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Problems
Contests
National and Regional Contests
Bosnia Herzegovina Contests
JBMO TST - Bosnia and Herzegovina
2003 Bosnia and Herzegovina Junior BMO TST
2003 Bosnia and Herzegovina Junior BMO TST
Part of
JBMO TST - Bosnia and Herzegovina
Subcontests
(4)
1
1
Hide problems
w =\frac{3b + 2c}{6a}+\frac{2c + 6a}{3b}+\frac{6a + 3b}{2c} if 1/a+2/b+3/c=0
Non-zero real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfy the condition
1
a
+
2
b
+
3
c
=
0
\frac{1}{a}+\frac{2}{b}+\frac{3}{c}= 0
a
1
+
b
2
+
c
3
=
0
.Determine the value of
w
=
3
b
+
2
c
6
a
+
2
c
+
6
a
3
b
+
6
a
+
3
b
2
c
w =\frac{3b + 2c}{6a}+\frac{2c + 6a}{3b}+\frac{6a + 3b}{2c}
w
=
6
a
3
b
+
2
c
+
3
b
2
c
+
6
a
+
2
c
6
a
+
3
b
.
2
1
Hide problems
2\sqrt{3(x + 1)^2} -3 \sqrt{2(y - 2)^2}= 4\sqrt2 + 5|\sqrt2 - \sqrt3|
Solve in the set of rational numbers the equation
2
3
(
x
+
1
)
2
−
3
2
(
y
−
2
)
2
=
4
2
+
5
∣
2
−
3
∣
2\sqrt{3(x + 1)^2} -3 \sqrt{2(y - 2)^2}= 4\sqrt2 + 5|\sqrt2 - \sqrt3|
2
3
(
x
+
1
)
2
−
3
2
(
y
−
2
)
2
=
4
2
+
5∣
2
−
3
∣
3
1
Hide problems
a^3 + b^3 + c^3 is divisible by 6
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be integers such that the number
a
2
+
b
2
+
c
2
a^2 +b^2 +c^2
a
2
+
b
2
+
c
2
is divisible by
6
6
6
and the number
a
b
+
b
c
+
c
a
ab + bc + ca
ab
+
b
c
+
c
a
is divisible by
3
3
3
. Prove that the number
a
3
+
b
3
+
c
3
a^3 + b^3 + c^3
a
3
+
b
3
+
c
3
is divisible by
6
6
6
.
4
1
Hide problems
a + c < b + d, ac < bd in trapezoid with m^2 + n^2 = (a + c)^2
In the trapezoid
A
B
C
D
ABCD
A
BC
D
(
A
B
∥
D
C
AB \parallel DC
A
B
∥
D
C
) the bases have lengths
a
a
a
and
c
c
c
(
c
<
a
c < a
c
<
a
), while the other sides have lengths
b
b
b
and
d
d
d
. The diagonals are of lengths
m
m
m
and
n
n
n
. It is known that
m
2
+
n
2
=
(
a
+
c
)
2
m^2 + n^2 = (a + c)^2
m
2
+
n
2
=
(
a
+
c
)
2
. a) Find the angle between the diagonals of the trapezoid. b) Prove that
a
+
c
<
b
+
d
a + c < b + d
a
+
c
<
b
+
d
. c) Prove that
a
c
<
b
d
ac < bd
a
c
<
b
d
.