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National and Regional Contests
Bosnia Herzegovina Contests
Bosnia Herzegovina Team Selection Test
1999 Bosnia and Herzegovina Team Selection Test
1999 Bosnia and Herzegovina Team Selection Test
Part of
Bosnia Herzegovina Team Selection Test
Subcontests
(6)
6
1
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Bosnia and Herzegovina TST 1999 Day 2 Problem 3
It is given polynomial
P
(
x
)
=
x
4
+
3
x
3
+
3
x
+
p
,
(
p
∈
R
)
P(x)=x^4+3x^3+3x+p, (p \in \mathbb{R})
P
(
x
)
=
x
4
+
3
x
3
+
3
x
+
p
,
(
p
∈
R
)
a
)
a)
a
)
Find
p
p
p
such that there exists polynomial with imaginary root
x
1
x_1
x
1
such that
∣
x
1
∣
=
1
\mid x_1 \mid =1
∣
x
1
∣=
1
and
2
R
e
(
x
1
)
=
1
2
(
17
−
3
)
2Re(x_1)=\frac{1}{2}\left(\sqrt{17}-3\right)
2
R
e
(
x
1
)
=
2
1
(
17
−
3
)
b
)
b)
b
)
Find all other roots of polynomial
P
P
P
c
)
c)
c
)
Prove that does not exist positive integer
n
n
n
such that
x
1
n
=
1
x_1^n=1
x
1
n
=
1
5
1
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Bosnia and Herzegovina TST 1999 Day 2 Problem 2
For any nonempty set
S
S
S
, we define
σ
(
S
)
\sigma(S)
σ
(
S
)
and
π
(
S
)
\pi(S)
π
(
S
)
as sum and product of all elements from set
S
S
S
, respectively. Prove that
a
)
a)
a
)
∑
1
π
(
S
)
=
n
\sum \limits_{} \frac{1}{\pi(S)} =n
∑
π
(
S
)
1
=
n
b
)
b)
b
)
∑
σ
(
S
)
π
(
S
)
=
(
n
2
+
2
n
)
−
(
1
+
1
2
+
1
3
+
.
.
.
+
1
n
)
(
n
+
1
)
\sum \limits_{} \frac{\sigma(S)}{\pi(S)} =(n^2+2n)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}\right)(n+1)
∑
π
(
S
)
σ
(
S
)
=
(
n
2
+
2
n
)
−
(
1
+
2
1
+
3
1
+
...
+
n
1
)
(
n
+
1
)
where
∑
\sum
∑
denotes sum by all nonempty subsets
S
S
S
of set
{
1
,
2
,
.
.
.
,
n
}
\{1,2,...,n\}
{
1
,
2
,
...
,
n
}
4
1
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Bosnia and Herzegovina TST 1999 Day 2 Problem 1
Let angle bisectors of angles
∠
B
A
C
\angle BAC
∠
B
A
C
and
∠
A
B
C
\angle ABC
∠
A
BC
of triangle
A
B
C
ABC
A
BC
intersect sides
B
C
BC
BC
and
A
C
AC
A
C
in points
D
D
D
and
E
E
E
, respectively. Let points
F
F
F
and
G
G
G
be foots of perpendiculars from point
C
C
C
on lines
A
D
AD
A
D
and
B
E
BE
BE
, respectively. Prove that
F
G
∣
∣
A
B
FG \mid \mid AB
FG
∣∣
A
B
3
1
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Bosnia and Herzegovina TST 1999 Day 1 Problem 3
Let
f
:
[
0
,
1
]
→
R
f : [0,1] \rightarrow \mathbb{R}
f
:
[
0
,
1
]
→
R
be injective function such that
f
(
0
)
+
f
(
1
)
=
1
f(0)+f(1)=1
f
(
0
)
+
f
(
1
)
=
1
. Prove that exists
x
1
x_1
x
1
,
x
2
∈
[
0
,
1
]
x_2 \in [0,1]
x
2
∈
[
0
,
1
]
,
x
1
≠
x
2
x_1 \neq x_2
x
1
=
x
2
such that
2
f
(
x
1
)
<
f
(
x
2
)
+
1
2
2f(x_1)<f(x_2)+\frac{1}{2}
2
f
(
x
1
)
<
f
(
x
2
)
+
2
1
. After that state at least one generalization of this result
2
1
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Bosnia and Herzegovina TST 1999 Day 1 Problem 2
Prove the inequality
a
2
b
+
c
−
a
+
b
2
a
+
c
−
b
+
c
2
a
+
b
−
c
≥
3
3
R
\frac{a^2}{b+c-a}+\frac{b^2}{a+c-b}+\frac{c^2}{a+b-c} \geq 3\sqrt{3}R
b
+
c
−
a
a
2
+
a
+
c
−
b
b
2
+
a
+
b
−
c
c
2
≥
3
3
R
in triangle
A
B
C
ABC
A
BC
where
a
a
a
,
b
b
b
and
c
c
c
are sides of triangle and
R
R
R
radius of circumcircle of
A
B
C
ABC
A
BC
1
1
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Bosnia and Herzegovina TST 1999 Day 1 Problem 1
Let
a
a
a
,
b
b
b
and
c
c
c
be lengths of sides of triangle
A
B
C
ABC
A
BC
. Prove that at least one of the equations
x
2
−
2
b
x
+
2
a
c
=
0
x^2-2bx+2ac=0
x
2
−
2
b
x
+
2
a
c
=
0
x
2
−
2
c
x
+
2
a
b
=
0
x^2-2cx+2ab=0
x
2
−
2
c
x
+
2
ab
=
0
x
2
−
2
a
x
+
2
b
c
=
0
x^2-2ax+2bc=0
x
2
−
2
a
x
+
2
b
c
=
0
does not have real solutions