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Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
2003 German National Olympiad
2003 German National Olympiad
Part of
German National Olympiad
Subcontests
(6)
3
1
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A difficult combinatorics problem with a (cute?) animal
Consider a
N
×
N
N\times N
N
×
N
square board where
N
≥
3
N\geq 3
N
≥
3
is an odd integer. The caterpillar Carl sits at the center of the square; all other cells contain distinct positive integers. An integer
n
n
n
weights 1\slash n kilograms. Carl wants to leave the board but can eat at most
2
2
2
kilograms. Determine whether Carl can always find a way out when a)
N
=
2003.
N=2003.
N
=
2003.
b)
N
N
N
is an arbitrary odd integer.
2
1
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Four circles of equal radius touching
There are four circles
k
1
,
k
2
,
k
3
k_1 , k_2 , k_3
k
1
,
k
2
,
k
3
and
k
4
k_4
k
4
of equal radius inside the triangle
A
B
C
ABC
A
BC
. The circle
k
1
k_1
k
1
touches the sides
A
B
,
C
A
AB, CA
A
B
,
C
A
and the circle
k
4
k_4
k
4
,
k
2
k_2
k
2
touches the sides
A
B
,
B
C
AB,BC
A
B
,
BC
and
k
4
k_4
k
4
, and
k
3
k_3
k
3
touches the sides
A
C
,
B
C
AC, BC
A
C
,
BC
and
k
4
.
k_4.
k
4
.
Prove that the center of
k
4
k_4
k
4
lies on the line connecting the incenter and circumcenter of
A
B
C
.
ABC.
A
BC
.
6
1
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Infinitely many pairs satisfy divisibility relation
Prove that there are infinitely many coprime, positive integers
a
,
b
a,b
a
,
b
such that
a
a
a
divides
b
2
−
5
b^2 -5
b
2
−
5
and
b
b
b
divides
a
2
−
5.
a^2 -5.
a
2
−
5.
1
1
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Chisinau MO p16 1949-56 VIII-IX system 2x2, x^3 + y^3= 7,x y (x + y) = -2
Solve the system of equations:
{
x
3
+
y
3
=
7
x
y
(
x
+
y
)
=
−
2
\begin{cases} x^3 + y^3= 7 \\ xy (x + y) = -2\end{cases}
{
x
3
+
y
3
=
7
x
y
(
x
+
y
)
=
−
2
4
1
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hexagon with 1/2 triangle area, ,_|_ midpoints (Oral Moscow Team MO 2004.1.10B4)
From the midpoints of the sides of an acute-angled triangle, perpendiculars are drawn to the adjacent sides. The resulting six straight lines bound the hexagon. Prove that its area is half the area of the original triangle.
5
1
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Interesting number theory problem
n
n
n
is a positive integer. Let
a
(
n
)
a(n)
a
(
n
)
be the smallest number for which
n
∣
a
(
n
)
!
n\mid a(n)!
n
∣
a
(
n
)!
Find all solutions of:
a
(
n
)
n
=
2
3
\frac{a(n)}{n}=\frac{2}{3}
n
a
(
n
)
=
3
2