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Problems
Contests
International Contests
Danube Competition in Mathematics
2008 Danube Mathematical Competition
2008 Danube Mathematical Competition
Part of
Danube Competition in Mathematics
Subcontests
(4)
3
1
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length of sum of 2n vectors from center of a semicircle is an odd integer
On a semicircle centred at
O
O
O
and with radius
1
1
1
choose the respective points
A
1
,
A
2
,
.
.
.
,
A
2
n
A_1,A_2,...,A_{2n}
A
1
,
A
2
,
...
,
A
2
n
, for
n
∈
N
∗
n \in N^*
n
∈
N
∗
. The lenght of the projection of the vector
u
→
=
O
A
1
→
+
O
A
2
→
+
.
.
.
+
O
A
2
n
→
\overrightarrow {u}=\overrightarrow{OA_1} +\overrightarrow{OA_2}+...+\overrightarrow{OA_{2n}}
u
=
O
A
1
+
O
A
2
+
...
+
O
A
2
n
on the diameter is an odd integer. Show that the projection of that vector on the diameter is at least
1
1
1
.
2
1
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danube 3 common chords' concurrency
In a triangle
A
B
C
ABC
A
BC
let
A
1
A_1
A
1
be the midpoint of side
B
C
BC
BC
. Draw circles with centers
A
,
A
1
A, A1
A
,
A
1
and radii
A
A
1
,
B
C
AA_1, BC
A
A
1
,
BC
respectively and let
A
′
A
′
′
A'A''
A
′
A
′′
be their common chord. Similarly denote the segments
B
′
B
′
′
B'B''
B
′
B
′′
and
C
′
C
′
′
C'C''
C
′
C
′′
. Show that lines
A
′
A
′
′
,
B
′
B
′
′
′
A'A'', B'B'''
A
′
A
′′
,
B
′
B
′′′
and
C
′
C
′
′
C'C''
C
′
C
′′
are concurrent.
1
1
Hide problems
Danube Cup 2008
x
,
y
,
z
,
t
∈
R
+
∗
x,y,z,t \in \mathbb R_+^*
x
,
y
,
z
,
t
∈
R
+
∗
:
(
x
y
)
1
/
2
+
(
y
z
)
1
/
2
+
(
z
t
)
1
/
2
+
(
t
x
)
1
/
2
+
(
x
z
)
1
/
2
+
(
y
t
)
1
/
2
≥
3
(
x
y
z
+
x
y
t
+
x
z
t
+
y
z
t
)
1
3
(xy)^{1/2}+(yz)^{1/2}+(zt)^{1/2}+(tx)^{1/2}+(xz)^{1/2}+(yt)^{1/2} \ge 3(xyz+xyt+xzt+yzt)^{\frac{1}{3}}
(
x
y
)
1/2
+
(
yz
)
1/2
+
(
z
t
)
1/2
+
(
t
x
)
1/2
+
(
x
z
)
1/2
+
(
y
t
)
1/2
≥
3
(
x
yz
+
x
y
t
+
x
z
t
+
yz
t
)
3
1
4
1
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Maximum number of segments with lenghts greater than 1
Let
n
≥
2
n\geq 2
n
≥
2
be a positive integer. Find the maximum number of segments with lenghts greater than
1
,
1,
1
,
determined by
n
n
n
points which lie on a closed disc with radius
1.
1.
1.