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Problems
Contests
National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2004 Junior Balkan Team Selection Tests - Moldova
2004 Junior Balkan Team Selection Tests - Moldova
Part of
JBMO TST - Moldova
Subcontests
(8)
2
1
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sum \sqrt{a_i^2+1}>= \sqrt{2n(a_1 + a_2 +...+ a_n}}
Let
n
∈
N
∗
n \in N^*
n
∈
N
∗
. Let
a
1
,
a
2
.
.
.
,
a
n
a_1, a_2..., a_n
a
1
,
a
2
...
,
a
n
be real such that
a
1
+
a
2
+
.
.
.
+
a
n
≥
0
a_1 + a_2 +...+ a_n \ge 0
a
1
+
a
2
+
...
+
a
n
≥
0
. Prove the inequality
a
1
2
+
1
+
a
2
2
+
1
+
.
.
.
+
a
1
2
+
1
≥
2
n
(
a
1
+
a
2
+
.
.
.
+
a
n
)
\sqrt{a_1^2+1}+\sqrt{a_2^2+1}+...+\sqrt{a_1^2+1}\ge \sqrt{2n(a_1 + a_2 +...+ a_n )}
a
1
2
+
1
+
a
2
2
+
1
+
...
+
a
1
2
+
1
≥
2
n
(
a
1
+
a
2
+
...
+
a
n
)
.
4
1
Hide problems
max {a_1, a_2,. . . , a_{12} } - min {a_1, a_2,. . . , a_{12}\} = 20
Different non-zero natural numbers a
1
,
a
2
,
.
.
.
,
a
12
_1, a_2,. . . , a_{12}
1
,
a
2
,
...
,
a
12
satisfy the condition: all positive differences other than two numbers
a
i
a_i
a
i
and
a
j
a_j
a
j
form many
20
20
20
consecutive natural numbers. a) Show that
max
{
a
1
,
a
2
,
.
.
.
,
a
12
}
−
min
{
a
1
,
a
2
,
.
.
.
,
a
12
}
=
20
\max \{a_1, a_2,. . . , a_{12}\} - \min \{a_1, a_2,. . . , a_{12}\} = 20
max
{
a
1
,
a
2
,
...
,
a
12
}
−
min
{
a
1
,
a
2
,
...
,
a
12
}
=
20
. b)Determine
12
12
12
natural numbers with the property from the statement.
8
1
Hide problems
ab in terms of inverse and sum / differences of a,b
The positive real numbers
a
a
a
and
b
b
b
(
a
>
b
a> b
a
>
b
) are written on the board. At every step, with numbers written on the board, one of the following operations can be performed: a) choose one of the numbers and write its square or its inverse. b) choose two numbers written on the board ¸and write their sum or their positive difference. Show how the product
a
⋅
b
a \cdot b
a
⋅
b
can be obtained with the help of the defined operations.
5
1
Hide problems
no in position 167 of sequence of naturals 1, 5, 6, 25, 26, 30, 31,...
The sequence of natural numbers
1
,
5
,
6
,
25
,
26
,
30
,
31
,
.
.
.
1, 5, 6, 25, 26, 30, 31,...
1
,
5
,
6
,
25
,
26
,
30
,
31
,
...
is made up of powers of
5
5
5
with natural exponents or sums of powers of
5
5
5
with different natural exponents, written in ascending order. Determine the term of the string written in position
167
167
167
.
1
1
Hide problems
x^2 + y^2 + z^2 <xy + 3y + 2z
Determine all triplets of integers
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
that validate the inequality
x
2
+
y
2
+
z
2
<
x
y
+
3
y
+
2
z
x^2 + y^2 + z^2 <xy + 3y + 2z
x
2
+
y
2
+
z
2
<
x
y
+
3
y
+
2
z
.
7
1
Hide problems
hexagon area wanted, angle bisectors, circumcircle related
Let the triangle
A
B
C
ABC
A
BC
have area
1
1
1
. The interior bisectors of the angles
∠
B
A
C
,
∠
A
B
C
,
∠
B
C
A
\angle BAC,\angle ABC, \angle BCA
∠
B
A
C
,
∠
A
BC
,
∠
BC
A
intersect the sides
(
B
C
)
,
(
A
C
)
,
(
A
B
)
(BC), (AC), (AB)
(
BC
)
,
(
A
C
)
,
(
A
B
)
and the circumscribed circle of the respective triangle
A
B
C
ABC
A
BC
at the points
L
L
L
and
G
,
N
G, N
G
,
N
and
F
,
Q
F, Q
F
,
Q
and
E
E
E
. The lines
E
F
,
F
G
,
G
E
EF, FG,GE
EF
,
FG
,
GE
intersect the bisectors
(
A
L
)
,
(
C
Q
)
,
(
B
N
)
(AL), (CQ) ,(BN)
(
A
L
)
,
(
CQ
)
,
(
BN
)
respectively at points
P
,
M
,
R
P, M, R
P
,
M
,
R
. Determine the area of the hexagon
L
M
N
P
R
LMNPR
L
MNPR
.
3
1
Hide problems
DP is angle bisector of <ADM iff PC = 4BC, ABCD parallelogram
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram and point
M
M
M
be the midpoint of
[
A
B
]
[AB]
[
A
B
]
so that the quadrilateral
M
B
C
D
MBCD
MBC
D
is cyclic. If
N
N
N
is the point of intersection of the lines
D
M
DM
D
M
and
B
C
BC
BC
, and
P
∈
B
C
P \in BC
P
∈
BC
, then prove that the ray
(
D
P
(DP
(
D
P
is the angle bisector of
∠
A
D
M
\angle ADM
∠
A
D
M
if and only if
P
C
=
4
B
C
PC = 4BC
PC
=
4
BC
.
6
1
Hide problems
Algebra polynomial
Represent the polynomial
P
(
X
)
=
X
100
+
X
20
+
1
P(X) = X^{100} + X^{20} + 1
P
(
X
)
=
X
100
+
X
20
+
1
as the product of 4 polynomials with integer coefficients.