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Problems
Contests
National and Regional Contests
North Macedonia Contests
JBMO TST - Macedonia
2007 Junior Macedonian Mathematical Olympiad
2007 Junior Macedonian Mathematical Olympiad
Part of
JBMO TST - Macedonia
Subcontests
(5)
5
1
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2007 JBMO TST- Macedonia, problem 5
We are given an arbitrary
△
A
B
C
\bigtriangleup ABC
△
A
BC
. a) Can we dissect
△
A
B
C
\bigtriangleup ABC
△
A
BC
in
4
4
4
pieces, from which we can make two triangle similar to
△
A
B
C
\bigtriangleup ABC
△
A
BC
(each piece can be used only once)? Justify your answer! b) Is it possible that for every positive integer
n
≥
2
n \ge 2
n
≥
2
, we are able to dissect
△
A
B
C
\bigtriangleup ABC
△
A
BC
in
2
n
2n
2
n
pieces, from which we can make two triangles similar to
△
A
B
C
\bigtriangleup ABC
△
A
BC
(each piece can be used only once)? Justify your answer!
4
1
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2007 JBMO TST- Macedonia, problem 4
The numbers
a
1
,
a
2
,
.
.
.
,
a
20
a_{1}, a_{2}, ..., a_{20}
a
1
,
a
2
,
...
,
a
20
satisfy the following conditions:
a
1
≥
a
2
≥
.
.
.
≥
a
20
≥
0
a_{1} \ge a_{2} \ge ... \ge a_{20} \ge 0
a
1
≥
a
2
≥
...
≥
a
20
≥
0
a
1
+
a
2
=
20
a_{1} + a_{2} = 20
a
1
+
a
2
=
20
a
3
+
a
4
+
.
.
.
+
a
20
≤
20
a_{3} + a_{4} + ... + a_{20} \le 20
a
3
+
a
4
+
...
+
a
20
≤
20
.What is maximum value of the expression:
a
1
2
+
a
2
2
+
.
.
.
+
a
20
2
a_{1}^2 + a_{2}^2 + ... + a_{20}^2
a
1
2
+
a
2
2
+
...
+
a
20
2
?For which values of
a
1
,
a
2
,
.
.
.
,
a
20
a_{1}, a_{2}, ..., a_{20}
a
1
,
a
2
,
...
,
a
20
is the maximum value achieved?
3
1
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2007 JBMO TST- Macedonia, problem 3
Let
a
a
a
,
b
b
b
,
c
c
c
be real numbers such that
0
<
a
≤
b
≤
c
0 < a \le b \le c
0
<
a
≤
b
≤
c
. Prove that
(
a
+
3
b
)
(
b
+
4
c
)
(
c
+
2
a
)
≥
60
a
b
c
(a + 3b)(b + 4c)(c + 2a) \ge 60abc
(
a
+
3
b
)
(
b
+
4
c
)
(
c
+
2
a
)
≥
60
ab
c
.When does equality hold?
2
1
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2007 JBMO TST- Macedonia, problem 2
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram and let
E
E
E
be a point on the side
A
D
AD
A
D
, such that
A
E
E
D
=
m
\frac{AE}{ED} = m
E
D
A
E
=
m
. Let
F
F
F
be a point on
C
E
CE
CE
, such that
B
F
⊥
C
E
BF \perp CE
BF
⊥
CE
, and the point
G
G
G
is symmetrical to
F
F
F
with respect to
A
B
AB
A
B
. If point
A
A
A
is the circumcenter of triangle
B
F
G
BFG
BFG
, find the value of
m
m
m
.
1
1
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2007 JBMO TST- Macedonia, problem 1
Does there exist a positive integer
n
n
n
, such that the number
n
(
n
+
1
)
(
n
+
2
)
n(n + 1)(n + 2)
n
(
n
+
1
)
(
n
+
2
)
is the square of a positive integer?