MathDB
Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2016 Federal Competition For Advanced Students, P1
2016 Federal Competition For Advanced Students, P1
Part of
Austrian MO National Competition
Subcontests
(4)
4
1
Hide problems
Problem 4 -- Proportional Divisors
Determine all composite positive integers
n
n
n
with the following property: If
1
=
d
1
<
d
2
<
⋯
<
d
k
=
n
1 = d_1 < d_2 < \cdots < d_k = n
1
=
d
1
<
d
2
<
⋯
<
d
k
=
n
are all the positive divisors of
n
n
n
, then
(
d
2
−
d
1
)
:
(
d
3
−
d
2
)
:
⋯
:
(
d
k
−
d
k
−
1
)
=
1
:
2
:
⋯
:
(
k
−
1
)
(d_2 - d_1) : (d_3 - d_2) : \cdots : (d_k - d_{k-1}) = 1:2: \cdots :(k-1)
(
d
2
−
d
1
)
:
(
d
3
−
d
2
)
:
⋯
:
(
d
k
−
d
k
−
1
)
=
1
:
2
:
⋯
:
(
k
−
1
)
(Walther Janous)
3
1
Hide problems
Problem 3 -- Hop to It
Consider 2016 points arranged on a circle. We are allowed to jump ahead by 2 or 3 points in clockwise direction.What is the minimum number of jumps required to visit all points and return to the starting point?(Gerd Baron)
2
1
Hide problems
Problem 2 -- Orthocenter and Circumcenter
We are given an acute triangle
A
B
C
ABC
A
BC
with
A
B
>
A
C
AB > AC
A
B
>
A
C
and orthocenter
H
H
H
. The point
E
E
E
lies symmetric to
C
C
C
with respect to the altitude
A
H
AH
A
H
. Let
F
F
F
be the intersection of the lines
E
H
EH
E
H
and
A
C
AC
A
C
. Prove that the circumcenter of the triangle
A
E
F
AEF
A
EF
lies on the line
A
B
AB
A
B
. (Karl Czakler)
1
1
Hide problems
Problem 1 -- Constant Inequality Chain
Determine the largest constant
C
C
C
such that
(
x
1
+
x
2
+
⋯
+
x
6
)
2
≥
C
⋅
(
x
1
(
x
2
+
x
3
)
+
x
2
(
x
3
+
x
4
)
+
⋯
+
x
6
(
x
1
+
x
2
)
)
(x_1 + x_2 + \cdots + x_6)^2 \ge C \cdot (x_1(x_2 + x_3) + x_2(x_3 + x_4) + \cdots + x_6(x_1 + x_2))
(
x
1
+
x
2
+
⋯
+
x
6
)
2
≥
C
⋅
(
x
1
(
x
2
+
x
3
)
+
x
2
(
x
3
+
x
4
)
+
⋯
+
x
6
(
x
1
+
x
2
))
holds for all real numbers
x
1
,
x
2
,
⋯
,
x
6
x_1, x_2, \cdots , x_6
x
1
,
x
2
,
⋯
,
x
6
.For this
C
C
C
, determine all
x
1
,
x
2
,
⋯
x
6
x_1, x_2, \cdots x_6
x
1
,
x
2
,
⋯
x
6
such that equality holds.(Walther Janous)