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Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
2004 German National Olympiad
2004 German National Olympiad
Part of
German National Olympiad
Subcontests
(6)
4
1
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A summation of integers close to square roots
For a positive integer
n
,
n,
n
,
let
a
n
a_n
a
n
be the integer closest to
n
.
\sqrt{n}.
n
.
Compute
1
a
1
+
1
a
2
+
⋯
+
1
a
2004
.
\frac{1}{a_1 } + \frac{1}{a_2 }+ \cdots + \frac{1}{a_{2004}}.
a
1
1
+
a
2
1
+
⋯
+
a
2004
1
.
2
1
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Some tangents are concurrent
Let
k
k
k
be a circle with center
M
.
M.
M
.
There is another circle
k
1
k_1
k
1
whose center
M
1
M_1
M
1
lies on
k
,
k,
k
,
and we denote the line through
M
M
M
and
M
1
M_1
M
1
by
g
.
g.
g
.
Let
T
T
T
be a point on
k
1
k_1
k
1
and inside
k
.
k.
k
.
The tangent
t
t
t
to
k
1
k_1
k
1
at
T
T
T
intersects
k
k
k
in two points
A
A
A
and
B
.
B.
B
.
Denote the tangents (diifferent from
t
t
t
) to
k
1
k_1
k
1
passing through
A
A
A
and
B
B
B
by
a
a
a
and
b
b
b
, respectively. Prove that the lines
a
,
b
,
a,b,
a
,
b
,
and
g
g
g
are either concurrent or parallel.
3
1
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Integer divisible by its sum of digits
Prove that for every positive integer
n
n
n
there is an
n
n
n
-digit number
z
z
z
with none of its digits
0
0
0
and such that
z
z
z
is divisible by its sum of digits.
1
1
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Symmetric system of equations
Find all real numbers
x
,
y
x,y
x
,
y
satisfying the following system of equations \begin{align*} x^4 +y^4 & =17(x+y)^2 \\ xy & =2(x+y). \end{align*}
5
1
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Standard inequality that I cannot find elsewhere
Prove that for four positive real numbers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
the following inequality holds and find all equality cases:
a
3
+
b
3
+
c
3
+
d
3
≥
a
2
b
+
b
2
c
+
c
2
d
+
d
2
a
.
a^3 +b^3 +c^3 +d^3 \geq a^2 b +b^2 c+ c^2 d +d^2 a.
a
3
+
b
3
+
c
3
+
d
3
≥
a
2
b
+
b
2
c
+
c
2
d
+
d
2
a
.
6
1
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circle with five points
Is there a circle which passes through five points with integer co-ordinates?