MathDB
Problems
Contests
International Contests
Rioplatense Mathematical Olympiad, Level 3
1992 Rioplatense Mathematical Olympiad, Level 3
1992 Rioplatense Mathematical Olympiad, Level 3
Part of
Rioplatense Mathematical Olympiad, Level 3
Subcontests
(6)
1
1
Hide problems
\log_{f(1)}f(x) is perfect square if f(x + y)f(x-y) = (f(x)f(y))^2
Let
f
:
Z
→
N
−
{
0
}
f:Z \to N -\{0\}
f
:
Z
→
N
−
{
0
}
such that:
f
(
x
+
y
)
f
(
x
−
y
)
=
(
f
(
x
)
f
(
y
)
)
2
f(x + y)f(x-y) = (f(x)f(y))^2
f
(
x
+
y
)
f
(
x
−
y
)
=
(
f
(
x
)
f
(
y
)
)
2
and
f
(
1
)
≠
1
f(1)\ne 1
f
(
1
)
=
1
.Provethat
log
f
(
1
)
f
(
z
)
\log_{f(1)}f(z)
lo
g
f
(
1
)
f
(
z
)
is a perfect square for every integer
z
z
z
.
2
1
Hide problems
2^n= a^2 + b^2 + c^2 + d^2
Determine the integers
0
≤
a
≤
b
≤
c
≤
d
0 \le a \le b \le c \le d
0
≤
a
≤
b
≤
c
≤
d
such that:
2
n
=
a
2
+
b
2
+
c
2
+
d
2
.
2^n= a^2 + b^2 + c^2 + d^2.
2
n
=
a
2
+
b
2
+
c
2
+
d
2
.
6
1
Hide problems
sum of its positive divisors is greater than its double
Definition: A natural number is abundant if the sum of its positive divisors is greater than its double.Find an odd abundant number and prove that there are infinitely many odd abundant numbers.
4
1
Hide problems
100 states that are in dispute on planet Mars
On the planet Mars there are
100
100
100
states that are in dispute. To achieve a peace situation, blocs must be formed that meet the following two conditions: (1) Each block must have at most
50
50
50
states. (2) Every pair of states must be together in at least one block. Find the minimum number of blocks that must be formed.
3
1
Hide problems
equal radii of circumcircles
Let
D
D
D
be the center of the circumcircle of the acute triangle
A
B
C
ABC
A
BC
. If the circumcircle of triangle
A
D
B
ADB
A
D
B
intersects
A
C
AC
A
C
(or its extension) at
M
M
M
and also
B
C
BC
BC
(or its extension) at
N
N
N
, show that the radii of the circumcircles of
△
A
D
B
\triangle ADB
△
A
D
B
and
△
M
N
C
\triangle MNC
△
MNC
are equal.
5
1
Hide problems
Locus of centers.
Let
A
B
C
ABC
A
BC
be an acute triangle. Find the locus of the centers of the rectangles which have their vertices on the sides of
A
B
C
ABC
A
BC
.