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Contests
National and Regional Contests
Bosnia Herzegovina Contests
Bosnia Herzegovina Team Selection Test
2011 Bosnia Herzegovina Team Selection Test
2011 Bosnia Herzegovina Team Selection Test
Part of
Bosnia Herzegovina Team Selection Test
Subcontests
(3)
3
2
Hide problems
Old and easy sums
Numbers
1
,
2
,
.
.
.
,
2
n
1,2, ..., 2n
1
,
2
,
...
,
2
n
are partitioned into two sequences
a
1
<
a
2
<
.
.
.
<
a
n
a_1<a_2<...<a_n
a
1
<
a
2
<
...
<
a
n
and
b
1
>
b
2
>
.
.
.
>
b
n
b_1>b_2>...>b_n
b
1
>
b
2
>
...
>
b
n
. Prove that number
W
=
∣
a
1
−
b
1
∣
+
∣
a
2
−
b
2
∣
+
.
.
.
+
∣
a
n
−
b
n
∣
W= |a_1-b_1|+|a_2-b_2|+...+|a_n-b_n|
W
=
∣
a
1
−
b
1
∣
+
∣
a
2
−
b
2
∣
+
...
+
∣
a
n
−
b
n
∣
is a perfect square.
AD/BC=AF/FB=DG/GC
In quadrilateral
A
B
C
D
ABCD
A
BC
D
sides
A
D
AD
A
D
and
B
C
BC
BC
aren't parallel. Diagonals
A
C
AC
A
C
and
B
D
BD
B
D
intersect in
E
E
E
.
F
F
F
and
G
G
G
are points on sides
A
B
AB
A
B
and
D
C
DC
D
C
such
A
F
F
B
=
D
G
G
C
=
A
D
B
C
\frac{AF}{FB}=\frac{DG}{GC}=\frac{AD}{BC}
FB
A
F
=
GC
D
G
=
BC
A
D
Prove that if
E
,
F
,
G
E, F, G
E
,
F
,
G
are collinear then
A
B
C
D
ABCD
A
BC
D
is cyclic.
2
2
Hide problems
semicircle with diameter d
On semicircle, with diameter
∣
A
B
∣
=
d
|AB|=d
∣
A
B
∣
=
d
, are given points
C
C
C
and
D
D
D
such that:
∣
B
C
∣
=
∣
C
D
∣
=
a
|BC|=|CD|=a
∣
BC
∣
=
∣
C
D
∣
=
a
and
∣
D
A
∣
=
b
|DA|=b
∣
D
A
∣
=
b
where
a
,
b
,
d
a, b, d
a
,
b
,
d
are different positive integers. Find minimum possible value of
d
d
d
a+b+c=1, trivial
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive reals such that
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
. Prove that the inequality
a
1
+
b
−
c
3
+
b
1
+
c
−
a
3
+
c
1
+
a
−
b
3
≤
1
a \sqrt[3]{1+b-c} + b\sqrt[3]{1+c-a} + c\sqrt[3]{1+a-b} \leq 1
a
3
1
+
b
−
c
+
b
3
1
+
c
−
a
+
c
3
1
+
a
−
b
≤
1
holds.
1
2
Hide problems
triangleABC BC=1/2(AB+AC)
In triangle
A
B
C
ABC
A
BC
it holds
∣
B
C
∣
=
1
2
(
∣
A
B
∣
+
∣
A
C
∣
)
|BC|= \frac{1}{2}(|AB|+|AC|)
∣
BC
∣
=
2
1
(
∣
A
B
∣
+
∣
A
C
∣
)
. Let
M
M
M
and
N
N
N
be midpoints of
A
B
AB
A
B
and
A
C
AC
A
C
, and let
I
I
I
be the incenter of
A
B
C
ABC
A
BC
. Prove that
A
,
M
,
I
,
N
A, M, I, N
A
,
M
,
I
,
N
are concyclic.
Numbers on circle
Find maximum value of number
a
a
a
such that for any arrangement of numbers
1
,
2
,
…
,
10
1,2,\ldots ,10
1
,
2
,
…
,
10
on a circle, we can find three consecutive numbers such their sum bigger or equal than
a
a
a
.