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Problems
Contests
International Contests
Danube Competition in Mathematics
2005 Danube Mathematical Olympiad
2005 Danube Mathematical Olympiad
Part of
Danube Competition in Mathematics
Subcontests
(4)
4
1
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Maximal k such that an acceptable 2-coloring exists
Let
k
k
k
and
n
n
n
be positive integers. Consider an array of
2
(
2
n
−
1
)
2\left(2^n-1\right)
2
(
2
n
−
1
)
rows by
k
k
k
columns. A
2
2
2
-coloring of the elements of the array is said to be acceptable if any two columns agree on less than
2
n
−
1
2^n-1
2
n
−
1
entries on the same row. Given
n
n
n
, determine the maximum value of
k
k
k
for an acceptable
2
2
2
-coloring to exist.
3
1
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Classical tangents to the circumcircle
Let
C
\mathcal{C}
C
be a circle with center
O
O
O
, and let
A
A
A
be a point outside the circle. Let the two tangents from the point
A
A
A
to the circle
C
\mathcal{C}
C
meet this circle at the points
S
S
S
and
T
T
T
, respectively. Given a point
M
M
M
on the circle
C
\mathcal{C}
C
which is different from the points
S
S
S
and
T
T
T
, let the line
M
A
MA
M
A
meet the perpendicular from the point
S
S
S
to the line
M
O
MO
MO
at
P
P
P
. Prove that the reflection of the point
S
S
S
in the point
P
P
P
lies on the line
M
T
MT
MT
.
2
1
Hide problems
Binomial sum is divisible by 2^(n-1) - not that surprising
Prove that the sum:
S
n
=
(
n
1
)
+
(
n
3
)
⋅
2005
+
(
n
5
)
⋅
200
5
2
+
.
.
.
=
∑
k
=
0
⌊
n
−
1
2
⌋
(
n
2
k
+
1
)
⋅
200
5
k
S_n=\binom{n}{1}+\binom{n}{3}\cdot 2005+\binom{n}{5}\cdot 2005^2+...=\sum_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor}\binom{n}{2k+1}\cdot 2005^k
S
n
=
(
1
n
)
+
(
3
n
)
⋅
2005
+
(
5
n
)
⋅
200
5
2
+
...
=
k
=
0
∑
⌊
2
n
−
1
⌋
(
2
k
+
1
n
)
⋅
200
5
k
is divisible by
2
n
−
1
2^{n-1}
2
n
−
1
for any positive integer
n
n
n
.
1
1
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4x^3 - 3x + 1 = 2y^2 has at least 31 solutions
Prove that the equation
4
x
3
−
3
x
+
1
=
2
y
2
4x^3-3x+1=2y^2
4
x
3
−
3
x
+
1
=
2
y
2
has at least
31
31
31
solutions in positive integers
x
x
x
and
y
y
y
with
x
≤
2005
x\leq 2005
x
≤
2005
.