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Problems
Contests
National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2009 Junior Balkan Team Selection Tests - Moldova
2009 Junior Balkan Team Selection Tests - Moldova
Part of
JBMO TST - Moldova
Subcontests
(7)
8
1
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Prove that the greatest solution $n$ of the inequation $c_n<2009$ is a prime
Side of an equilatreal triangle has the length
n
∈
N
.
n\in\mathbb{N}.
n
∈
N
.
Each side is divided in
n
n
n
equal segments by division points. A line parallel with the third side of the triangle is drawn through the division points of every two sides. Let
c
n
c_n
c
n
be the number of all rhombuses with sidelength
1
1
1
inside the initial triangle. Prove that the greatest solution
n
n
n
of the inequation
c
n
<
2009
c_n<2009
c
n
<
2009
is a prime number.
7
1
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Prove that $BF=CD$.
In triangle
A
B
C
ABC
A
BC
there are points
D
∈
(
A
C
)
D\in(AC)
D
∈
(
A
C
)
and
F
∈
(
A
B
)
F\in(AB)
F
∈
(
A
B
)
such that
A
D
=
A
B
AD=AB
A
D
=
A
B
and line
B
C
BC
BC
splits the segment
[
C
F
]
[CF]
[
CF
]
in half. Prove that
B
F
=
C
D
BF=CD
BF
=
C
D
.
6
1
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x^3-y^3=x-y+2^{x-y}
Prove that there are no pairs of nonnegative integers
(
x
,
y
)
(x,y)
(
x
,
y
)
that satisfy the equality
x
3
−
y
3
=
x
−
y
+
2
x
−
y
.
x^3-y^3=x-y+2^{x-y}.
x
3
−
y
3
=
x
−
y
+
2
x
−
y
.
5
1
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divisible by $d^2$ and $a+b+c+d+e+f$
Find the lowest odd positive integer with an odd number of divisors and is divisible by
d
2
d^2
d
2
and
a
+
b
+
c
+
d
+
e
+
f
a+b+c+d+e+f
a
+
b
+
c
+
d
+
e
+
f
, where
a
,
b
,
c
,
d
,
e
,
f
a, b, c, d, e, f
a
,
b
,
c
,
d
,
e
,
f
are consecutive prime numbers.
4
1
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What are the lowest and highest possible scores of Tudor?
Petrică, Vasile and Tudor participated at a math contest. At the contest
5
5
5
problems where proposed, each worth distinct integer numbers of points. Petrică solved
4
4
4
problems completely and got
21
21
21
points and Vasile solved
3
3
3
problems completely and got
22
22
22
points. Tudor solved all problems completely. What are the lowest and highest possible scores of Tudor?
3
1
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Find $\angle ADF.$
Let
A
B
C
ABC
A
BC
be a triangle with
∠
B
C
A
=
20.
\angle BCA=20.
∠
BC
A
=
20.
Let points
D
∈
(
B
C
)
,
F
∈
(
A
C
)
D\in(BC), F\in(AC)
D
∈
(
BC
)
,
F
∈
(
A
C
)
be such that
C
D
=
D
F
=
F
B
=
B
A
.
CD=DF=FB=BA.
C
D
=
D
F
=
FB
=
B
A
.
Find
∠
A
D
F
.
\angle ADF.
∠
A
D
F
.
2
1
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\frac{a^2+b^2}{a^4+b^4}+\frac{b^2+c^2}{b^4+c^4}+\frac{c^2+a^2}{c^4+a^4}
Real positive numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfy
a
b
c
=
1
abc=1
ab
c
=
1
. Prove the inequality
a
2
+
b
2
a
4
+
b
4
+
b
2
+
c
2
b
4
+
c
4
+
c
2
+
a
2
c
4
+
a
4
≤
a
+
b
+
c
.
\frac{a^2+b^2}{a^4+b^4}+\frac{b^2+c^2}{b^4+c^4}+\frac{c^2+a^2}{c^4+a^4}\leq a+b+c.
a
4
+
b
4
a
2
+
b
2
+
b
4
+
c
4
b
2
+
c
2
+
c
4
+
a
4
c
2
+
a
2
≤
a
+
b
+
c
.