MathDB
Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO Beginners' Competition
2020 Austrian Junior Regional Competition
2020 Austrian Junior Regional Competition
Part of
Austrian MO Beginners' Competition
Subcontests
(4)
4
1
Hide problems
pos. integer a , such that 7an -3n! = 2020 has a positive integer solutions
Find all positive integers
a
a
a
for which the equation
7
a
n
−
3
n
!
=
2020
7an -3n! = 2020
7
an
−
3
n
!
=
2020
has a positive integer solution
n
n
n
.(Richard Henner)
3
1
Hide problems
EM // AC wanted, isosceles trapezoid related
Given is an isosceles trapezoid
A
B
C
D
ABCD
A
BC
D
with
A
B
∥
C
D
AB \parallel CD
A
B
∥
C
D
and
A
B
>
C
D
AB> CD
A
B
>
C
D
. The projection from
D
D
D
on
A
B
AB
A
B
is
E
E
E
. The midpoint of the diagonal
B
D
BD
B
D
is
M
M
M
. Prove that
E
M
EM
EM
is parallel to
A
C
AC
A
C
.(Karl Czakler)
2
1
Hide problems
5-digit numbers with product of digits 900
How many positive five-digit integers are there that have the product of their five digits equal to
900
900
900
?(Karl Czakler)
1
1
Hide problems
Find all pairs (a, b)
Let
a
a
a
be a real number and
b
b
b
a real number with
b
≠
−
1
b\neq-1
b
=
−
1
and
b
≠
0.
b\neq0.
b
=
0.
Find all pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
such that
(
1
+
a
)
2
1
+
b
≤
1
+
a
2
b
.
\frac{(1 + a)^2 }{1 + b}\leq 1 + \frac{a^2}{b}.
1
+
b
(
1
+
a
)
2
≤
1
+
b
a
2
.
For which pairs (a, b) does equality apply? (Walther Janous)