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International Contests
Junior Balkan MO
2008 Junior Balkan MO
2008 Junior Balkan MO
Part of
Junior Balkan MO
Subcontests
(4)
3
1
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Equation with primes
Find all prime numbers
p
,
q
,
r
p,q,r
p
,
q
,
r
, such that \frac{p}{q}\minus{}\frac{4}{r\plus{}1}\equal{}1
4
1
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4x4 table and 16 squares
A
4
×
4
4\times 4
4
×
4
table is divided into
16
16
16
white unit square cells. Two cells are called neighbors if they share a common side. A move consists in choosing a cell and the colors of neighbors from white to black or from black to white. After exactly
n
n
n
moves all the
16
16
16
cells were black. Find all possible values of
n
n
n
.
2
1
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Equilateral triangle and an unit circle
The vertices
A
A
A
and
B
B
B
of an equilateral triangle
A
B
C
ABC
A
BC
lie on a circle
k
k
k
of radius
1
1
1
, and the vertex
C
C
C
is in the interior of the circle
k
k
k
. A point
D
D
D
, different from
B
B
B
, lies on
k
k
k
so that AD\equal{}AB. The line
D
C
DC
D
C
intersects
k
k
k
for the second time at point
E
E
E
. Find the length of the line segment
C
E
CE
CE
.
1
1
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Find a,b,c,d
Find all real numbers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
such that \left\{\begin{array}{cc}a \plus{} b \plus{} c \plus{} d \equal{} 20, \\ ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd \equal{} 150. \end{array} \right.