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Problems
Contests
National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2002 Junior Balkan Team Selection Tests - Moldova
2002 Junior Balkan Team Selection Tests - Moldova
Part of
JBMO TST - Moldova
Subcontests
(12)
5
1
Hide problems
E (x, m) = \frac{sum [(2m -1)^ 4 + x + x]}{sum[(2m )^ 4 + x]}
For any natural number
m
≥
1
m \ge 1
m
≥
1
and any real number
x
≥
0
x \ge 0
x
≥
0
we define expression
E
(
x
,
m
)
=
(
1
4
+
x
)
(
3
4
+
x
)
(
5
4
+
x
)
.
.
.
[
(
2
m
−
1
)
4
+
x
]
(
2
4
+
x
)
(
4
4
+
x
)
(
6
4
+
x
)
.
.
.
[
(
2
m
)
4
+
x
]
.
E (x, m) = \frac{(1^4 + x) (3^4 + x) (5^4 + x) ... [(2m -1)^ 4 + x]}{(2^4 + x) (4^4 + x) (6^4 + x) ... [(2m )^ 4 + x]}.
E
(
x
,
m
)
=
(
2
4
+
x
)
(
4
4
+
x
)
(
6
4
+
x
)
...
[(
2
m
)
4
+
x
]
(
1
4
+
x
)
(
3
4
+
x
)
(
5
4
+
x
)
...
[(
2
m
−
1
)
4
+
x
]
.
It is known that
E
(
1
4
,
m
)
=
1
1013
.
E\left(\frac{1}{4},m\right)=\frac{1}{1013}.
E
(
4
1
,
m
)
=
1013
1
.
. Determine the value of
m
m
m
12
1
Hide problems
f (-x) = -f (x) if f (g (x)) = g (f (x)) = x, f (x) + g (x) = x
Let
M
M
M
be an empty set of real numbers. For any
x
∈
M
x \in M
x
∈
M
the functions
f
:
M
→
M
f: M\to M
f
:
M
→
M
and
g
:
M
→
M
g: M\to M
g
:
M
→
M
satisfy the relations
f
(
g
(
x
)
)
=
g
(
f
(
x
)
)
=
x
f (g (x)) = g (f (x)) = x
f
(
g
(
x
))
=
g
(
f
(
x
))
=
x
and
f
(
x
)
+
g
(
x
)
=
x
f (x) + g (x) = x
f
(
x
)
+
g
(
x
)
=
x
. Show that
−
x
∈
M
- x \in M
−
x
∈
M
¸ and
f
(
−
x
)
=
−
f
(
x
)
f (-x) = -f (x)
f
(
−
x
)
=
−
f
(
x
)
whatever
x
∈
M
x \in M
x
∈
M
.
11
1
Hide problems
in the same direction for 2 hours 2 bodies move evenly on a circular route
Simultaneously from the same point of a circular route and in the same direction for two hours two bodies move evenly. The first body performs a complete rotation three minutes faster than the second body and exceeds it every
9
9
9
minutes and
20
20
20
seconds. Whenever the first body will overtake the other the second exactly at the starting point?
9
1
Hide problems
8 (a^4 + b^4) >=1 1 if a + b >=1
The real numbers
a
a
a
and
b
b
b
satisfy the relation
a
+
b
≥
1
a + b \ge 1
a
+
b
≥
1
. Show that
8
(
a
4
+
b
4
)
≥
1
8 (a^4 + b^4) \ge 1
8
(
a
4
+
b
4
)
≥
1
.
4
1
Hide problems
9 chess players
9
9
9
chess players participate in a chess tournament. According to the regulation, each participant plays a single game with each of the others. At a certain moment of the competition it was found that exactly two participants played the same number of party. To prove that in this case, not a single chess player played any the game, or just one chess player played with everyone else.
2
1
Hide problems
among 64 points of 2003 different lines there are at least 4 collinear points
64
64
64
distinct points are positioned in the plane so that they determine exactly
2003
2003
2003
different lines. Prove that among the
64
64
64
points there are at least
4
4
4
collinear points.
1
1
Hide problems
a = n^5 + 6n^3 + 8n ¸ b = n^4 + 4n^2 + 3 are relative prime of have gcd 3
For any integer
n
n
n
we define the numbers
a
=
n
5
+
6
n
3
+
8
n
a = n^5 + 6n^3 + 8n
a
=
n
5
+
6
n
3
+
8
n
¸
b
=
n
4
+
4
n
2
+
3
b = n^4 + 4n^2 + 3
b
=
n
4
+
4
n
2
+
3
. Prove that the numbers
a
a
a
and
b
b
b
are relatively prime or have the greatest common factor of
3
3
3
.
10
1
Hide problems
<MNA=<MNB wanted, intersecting circles related
The circles
C
1
C_1
C
1
and
C
2
C_2
C
2
intersect at the distinct points
M
M
M
and
N
N
N
. Points
A
A
A
and
B
B
B
belong respectively to the circles
C
1
C_1
C
1
and
C
2
C_2
C
2
so that the chords
[
M
A
]
[MA]
[
M
A
]
and
[
M
B
]
[MB]
[
MB
]
are tangent at point
M
M
M
to the circles
C
2
C_2
C
2
and
C
1
C_1
C
1
, respectively. To prove it that the angles
∠
M
N
A
\angle MNA
∠
MN
A
and
∠
M
N
B
\angle MNB
∠
MNB
are equal.
7
1
Hide problems
locus of projection inside a square, fixed triangle perimeter
The side of the square
A
B
C
D
ABCD
A
BC
D
has a length equal to
1
1
1
. On the sides
(
B
C
)
(BC)
(
BC
)
¸and
(
C
D
)
(CD)
(
C
D
)
take respectively the arbitrary points
M
M
M
and
N
N
N
so that the perimeter of the triangle
M
C
N
MCN
MCN
is equal to
2
2
2
. a) Determine the measure of the angle
∠
M
A
N
\angle MAN
∠
M
A
N
. b) If the point
P
P
P
is the foot of the perpendicular taken from point
A
A
A
to the line
M
N
MN
MN
, determine the locus of the points
P
P
P
.
3
1
Hide problems
perpendicular wanted, altitudes related
Let
A
B
C
ABC
A
BC
be a an acute triangle. Points
A
1
,
B
1
A_1, B_1
A
1
,
B
1
and
C
1
C_1
C
1
are respectively the projections of the vertices
A
,
B
A, B
A
,
B
and
C
C
C
on the opposite sides of the triangle, the point
H
H
H
is the orthocenter of the triangle, and the point
P
P
P
is the middle of the segment
[
A
H
]
[AH]
[
A
H
]
. The lines
B
H
BH
B
H
and
A
1
C
1
A_1C_1
A
1
C
1
,
P
B
1
P B_1
P
B
1
and
A
B
AB
A
B
intersect respectively at the points
M
M
M
and
N
N
N
. Prove that the lines
M
N
MN
MN
and
B
C
BC
BC
are perpendicular.
8
1
Hide problems
Number Theory
Find all triplets (a, b, c) of positive integers so that
a
2
b
a^2b
a
2
b
,
b
2
c
b^2c
b
2
c
and
c
2
a
c^2a
c
2
a
devide
a
3
+
b
3
+
c
3
a^3+b^3+c^3
a
3
+
b
3
+
c
3
6
1
Hide problems
Combinatorics
Determine the smallest positive integer n for that there are positive integers
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2,. . . , x_n
x
1
,
x
2
,
...
,
x
n
so that each natural number from 1001 to 2021 inclusive can be written as sum of one or more different terms
x
i
x_i
x
i
(i = 1, 2,..., n).