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National and Regional Contests
Bosnia Herzegovina Contests
Bosnia Herzegovina Team Selection Test
2001 Bosnia and Herzegovina Team Selection Test
2001 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 2001 Day 2 Problem 3
Prove that there exists infinitely many positive integers
n
n
n
such that equation
(
x
+
y
+
z
)
3
=
n
2
x
y
z
(x+y+z)^3=n^2xyz
(
x
+
y
+
z
)
3
=
n
2
x
yz
has solution
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
in set
N
3
\mathbb{N}^3
N
3
5
1
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Bosnia and Herzegovina TST 2001 Day 2 Problem 2
Let
n
n
n
be a positive integer,
n
≥
1
n \geq 1
n
≥
1
and
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
positive real numbers such that
x
1
+
x
2
+
.
.
.
+
x
n
=
1
x_1+x_2+...+x_n=1
x
1
+
x
2
+
...
+
x
n
=
1
. Does the following inequality hold
∑
i
=
1
n
x
i
1
−
x
1
⋅
.
.
.
⋅
x
i
−
1
⋅
x
i
+
1
⋅
.
.
.
x
n
≤
1
1
−
(
1
n
)
n
−
1
\sum_{i=1}^{n} {\frac{x_i}{1-x_1\cdot...\cdot x_{i-1} \cdot x_{i+1} \cdot ... x_n}} \leq \frac{1}{1-\left(\frac{1}{n}\right)^{n-1}}
i
=
1
∑
n
1
−
x
1
⋅
...
⋅
x
i
−
1
⋅
x
i
+
1
⋅
...
x
n
x
i
≤
1
−
(
n
1
)
n
−
1
1
4
1
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Bosnia and Herzegovina TST 2001 Day 2 Problem 1
In plane there are two circles with radiuses
r
1
r_1
r
1
and
r
2
r_2
r
2
, one outside the other. There are two external common tangents on those circles and one internal common tangent. The internal one intersects external ones in points
A
A
A
and
B
B
B
and touches one of the circles in point
C
C
C
. Prove that
A
C
⋅
B
C
=
r
1
⋅
r
2
AC \cdot BC=r_1\cdot r_2
A
C
⋅
BC
=
r
1
⋅
r
2
3
1
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Bosnia and Herzegovina TST 2001 Day 1 Problem 3
Find maximal value of positive integer
n
n
n
such that there exists subset of
S
=
{
1
,
2
,
.
.
.
,
2001
}
S=\{1,2,...,2001\}
S
=
{
1
,
2
,
...
,
2001
}
with
n
n
n
elements, such that equation
y
=
2
x
y=2x
y
=
2
x
does not have solutions in set
S
×
S
S \times S
S
×
S
2
1
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Bosnia and Herzegovina TST 2001 Day 1 Problem 2
For positive integers
x
x
x
,
y
y
y
and
z
z
z
holds
1
x
2
+
1
y
2
=
1
z
2
\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{z^2}
x
2
1
+
y
2
1
=
z
2
1
. Prove that
x
y
z
≥
3600
xyz\geq 3600
x
yz
≥
3600
1
1
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Bosnia and Herzegovina TST 2001 Day 1 Problem 1
On circle there are points
A
A
A
,
B
B
B
and
C
C
C
such that they divide circle in ratio
3
:
5
:
7
3:5:7
3
:
5
:
7
. Find angles of triangle
A
B
C
ABC
A
BC