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Contests
National and Regional Contests
Bosnia Herzegovina Contests
JBMO TST - Bosnia and Herzegovina
2004 Bosnia and Herzegovina Junior BMO TST
2004 Bosnia and Herzegovina Junior BMO TST
Part of
JBMO TST - Bosnia and Herzegovina
Subcontests
(5)
2
1
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a rectangle is divided into 9 smaller rectangles, 4 areas given
A rectangle is divided into
9
9
9
smaller rectangles. The area of four of them is
5
,
3
,
9
5, 3, 9
5
,
3
,
9
and
2
2
2
, as in the picture below. (The picture is not at scale.) https://cdn.artofproblemsolving.com/attachments/8/e/0ccd6f41073f776b62e9ef4522df1f1639ee31.png Determine the minimum area of the rectangle. Under what circumstances is it achieved?
3
1
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value of w =a/b+c/d
Let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
be reals such that
a
b
+
b
c
+
c
d
+
d
a
=
7
\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}= 7
b
a
+
c
b
+
d
c
+
a
d
=
7
and
a
c
+
b
d
+
c
a
+
d
b
=
12
\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}= 12
c
a
+
d
b
+
a
c
+
b
d
=
12
. Find the value of
w
=
a
b
+
c
d
w =\frac{a}{b}+\frac{c}{d}
w
=
b
a
+
d
c
.
1
1
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1/x +1/ y= 1/p diophantine
In the set of integers solve the equation
1
x
+
1
y
=
1
p
\frac{1}{x}+\frac{1}{y}=\frac{1}{p}
x
1
+
y
1
=
p
1
, where
p
p
p
is a prime number.
5
1
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D belongs to the angle bisector of <AGF
In the isosceles triangle
A
B
C
ABC
A
BC
(
A
C
=
B
C
AC = BC
A
C
=
BC
),
A
B
=
3
AB =\sqrt3
A
B
=
3
and the altitude
C
D
=
2
CD =\sqrt2
C
D
=
2
. Let
E
E
E
and
F
F
F
be the midpoints of
B
C
BC
BC
and
D
B
DB
D
B
, respectively and
G
G
G
be the intersection of
A
E
AE
A
E
and
C
F
CF
CF
. Prove that
D
D
D
belongs to the angle bisector of
∠
A
G
F
\angle AGF
∠
A
GF
.
4
1
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AB / BF - AC /AE = 1, parallelogram related
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram. On the ray
(
D
B
(DB
(
D
B
a point
E
E
E
is given such that the ray
(
A
B
(AB
(
A
B
is the angle bisector of
∠
C
A
E
\angle CAE
∠
C
A
E
. Let
F
F
F
be the intersection of
C
E
CE
CE
and
A
B
AB
A
B
. Prove that
A
B
B
F
−
A
C
A
E
=
1
\frac{AB}{BF} - \frac{AC}{AE} = 1
BF
A
B
−
A
E
A
C
=
1