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Problems
Contests
National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2012 Junior Balkan Team Selection Tests - Moldova
2012 Junior Balkan Team Selection Tests - Moldova
Part of
JBMO TST - Moldova
Subcontests
(4)
4
2
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linear Sequence
Let there be an infinite sequence
a
k
a_{k}
a
k
with
k
≥
1
k\geq 1
k
≥
1
defined by:
a
k
+
2
=
a
k
+
14
a_{k+2} = a_{k} + 14
a
k
+
2
=
a
k
+
14
and
a
1
=
12
a_{1} = 12
a
1
=
12
,
a
2
=
24
a_{2} = 24
a
2
=
24
.a) Does
2012
2012
2012
belong to the sequence? b) Prove that the sequence doesn't contain perfect squares.
how many solutions
How many solutions does the system have:
{
(
3
x
+
2
y
)
∗
(
3
x
+
1
y
)
=
2
x
2
+
y
2
≤
2012
\{\begin{matrix}&(3x+2y) *(\frac{3}{x}+\frac{1}{y})=2\\ & x^2+y^2\leq 2012\\ \end{matrix}
{
(
3
x
+
2
y
)
∗
(
x
3
+
y
1
)
=
2
x
2
+
y
2
≤
2012
where
x
,
y
x,y
x
,
y
are non-zero integers
3
2
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bisector inside equilateral triangle
Let
A
B
C
ABC
A
BC
be an equilateral triangle, take line
t
t
t
such that
t
∥
B
C
t\parallel BC
t
∥
BC
and
t
t
t
passes through
A
A
A
. Let point
D
D
D
be on side
A
C
AC
A
C
, the bisector of angle
A
B
D
ABD
A
B
D
intersects line
t
t
t
in point
E
E
E
. Prove that
B
D
=
C
D
+
A
E
BD = CD + AE
B
D
=
C
D
+
A
E
.
Isosceles triangle bisectors
Let
A
B
C
ABC
A
BC
be an isosceles triangle with
A
C
=
B
C
AC=BC
A
C
=
BC
. Take points
D
D
D
on side
A
C
AC
A
C
and
E
E
E
on side
B
C
BC
BC
and
F
F
F
the intersection of bisectors of angles
D
E
B
DEB
D
EB
and
A
D
E
ADE
A
D
E
such that
F
F
F
lies on side
A
B
AB
A
B
. Prove that
F
F
F
is the midpoint of
A
B
AB
A
B
.
2
2
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Simple Inequality
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers, prove the inequality:
(
a
+
b
+
c
)
2
+
a
b
+
b
c
+
a
c
≥
6
a
b
c
(
a
+
b
+
c
)
(a+b+c)^2+ab+bc+ac\geq 6\sqrt{abc(a+b+c)}
(
a
+
b
+
c
)
2
+
ab
+
b
c
+
a
c
≥
6
ab
c
(
a
+
b
+
c
)
Does there exist
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be positive real numbers and
c
d
=
1
cd=1
c
d
=
1
. Prove that there exists a positive integer
n
n
n
such that
a
b
≤
n
2
≤
(
a
+
c
)
(
b
+
d
)
ab\leq n^2\leq (a+c)(b+d)
ab
≤
n
2
≤
(
a
+
c
)
(
b
+
d
)
1
2
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minimum of a*b*c+d*e*f+g*h*k
Let
1
≤
a
,
b
,
c
,
d
,
e
,
f
,
g
,
h
,
k
≤
9
1\leq a,b,c,d,e,f,g,h,k \leq 9
1
≤
a
,
b
,
c
,
d
,
e
,
f
,
g
,
h
,
k
≤
9
and
a
,
b
,
c
,
d
,
e
,
f
,
g
,
h
,
k
a,b,c,d,e,f,g,h,k
a
,
b
,
c
,
d
,
e
,
f
,
g
,
h
,
k
are different integers, find the minimum value of the expression
E
=
a
∗
b
∗
c
+
d
∗
e
∗
f
+
g
∗
h
∗
k
E = a*b*c+d*e*f+g*h*k
E
=
a
∗
b
∗
c
+
d
∗
e
∗
f
+
g
∗
h
∗
k
and prove that it is minimum.
Sum and Product of sequence
Find a sequence of
2012
2012
2012
distinct integers bigger than
0
0
0
such that their sum is a perfect square and their product is a perfect cube.