MathDB
Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2014 Federal Competition For Advanced Students, P2
2014 Federal Competition For Advanced Students, P2
Part of
Austrian MO National Competition
Subcontests
(6)
6
1
Hide problems
triangle by 3 circumcenters similar to the starting triangle
Let
U
U
U
be the center of the circumcircle of the acute-angled triangle
A
B
C
ABC
A
BC
. Let
M
A
,
M
B
M_A, M_B
M
A
,
M
B
and
M
C
M_C
M
C
be the circumcenters of triangles
U
B
C
,
U
A
C
UBC, UAC
U
BC
,
U
A
C
and
U
A
B
UAB
U
A
B
respecrively. For which triangles
A
B
C
ABC
A
BC
is the triangle
M
A
M
B
M
C
M_AM_BM_C
M
A
M
B
M
C
similar to the starting triangle (with a suitable order of the vertices)?
5
1
Hide problems
in Z^3: (x^2 + y^2z^2) (y^2 + x^2z^2) (z^2 + x^2y^2) \ge 8xy^2z^3
Show that the inequality
(
x
2
+
y
2
z
2
)
(
y
2
+
x
2
z
2
)
(
z
2
+
x
2
y
2
)
≥
8
x
y
2
z
3
(x^2 + y^2z^2) (y^2 + x^2z^2) (z^2 + x^2y^2) \ge 8xy^2z^3
(
x
2
+
y
2
z
2
)
(
y
2
+
x
2
z
2
)
(
z
2
+
x
2
y
2
)
≥
8
x
y
2
z
3
is valid for all integers
x
,
y
x, y
x
,
y
and
z
z
z
.When does equality apply?
4
1
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sum of squares divides product of squares of {n, n + 1, n + 2, n + 3, n + 4}
For an integer
n
n
n
let
M
(
n
)
=
{
n
,
n
+
1
,
n
+
2
,
n
+
3
,
n
+
4
}
M (n) = \{n, n + 1, n + 2, n + 3, n + 4\}
M
(
n
)
=
{
n
,
n
+
1
,
n
+
2
,
n
+
3
,
n
+
4
}
. Furthermore, be
S
(
n
)
S (n)
S
(
n
)
sum of squares and
P
(
n
)
P (n)
P
(
n
)
the product of the squares of the elements of
M
(
n
)
M (n)
M
(
n
)
. For which integers
n
n
n
is
S
(
n
)
S (n)
S
(
n
)
a divisor of
P
(
n
)
P (n)
P
(
n
)
?
3
1
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triangle inequality =>a^n + b^n> c^n, b^n + c^n> a^n and a^n + c^n> b^n
(i) For which triangles with side lengths
a
,
b
a, b
a
,
b
and
c
c
c
apply besides the triangle inequalities
a
+
b
>
c
,
b
+
c
>
a
a + b> c, b + c> a
a
+
b
>
c
,
b
+
c
>
a
and
c
+
a
>
b
c + a> b
c
+
a
>
b
also the inequalities
a
2
+
b
2
>
c
2
,
b
2
+
c
2
>
a
2
a^2 + b^2> c^2, b^2 + c^2> a^2
a
2
+
b
2
>
c
2
,
b
2
+
c
2
>
a
2
and
a
2
+
c
2
>
b
2
a^2 + c^2> b^2
a
2
+
c
2
>
b
2
?(ii) For which triangles with side lengths
a
,
b
a, b
a
,
b
and
c
c
c
apply besides the triangle inequalities
a
+
b
>
c
,
b
+
c
>
a
a + b> c, b + c> a
a
+
b
>
c
,
b
+
c
>
a
and
c
+
a
>
b
c + a> b
c
+
a
>
b
also for all positive natural
n
n
n
the inequalities
a
n
+
b
n
>
c
n
,
b
n
+
c
n
>
a
n
a^n + b^n> c^n, b^n + c^n> a^n
a
n
+
b
n
>
c
n
,
b
n
+
c
n
>
a
n
and
a
n
+
c
n
>
b
n
a^n + c^n> b^n
a
n
+
c
n
>
b
n
?
2
1
Hide problems
functional in [1,oo) , f (x^2 -y^2) = f (xy)
Let
S
S
S
be the set of all real numbers greater than or equal to
1
1
1
. Determine all functions
f
:
S
→
S
f: S \to S
f
:
S
→
S
, so that for all real numbers
x
,
y
∈
S
x ,y \in S
x
,
y
∈
S
with
x
2
−
y
2
∈
S
x^2 -y^2 \in S
x
2
−
y
2
∈
S
the condition
f
(
x
2
−
y
2
)
=
f
(
x
y
)
f (x^2 -y^2) = f (xy)
f
(
x
2
−
y
2
)
=
f
(
x
y
)
is fulfilled.
1
1
Hide problems
no of divisors function, for every divisor t of n , d(t) is a divisor of d(n)
For each positive natural number
n
n
n
let
d
(
n
)
d (n)
d
(
n
)
be the number of its divisors including
1
1
1
and
n
n
n
. For which positive natural numbers
n
n
n
, for every divisor
t
t
t
of
n
n
n
, that
d
(
t
)
d (t)
d
(
t
)
is a divisor of
d
(
n
)
d (n)
d
(
n
)
?