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Problems
Contests
International Contests
Danube Competition in Mathematics
2010 Danube Mathematical Olympiad
2010 Danube Mathematical Olympiad
Part of
Danube Competition in Mathematics
Subcontests
(5)
3
1
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Number of pairwise disjoint monochromatic segments
All sides and diagonals of a convex
n
n
n
-gon,
n
≥
3
n\ge 3
n
≥
3
, are coloured one of two colours. Show that there exist
[
n
+
1
3
]
\left[\frac{n+1}{3}\right]
[
3
n
+
1
]
pairwise disjoint monochromatic segments.(Two segments are disjoint if they do not share an endpoint or an interior point).
2
1
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AA", BB" and CC" are concurrent
Given a triangle
A
B
C
ABC
A
BC
, let
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
be the perpendicular feet dropped from the centroid
G
G
G
of the triangle
A
B
C
ABC
A
BC
onto the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
respectively. Reflect
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
through
G
G
G
to
A
′
′
,
B
′
′
,
C
′
′
A'',B'',C''
A
′′
,
B
′′
,
C
′′
respectively. Prove that the lines
A
A
′
′
,
B
B
′
′
,
C
C
′
′
AA'',BB'',CC''
A
A
′′
,
B
B
′′
,
C
C
′′
are concurrent.
1
1
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n-gon decomposed into isosceles triangles
Determine all integer numbers
n
≥
3
n\ge 3
n
≥
3
such that the regular
n
n
n
-gon can be decomposed into isosceles triangles by non-intersecting diagonals.
5
1
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please help!!!! problem from danube cup romania 2010
Let
n
≥
3
n\ge3
n
≥
3
be a positive integer. Find the real numbers
x
1
≥
0
,
…
,
x
n
≥
0
x_1\ge0,\ldots,x_n\ge 0
x
1
≥
0
,
…
,
x
n
≥
0
, with
x
1
+
x
2
+
…
+
x
n
=
n
x_1+x_2+\ldots +x_n=n
x
1
+
x
2
+
…
+
x
n
=
n
, for which the expression
(
n
−
1
)
(
x
1
2
+
x
2
2
+
…
+
x
n
2
)
+
n
x
1
x
2
…
x
n
(n-1)(x_1^2+x_2^2+\ldots+x_n^2)+nx_1x_2\ldots x_n
(
n
−
1
)
(
x
1
2
+
x
2
2
+
…
+
x
n
2
)
+
n
x
1
x
2
…
x
n
takes a minimal value.
4
1
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Prime number and equation without solution
Let
p
p
p
be a prime number of the form
4
k
+
3
4k+3
4
k
+
3
. Prove that there are no integers
w
,
x
,
y
,
z
w,x,y,z
w
,
x
,
y
,
z
whose product is not divisible by
p
p
p
, such that:
w
2
p
+
x
2
p
+
y
2
p
=
z
2
p
.
w^{2p}+x^{2p}+y^{2p}=z^{2p}.
w
2
p
+
x
2
p
+
y
2
p
=
z
2
p
.