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National and Regional Contests
Bosnia Herzegovina Contests
JBMO TST - Bosnia and Herzegovina
2011 Bosnia and Herzegovina Junior BMO TST
2011 Bosnia and Herzegovina Junior BMO TST
Part of
JBMO TST - Bosnia and Herzegovina
Subcontests
(4)
4
1
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Bosnia and Herzegovina JBMO TST 2011 Problem 4
Let us consider mathematical crossword which we fill with numbers
0
0
0
,
1
1
1
,
2
2
2
,
3
3
3
,
4
4
4
,
5
5
5
,
6
6
6
,
7
7
7
,
8
8
8
,
9
9
9
such that: 1) All digits occur exactly twice 2)
10
10
10
horizontally divides
4
4
4
vertically 3)
4
⋅
4 \cdot
4
⋅
(
4
4
4
horizontally -
4
4
4
vertically +
5
5
5
) equals
1
1
1
vertically 4)
36
36
36
divides
1
1
1
horizontally and
5
5
5
vertically 5)
9
9
9
vertically divides
5
5
5
verticallyIn how many ways we can solve this mathematical crossword?https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvOC85LzgyNjUzYmNkNTVmNDE1YTg4OWVkNzAzYzE1M2JkZWE0MThiYWY1LnBuZw==&rn=Y3Jvc3N3b3JkLnBuZw==
3
1
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Bosnia and Herzegovina JBMO TST 2011 Problem 3
In isosceles triangle
A
B
C
ABC
A
BC
(
A
C
=
B
C
AC=BC
A
C
=
BC
), angle bisector
∠
B
A
C
\angle BAC
∠
B
A
C
and altitude
C
D
CD
C
D
from point
C
C
C
intersect at point
O
O
O
, such that
C
O
=
3
⋅
O
D
CO=3 \cdot OD
CO
=
3
⋅
O
D
. In which ratio does altitude from point
A
A
A
on side
B
C
BC
BC
divide altitude
C
D
CD
C
D
of triangle
A
B
C
ABC
A
BC
2
1
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Bosnia and Herzegovina JBMO TST 2011 Problem 2
Prove inequality, with
a
a
a
and
b
b
b
nonnegative real numbers:
a
+
b
1
+
a
+
b
≤
a
1
+
a
+
b
1
+
b
≤
2
(
a
+
b
)
2
+
a
+
b
\frac{a+b}{1+a+b}\leq \frac{a}{1+a} + \frac{b}{1+b} \leq \frac{2(a+b)}{2+a+b}
1
+
a
+
b
a
+
b
≤
1
+
a
a
+
1
+
b
b
≤
2
+
a
+
b
2
(
a
+
b
)
1
1
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Bosnia and Herzegovina JBMO TST 2011 Problem 1
Solve equation
1
x
−
1
y
=
1
5
−
1
x
y
\frac{1}{x}-\frac{1}{y}=\frac{1}{5}-\frac{1}{xy}
x
1
−
y
1
=
5
1
−
x
y
1
, where
x
x
x
and
y
y
y
are positive integers.