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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2013 Federal Competition For Advanced Students, Part 2
2013 Federal Competition For Advanced Students, Part 2
Part of
Austrian MO National Competition
Subcontests
(6)
6
1
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Regular octahedron - equal angles
Consider a regular octahedron
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
with lower vertex
E
E
E
, upper vertex
F
F
F
, middle cross-section
A
B
C
D
ABCD
A
BC
D
, midpoint
M
M
M
and circumscribed sphere
k
k
k
. Further, let
X
X
X
be an arbitrary point inside the face
A
B
F
ABF
A
BF
. Let the line
E
X
EX
EX
intersect
k
k
k
in
E
E
E
and
Z
Z
Z
, and the plane
A
B
C
D
ABCD
A
BC
D
in
Y
Y
Y
. Show that
∢
E
M
Z
=
∢
E
Y
F
\sphericalangle{EMZ}=\sphericalangle{EYF}
∢
EMZ
=
∢
E
Y
F
.
5
1
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Number of different orders of points
Let
n
⩾
3
n\geqslant3
n
⩾
3
be an integer. Let
A
1
A
2
…
A
n
A_1A_2\ldots A_n
A
1
A
2
…
A
n
be a convex
n
n
n
-gon. Consider a line
g
g
g
through
A
1
A_1
A
1
that does not contain a further vertice of the
n
n
n
-gon. Let
h
h
h
be the perpendicular to
g
g
g
through
A
1
A_1
A
1
. Project the
n
n
n
-gon orthogonally on
h
h
h
. For
j
=
1
,
…
,
n
j=1,\ldots,n
j
=
1
,
…
,
n
, let
B
j
B_j
B
j
be the image of
A
j
A_j
A
j
under this projection. The line
g
g
g
is called admissible if the points
B
j
B_j
B
j
are pairwise distinct. Consider all convex
n
n
n
-gons and all admissible lines
g
g
g
. How many different orders of the points
B
1
,
…
,
B
n
B_1,\ldots,B_n
B
1
,
…
,
B
n
are possible?
4
1
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Smaller than [n(n+1)/2]^2
For a positive integer
n
n
n
, let
a
1
,
a
2
,
…
a
n
a_1, a_2, \ldots a_n
a
1
,
a
2
,
…
a
n
be nonnegative real numbers such that for all real numbers
x
1
>
x
2
>
…
>
x
n
>
0
x_1>x_2>\ldots>x_n>0
x
1
>
x
2
>
…
>
x
n
>
0
with
x
1
+
x
2
+
…
+
x
n
<
1
x_1+x_2+\ldots+x_n<1
x
1
+
x
2
+
…
+
x
n
<
1
, the inequality
∑
k
=
1
n
a
k
x
k
3
<
1
\sum_{k=1}^na_kx_k^3<1
∑
k
=
1
n
a
k
x
k
3
<
1
holds. Show that
n
a
1
+
(
n
−
1
)
a
2
+
…
+
(
n
−
j
+
1
)
a
j
+
…
+
a
n
⩽
n
2
(
n
+
1
)
2
4
.
na_1+(n-1)a_2+\ldots+(n-j+1)a_j+\ldots+a_n\leqslant\frac{n^2(n+1)^2}{4}.
n
a
1
+
(
n
−
1
)
a
2
+
…
+
(
n
−
j
+
1
)
a
j
+
…
+
a
n
⩽
4
n
2
(
n
+
1
)
2
.
3
1
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Heptagon with extremal area
A square and an equilateral triangle are inscribed in a same circle. The seven vertices form a convex heptagon
S
S
S
inscribed in the circle (
S
S
S
might be a hexagon if two vertices coincide). For which positions of the triangle relative to the square does
S
S
S
have the largest and smallest area, respectively?
2
1
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f(x^k*y^k) = x*y*f(x)*f(y)
Let
k
k
k
be an integer. Determine all functions
f
:
R
→
R
f\colon \mathbb{R}\to\mathbb{R}
f
:
R
→
R
with
f
(
0
)
=
0
f(0)=0
f
(
0
)
=
0
and f(x^ky^k)=xyf(x)f(y)\qquad \mbox{for } x,y\neq 0.
1
1
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Floor and Ceiling
For each pair
(
a
,
b
)
(a,b)
(
a
,
b
)
of positive integers, determine all non-negative integers
n
n
n
such that
b
+
⌊
n
a
⌋
=
⌈
n
+
b
a
⌉
.
b+\left\lfloor{\frac{n}{a}}\right\rfloor=\left\lceil{\frac{n+b}{a}}\right\rceil.
b
+
⌊
a
n
⌋
=
⌈
a
n
+
b
⌉
.