MathDB
Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO Regional Competition
2019 Regional Competition For Advanced Students
2019 Regional Competition For Advanced Students
Part of
Austrian MO Regional Competition
Subcontests
(4)
3
1
Hide problems
sum of -1 and 1 in a nxn grid , when is zero
Let
n
≥
2
n\ge 2
n
≥
2
be a natural number. An
n
×
n
n \times n
n
×
n
grid is drawn on a blackboard and each field with one of the numbers
−
1
-1
−
1
or
+
1
+1
+
1
labeled. Then the
n
n
n
row and also the
n
n
n
column sums calculated and the sum
S
n
S_n
S
n
of all these
2
n
2n
2
n
sums determined. (a) Show that for no odd number
n
n
n
there is a label with
S
n
=
0
S_n = 0
S
n
=
0
. (b) Show that if
n
n
n
is an even number, there are at least six different labels with
S
n
=
0
S_n = 0
S
n
=
0
.
4
1
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find n smaller than 128^{97} with exactly 2019 divisors
Find all natural numbers
n
n
n
that are smaller than
12
8
97
128^{97}
12
8
97
and have exactly
2019
2019
2019
divisors.
2
1
Hide problems
parallel wanted, cyclic convex pentagon given
The convex pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
is cyclic and
A
B
=
B
D
AB = BD
A
B
=
B
D
. Let point
P
P
P
be the intersection of the diagonals
A
C
AC
A
C
and
B
E
BE
BE
. Let the straight lines
B
C
BC
BC
and
D
E
DE
D
E
intersect at point
Q
Q
Q
. Prove that the straight line
P
Q
PQ
PQ
is parallel to the diagonal
A
D
AD
A
D
.
1
1
Hide problems
Simple inequality
Let
x
,
y
x,y
x
,
y
be real numbers such that
(
x
+
1
)
(
y
+
2
)
=
8.
(x+1)(y+2)=8.
(
x
+
1
)
(
y
+
2
)
=
8.
Prove that
(
x
y
−
10
)
2
≥
64.
(xy-10)^2\ge 64.
(
x
y
−
10
)
2
≥
64.