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Problems
Contests
National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2001 Junior Balkan Team Selection Tests - Moldova
2001 Junior Balkan Team Selection Tests - Moldova
Part of
JBMO TST - Moldova
Subcontests
(8)
6
1
Hide problems
2 <= i <= 8 there is a number $a_i$, such that a_{i - 1} + a_{i + 1} <ka_i
Let the nonnegative numbers
a
1
,
a
2
,
.
.
.
a
9
a_1, a_2,... a_9
a
1
,
a
2
,
...
a
9
, where
a
1
=
a
9
=
0
a_1 = a_9 = 0
a
1
=
a
9
=
0
and let at least one of the numbers is nonzero. Denote the sentence
(
P
)
(P)
(
P
)
: '' For
2
≤
i
≤
8
2 \le i \le 8
2
≤
i
≤
8
there is a number
a
i
a_i
a
i
, such that
a
i
−
1
+
a
i
+
1
<
k
a
i
a_{i - 1} + a_{i + 1} <ka_i
a
i
−
1
+
a
i
+
1
<
k
a
i
”. a) Show that the sentence
(
P
)
(P)
(
P
)
is true for
k
=
2
k = 2
k
=
2
. b) Determine whether is the sentence
(
P
)
(P)
(
P
)
true for
k
=
19
10
k = \frac{19}{10}
k
=
10
19
7
1
Hide problems
Noah has on his ark 4 large coffins in which to place 8 animals
Noah has on his ark
4
4
4
large coffins in which to place
8
8
8
animals. It is known that for any animal there are at most
5
5
5
animals with which it is incompatible (those can't live together). Show that: a) Noah can place the animals in the cages according to their compatibility. b) Noah can place two animals in each cage.
5
1
Hide problems
\sqrt{n + 1} + \sqrt{n - 1} is rational for non natural n
Determine if there is a non-natural natural number
n
n
n
with the property that
n
+
1
+
n
−
1
\sqrt{n + 1} + \sqrt{n - 1}
n
+
1
+
n
−
1
is rational.
2
1
Hide problems
[x] {x } = 2001 x
Solve in
R
R
R
equation
[
x
]
⋅
{
x
}
=
2001
x
[x] \cdot \{x\} = 2001 x
[
x
]
⋅
{
x
}
=
2001
x
, where
[
.
]
[ .]
[
.
]
and
{
.
}
\{ .\}
{
.
}
represent respectively the floor and the integer functions.
1
1
Hide problems
a set of M points on a circle, with only one colored red
On a circle we consider a set
M
M
M
consisting of
n
n
n
(
n
≥
3
n \ge 3
n
≥
3
) points, of which only one is colored red. Determine of which polygons inscribed in a circle having the vertices in the set
M
M
M
are more: those that contain the red dot or those that do not contain those points? How many more are there than others?
3
1
Hide problems
equilateral ABP wanted, AD = BC, <A + <B = 120^o, DCP equilateral given
Let the convex quadrilateral
A
B
C
D
ABCD
A
BC
D
with
A
D
=
B
C
AD = BC
A
D
=
BC
¸and
∠
A
+
∠
B
=
12
0
o
\angle A + \angle B = 120^o
∠
A
+
∠
B
=
12
0
o
. Take a point
P
P
P
in the plane so that the line
C
D
CD
C
D
separates the points
A
A
A
and
P
P
P
, and the
D
C
P
DCP
D
CP
triangle is equilateral. Show that the triangle
A
B
P
ABP
A
BP
is equilateral. It is the true statement for a non-convex quadrilateral?
8
1
Hide problems
Number Theory
Let a, b, c be natural numbers , so that c> b> a> 0. Show that, among any 2c consecutive natural numbers, there are three distinct numbers x, y, z so abc divides xyz.
4
1
Hide problems
Combinatorics
Determine the smallest natural number
n
=
>
2
n =>2
n
=>
2
with the property: For every positive integers
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2,. . . , a_n
a
1
,
a
2
,
...
,
a
n
the product of all differences
a
j
−
a
i
a_j-a_i
a
j
−
a
i
,
1
<
=
i
<
j
<
=
n
1 <=i <j <=n
1
<=
i
<
j
<=
n
, is divisible by 2001.