interior to triangle
A
D
E
ADE
A
D
E
, and let
F
F
F
and
G
G
G
be the intersections of
D
E
DE
D
E
with the lines
B
P
BP
BP
and
C
P
CP
CP
, respectively. Let
Q
Q
Q
be the second intersection point of the circumcircles of triangles
P
D
G
PDG
P
D
G
and
P
F
E
PFE
PFE
. Prove that the points
A
,
P
,
A,P,
A
,
P
,
and
Q
Q
Q
are collinear.
5
1
Hide problems
A very nice problem
Find
a
,
b
,
c
∈
N
a,b,c \in N
a
,
b
,
c
∈
N
such that
a
b
ab
ab
divides a^2\plus{}b^2\plus{}1.
1
1
Hide problems
Sum of integer powers belongs in A, while their sum doesn't
Let
n
n
n
be a positive integer of the form
4
k
+
1
4k+1
4
k
+
1
,
k
∈
N
k\in \mathbb N
k
∈
N
and
A
=
{
a
2
+
n
b
2
∣
a
,
b
∈
Z
}
A = \{ a^2 + nb^2 \mid a,b \in \mathbb Z\}
A
=
{
a
2
+
n
b
2
∣
a
,
b
∈
Z
}
. Prove that there exist integers
x
,
y
x,y
x
,
y
such that
x
n
+
y
n
∈
A
x^n+y^n \in A
x
n
+
y
n
∈
A
and
x
+
y
∉
A
x+y \notin A
x
+
y
∈
/
A
.