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Problems
Contests
National and Regional Contests
Slovenia Contests
Slovenia Team Selection Tests
2005 Slovenia Team Selection Test
2005 Slovenia Team Selection Test
Part of
Slovenia Team Selection Tests
Subcontests
(4)
1
1
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if MA\cdot MC + MA\cdot CD = MB \cdot MD then <BKC =< BDC
The diagonals of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
intersect at
M
M
M
. The bisector of
∠
A
C
D
\angle ACD
∠
A
C
D
intersects the ray
B
A
BA
B
A
at
K
K
K
. Prove that if
M
A
⋅
M
C
+
M
A
⋅
C
D
=
M
B
⋅
M
D
MA\cdot MC + MA\cdot CD = MB \cdot MD
M
A
⋅
MC
+
M
A
⋅
C
D
=
MB
⋅
M
D
, then
∠
B
K
C
=
∠
B
D
C
\angle BKC = \angle BDC
∠
B
K
C
=
∠
B
D
C
2
1
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x^2(f(x)+ f(y)) = (x+y)f (f(x)y) in R^+
Find all functions
f
:
R
+
→
R
+
f : R^+ \to R^+
f
:
R
+
→
R
+
such that
x
2
(
f
(
x
)
+
f
(
y
)
)
=
(
x
+
y
)
f
(
f
(
x
)
y
)
x^2(f(x)+ f(y)) = (x+y)f (f(x)y)
x
2
(
f
(
x
)
+
f
(
y
))
=
(
x
+
y
)
f
(
f
(
x
)
y
)
for any
x
,
y
>
0
x,y > 0
x
,
y
>
0
.
4
1
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number of sequences of 2005 terms with terms 1 or -1
Find the number of sequences of
2005
2005
2005
terms with the following properties: (i) No three consecutive terms of the sequence are equal, (ii) Every term equals either
1
1
1
or
−
1
-1
−
1
, (iii) The sum of all terms of the sequence is at least
666
666
666
.
6
1
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3\sqrt[3]{\frac{1}{abc} +6(a+b+c) }\le \frac{\sqrt[3]3}{abc}
Let
a
,
b
,
c
>
0
a,b,c > 0
a
,
b
,
c
>
0
and
a
b
+
b
c
+
c
a
=
1
ab+bc+ca = 1
ab
+
b
c
+
c
a
=
1
. Prove the inequality
3
1
a
b
c
+
6
(
a
+
b
+
c
)
3
≤
3
3
a
b
c
3\sqrt[3]{\frac{1}{abc} +6(a+b+c) }\le \frac{\sqrt[3]3}{abc}
3
3
ab
c
1
+
6
(
a
+
b
+
c
)
≤
ab
c
3
3