MathDB
Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
2021 German National Olympiad
2021 German National Olympiad
Part of
German National Olympiad
Subcontests
(6)
6
1
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u,v,w in arithmetic progression such that uv+1, vw+1, wu+1 perfect squares
Determine whether there are infinitely many triples
(
u
,
v
,
w
)
(u,v,w)
(
u
,
v
,
w
)
of positive integers such that
u
,
v
,
w
u,v,w
u
,
v
,
w
form an arithmetic progression and the numbers
u
v
+
1
,
v
w
+
1
uv+1, vw+1
uv
+
1
,
v
w
+
1
and
w
u
+
1
wu+1
w
u
+
1
are all perfect squares.
5
1
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Symmetric inhomogeneous inequality with fractions
a) Determine the largest real number
A
A
A
with the following property: For all non-negative real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
, one has
1
+
y
z
1
+
x
2
+
1
+
z
x
1
+
y
2
+
1
+
x
y
1
+
z
2
≥
A
.
\frac{1+yz}{1+x^2}+\frac{1+zx}{1+y^2}+\frac{1+xy}{1+z^2} \ge A.
1
+
x
2
1
+
yz
+
1
+
y
2
1
+
z
x
+
1
+
z
2
1
+
x
y
≥
A
.
b) For this real number
A
A
A
, find all triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of non-negative real numbers for which equality holds in the above inequality.
4
1
Hide problems
Two similar triangles and a parallelogram called FANO
Let
O
F
T
OFT
OFT
and
N
O
T
NOT
NOT
be two similar triangles (with the same orientation) and let
F
A
N
O
FANO
F
A
NO
be a parallelogram. Show that
∣
O
F
∣
⋅
∣
O
N
∣
=
∣
O
A
∣
⋅
∣
O
T
∣
.
\vert OF\vert \cdot \vert ON\vert=\vert OA\vert \cdot \vert OT\vert.
∣
OF
∣
⋅
∣
ON
∣
=
∣
O
A
∣
⋅
∣
OT
∣.
3
1
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Sums of cubes of numbers with certain digits are equal
For a fixed
k
k
k
with
4
≤
k
≤
9
4 \le k \le 9
4
≤
k
≤
9
consider the set of all positive integers with
k
k
k
decimal digits such that each of the digits from
1
1
1
to
k
k
k
occurs exactly once.Show that it is possible to partition this set into two disjoint subsets such that the sum of the cubes of the numbers in the first set is equal to the sum of the cubes in the second set.
2
1
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Interior points of the sides minimizing the perimeter iff cyclic
Let
P
P
P
on
A
B
AB
A
B
,
Q
Q
Q
on
B
C
BC
BC
,
R
R
R
on
C
D
CD
C
D
and
S
S
S
on
A
D
AD
A
D
be points on the sides of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
. Show that the following are equivalent:(1) There is a choice of
P
,
Q
,
R
,
S
P,Q,R,S
P
,
Q
,
R
,
S
, for which all of them are interior points of their side, such that
P
Q
R
S
PQRS
PQRS
has minimal perimeter.(2)
A
B
C
D
ABCD
A
BC
D
is a cyclic quadrilateral with circumcenter in its interior.
1
1
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Roots of two quadratic equations
Determine all real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
and
d
d
d
with the following property: The numbers
a
a
a
and
b
b
b
are distinct roots of
2
x
2
−
3
c
x
+
8
d
2x^2-3cx+8d
2
x
2
−
3
c
x
+
8
d
and the numbers
c
c
c
and
d
d
d
are distinct roots of
2
x
2
−
3
a
x
+
8
b
2x^2-3ax+8b
2
x
2
−
3
a
x
+
8
b
.