MathDB
Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
2024 German National Olympiad
2024 German National Olympiad
Part of
German National Olympiad
Subcontests
(6)
6
1
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Which power-means are majorized by (a^2/b+b^2/a)/2?
Decide whether there exists a largest positive integer
n
n
n
such that the inequality
a
2
b
+
b
2
a
2
≥
a
n
+
b
n
2
n
\frac{\frac{a^2}{b}+\frac{b^2}{a}}{2} \ge \sqrt[n]{\frac{a^n+b^n}{2}}
2
b
a
2
+
a
b
2
≥
n
2
a
n
+
b
n
holds for all positive real numbers
a
a
a
and
b
b
b
. If such a largest positive integer
n
n
n
exists, determine it.
4
1
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When is 133...3 prime?
Let
k
>
2
k>2
k
>
2
be a positive integer such that the
k
k
k
-digit number
n
k
=
133
…
3
n_k=133\dots 3
n
k
=
133
…
3
, consisting of a digit
1
1
1
followed by
k
−
1
k-1
k
−
1
digits
3
3
3
is prime. Show that
24
∣
k
(
k
+
2
)
24 \mid k(k+2)
24
∣
k
(
k
+
2
)
.
3
1
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The 24/7 party of 25 elves
At a party,
25
25
25
elves give each other presents. No elf gives a present to herself. Each elf gives a present to at least one other elf, but no elf gives a present to all other elves. Show that it is possible to choose a group of three elves including at least two elves who give a present to exactly one of the other two elves in the group.
2
1
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3D Billard in a mirrored cube
Six quadratic mirrors are put together to form a cube
A
B
C
D
E
F
G
H
ABCDEFGH
A
BC
D
EFG
H
with a mirrored interior. At each of the eight vertices, there is a tiny hole through which a laser beam can enter and leave the cube. A laser beam enters the cube at vertex
A
A
A
in a direction not parallel to any of the cube's sides. If the beam hits a side, it is reflected; if it hits an edge, the light is absorbed, and if it hits a vertex, it leaves the cube. For each positive integer
n
n
n
, determine the set of vertices where the laser beam can leave the cube after exactly
n
n
n
reflections.
1
1
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Symmetric equations in five variables with many solutions
The five real numbers
v
,
w
,
x
,
y
,
s
v,w,x,y,s
v
,
w
,
x
,
y
,
s
satisfy the system of equations \begin{align*} v&=wx+ys,\\ v^2&=w^2x+y^2s,\\ v^3&=w^3x+y^3s. \end{align*} Show that at least two of them are equal.
5
1
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prove that Q, R, V, and W lie on a circle
Let
△
A
B
C
\triangle ABC
△
A
BC
be a triangle and let
X
X
X
be a point in the interior of the triangle. The second intersection points of the lines
X
A
,
X
B
XA,XB
X
A
,
XB
and
X
C
XC
XC
with the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
are
P
,
Q
P,Q
P
,
Q
and
R
R
R
. Let
U
U
U
be a point on the ray
X
P
XP
XP
(these are the points on the line
X
P
XP
XP
such that
P
P
P
and
U
U
U
lie on the same side of
X
X
X
). The line through
U
U
U
parallel to
A
B
AB
A
B
intersects
B
Q
BQ
BQ
in
V
V
V
. The line through
U
U
U
parallel to
A
C
AC
A
C
intersects
C
R
CR
CR
in
W
W
W
. Prove that
Q
,
R
,
V
Q, R, V
Q
,
R
,
V
, and
W
W
W
lie on a circle.