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National and Regional Contests
North Macedonia Contests
JBMO TST - Macedonia
2012 JBMO TST - Macedonia
2012 JBMO TST - Macedonia
Part of
JBMO TST - Macedonia
Subcontests
(5)
1
1
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Macedonian JBMO TST 2012 - Problem 1
Find all prime numbers of the form
1
11
⋅
11
…
1
⏟
2
n
ones
\tfrac{1}{11} \cdot \underbrace{11\ldots 1}_{2n \textrm{ ones}}
11
1
⋅
2
n
ones
11
…
1
, where
n
n
n
is a natural number.
2
1
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Macedonian JBMO TST 2012 - Problem 2
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral inscribed in a circle of radius
1
1
1
. Prove that
0
<
(
A
B
+
B
C
+
C
D
+
A
D
)
−
(
A
C
+
B
D
)
<
4.
0< (AB+BC+CD+AD)-(AC+BD) < 4.
0
<
(
A
B
+
BC
+
C
D
+
A
D
)
−
(
A
C
+
B
D
)
<
4.
5
1
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Macedonian JBMO TST 2012 - Problem 5
n
≥
4
n\geq 4
n
≥
4
points are given in a plane such that any 3 of them are not collinear. Prove that a triangle exist such that all the points are in its interior and there is exactly one point laying on each side.
4
1
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Macedonia JBMO TST 2012 - Problem 4
Find all primes
p
p
p
and
q
q
q
such that
(
p
+
q
)
p
=
(
q
−
p
)
(
2
q
−
1
)
(p+q)^p = (q-p)^{(2q-1)}
(
p
+
q
)
p
=
(
q
−
p
)
(
2
q
−
1
)
3
1
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Macedonian JBMO TST 2012 - Problem 3
Let
a
a
a
,
b
b
b
,
c
c
c
be positive real numbers and
a
+
b
+
c
+
2
=
a
b
c
a+b+c+2=abc
a
+
b
+
c
+
2
=
ab
c
. Prove that
a
b
+
1
+
b
c
+
1
+
c
a
+
1
≥
2.
\frac{a}{b+1}+\frac{b}{c+1}+\frac{c}{a+1}\geq{2}.
b
+
1
a
+
c
+
1
b
+
a
+
1
c
≥
2
.