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National and Regional Contests
Serbia Contests
Serbia JBMO TST
2011 Serbia JBMO TST
2011 Serbia JBMO TST
Part of
Serbia JBMO TST
Subcontests
(4)
2
1
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easy problem from Serbia JBMO TST 2011
p
(
n
)
p(n)
p
(
n
)
is a product of all digits of n.Calculate:
p
(
1001
)
+
p
(
1002
)
+
.
.
.
+
p
(
2011
)
p(1001) + p(1002) + ... + p(2011)
p
(
1001
)
+
p
(
1002
)
+
...
+
p
(
2011
)
1
1
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Tetromino
A
t
e
t
r
o
m
i
n
o
tetromino
t
e
t
ro
min
o
is a figure made up of four unit squares connected by common edges. [List=i] [*] If we do not distinguish between the possible rotations of a tetromino within its plane, prove that there are seven distinct tetrominos. [*]Prove or disprove the statement: It is possible to pack all seven distinct tetrominos into
4
×
7
4\times 7
4
×
7
rectangle without overlapping.
3
1
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Geometry problem
Let
△
A
B
C
\triangle ABC
△
A
BC
be a right-angled triangle and
B
C
>
A
C
BC > AC
BC
>
A
C
.
M
M
M
is a point on
B
C
BC
BC
such that
B
M
=
A
C
BM = AC
BM
=
A
C
and
N
N
N
is a point on
A
C
AC
A
C
such that
A
N
=
C
M
AN = CM
A
N
=
CM
. Find the angle between
B
N
BN
BN
and
A
M
AM
A
M
.
4
1
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minimum
If a, b, c are positive real numbers with
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
. Find the minimum value of
a
+
b
+
c
+
1
a
b
c
\sqrt{a}+\sqrt{b}+\sqrt{c}+\frac{1}{\sqrt{abc}}
a
+
b
+
c
+
ab
c
1