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Problems
Contests
National and Regional Contests
Bosnia Herzegovina Contests
JBMO TST - Bosnia and Herzegovina
2006 Bosnia and Herzegovina Junior BMO TST
2006 Bosnia and Herzegovina Junior BMO TST
Part of
JBMO TST - Bosnia and Herzegovina
Subcontests
(4)
1
1
Hide problems
1/x + 2/y + 3/z = 1, diophantine, x > y > z
. Find all triplets
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
,
x
>
y
>
z
x > y > z
x
>
y
>
z
of positive integers such that
1
x
+
2
y
+
3
z
=
1
\frac{1}{x}+\frac{2}{y}+\frac{3}{z}= 1
x
1
+
y
2
+
z
3
=
1
3
1
Hide problems
w = a^{2006} + b^{2006} + c^{2006} + d^{2006}$ is composite when ab = cd
Let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
be positive integers such that
a
b
=
c
d
ab = cd
ab
=
c
d
. Prove that
w
=
a
2006
+
b
2006
+
c
2006
+
d
2006
w = a^{2006} + b^{2006} + c^{2006} + d^{2006}
w
=
a
2006
+
b
2006
+
c
2006
+
d
2006
is composite.
4
1
Hide problems
numbers of Tetris Figures, fill 4 x7 rectangle
A Tetris Figure is every figure in the plane which consists of
4
4
4
unit squares connected by their sides (and don’t overlap). Two Tetris Figures are the same if one can be rotated in the plane to become the other. a) Prove that there exist exactly
7
7
7
different Tetris Figures. b) Is it possible to fill a
4
×
7
4 \times 7
4
×
7
rectangle by using once each of the
7
7
7
different Tetris Figures?
2
1
Hide problems
equilateral wanted, <C=60^o, altitudes and midpoint
In an acute triangle
A
B
C
ABC
A
BC
,
∠
C
=
6
0
o
\angle C = 60^o
∠
C
=
6
0
o
. If
A
A
′
AA'
A
A
′
and
B
B
′
BB'
B
B
′
are two of the altitudes and
C
1
C_1
C
1
is the midpoint of
A
B
AB
A
B
, prove that triangle
C
1
A
′
B
′
C_1A'B'
C
1
A
′
B
′
is equilateral.