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Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
2008 German National Olympiad
2008 German National Olympiad
Part of
German National Olympiad
Subcontests
(6)
5
1
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Sum of circumferences is 10 => Line intersects 4 disks
Inside a square of sidelength
1
1
1
there are finitely many disks that are allowed to overlap. The sum of all circumferences is
10
10
10
. Show that there is a line intersecting or touching at least
4
4
4
disks.
2
1
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Sum of radii of incircle and excircle is the length of [FA]
The triangle
△
S
F
A
\triangle SFA
△
SF
A
has a right angle at
F
F
F
. The points
P
P
P
and
Q
Q
Q
lie on the line
S
F
SF
SF
such that the point
P
P
P
lies between
S
S
S
and
F
F
F
and the point
F
F
F
is the midpoint of the segment
[
P
Q
]
[PQ]
[
PQ
]
. The circle
k
1
k_1
k
1
is th incircle of the triangle
△
S
P
A
\triangle SPA
△
SP
A
. The circle
k
2
k_2
k
2
lies outside the triangle
△
S
Q
A
\triangle SQA
△
SQ
A
and touches the segment
[
Q
A
]
[QA]
[
Q
A
]
and the lines
S
Q
SQ
SQ
and
S
A
SA
S
A
. Prove that the sum of the radii of the circles
k
1
k_1
k
1
and
k
2
k_2
k
2
equals the length of
[
F
A
]
[FA]
[
F
A
]
.
4
1
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Find smallest C such that 1+(x+y)² <= C·(1+x²)·(1+y²)
Find the smallest constant
C
C
C
such that for all real
x
,
y
x,y
x
,
y
1\plus{}(x\plus{}y)^2 \leq C \cdot (1\plus{}x^2) \cdot (1\plus{}y^2) holds.
1
1
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Solve sqrt(x+1)+sqrt(x+3) = sqrt(2x-1)+sqrt(2x+1)
Find all real numbers
x
x
x
such that \sqrt{x\plus{}1}\plus{}\sqrt{x\plus{}3} \equal{} \sqrt{2x\minus{}1}\plus{}\sqrt{2x\plus{}1}.
3
1
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Functional equation: f(y·f(x))·f(x) = f(x+y)
Find all functions
f
f
f
defined on non-negative real numbers having the following properties: (i) For all non-negative
x
x
x
it is
f
(
x
)
≥
0
f(x) \geq 0
f
(
x
)
≥
0
. (ii) It is f\left(1\right)\equal{}\frac 12. (iii) For all non-negative numbers
x
,
y
x,y
x
,
y
it is f\left( y \cdot f(x) \right) \cdot f(x) \equal{} f(x\plus{}y).
6
1
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4x^5-7 and 4x^13-7 being perfect squares
Find all real numbers
x
x
x
such that 4x^5 \minus{} 7 and 4x^{13} \minus{} 7 are both perfect squares.