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Problems
Contests
National and Regional Contests
North Macedonia Contests
JBMO TST - Macedonia
2011 Junior Macedonian Mathematical Olympiad
2011 Junior Macedonian Mathematical Olympiad
Part of
JBMO TST - Macedonia
Subcontests
(5)
5
1
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We say that a set $M$ containing $4$ elements is "evenly connected"
We say that a set
M
M{}
M
containing
4
4
4
elements is "evenly connected" if for each element
x
∈
M
x\in M
x
∈
M
, at least one of the numbers
x
−
2
x-2
x
−
2
or
x
+
2
x+2
x
+
2
belongs to the set
M
.
M.
M
.
Let
S
n
S_n
S
n
be the number of "evenly connected" subsets of
{
1
,
2
,
3
…
,
n
}
\{1,2,3\ldots,n\}
{
1
,
2
,
3
…
,
n
}
. Find the smallest
n
n{}
n
such that
S
n
≥
2011.
S_n \geq 2011.
S
n
≥
2011.
3
1
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All numbers from $1$ to $32$ a written on the stars
All numbers from
1
1
1
to
32
32
32
a written on the stars from the picture below, such that each number is written once. Can all sums of the numbers written in each square that is not divided in smaller squares be equal?
1
1
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classic problem
Let
S
(
n
)
S(n)
S
(
n
)
be the sum of digits of natural number
n
n{}
n
. Is there a natural number
n
n{}
n
for which
n
+
S
(
n
)
+
S
(
S
(
n
)
)
=
2011
?
n+S(n)+S(S(n))=2011?
n
+
S
(
n
)
+
S
(
S
(
n
))
=
2011
?
4
1
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find all integers m
Find all integers
m
m
m
such that
m
3
+
m
2
+
7
m^3+m^2+7
m
3
+
m
2
+
7
is divisible by
m
2
−
m
+
1
m^2-m+1
m
2
−
m
+
1
.
2
1
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nice
Two circles
k
1
k_1
k
1
and
k
2
k_2
k
2
are given with centers
P
P
P
and
R
R
R
respectively, touching externally at point
A
A
A
. Let
p
p
p
be their common tangent line which does not pass trough
A
A
A
and touch
k
1
k_1
k
1
at
B
B
B
and
k
2
k_2
k
2
at
C
C
C
.
P
R
PR
PR
cuts
B
C
BC
BC
at point
E
E
E
and
k
2
k_2
k
2
at
A
A
A
and
D
D
D
. If
A
B
=
2
A
C
AB=2AC
A
B
=
2
A
C
find
B
C
D
E
\frac{BC}{DE}
D
E
BC
.