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Problems
Contests
International Contests
Cono Sur Olympiad
1994 Cono Sur Olympiad
1994 Cono Sur Olympiad
Part of
Cono Sur Olympiad
Subcontests
(3)
3
2
Hide problems
Find minimun value
Let
p
p
p
be a positive real number given. Find the minimun vale of
x
3
+
y
3
x^3+y^3
x
3
+
y
3
, knowing that
x
x
x
and
y
y
y
are positive real numbers such that
x
y
(
x
+
y
)
=
p
xy(x+y)=p
x
y
(
x
+
y
)
=
p
.
Prove area
Consider a
△
A
B
C
\triangle {ABC}
△
A
BC
, with
A
C
⊥
B
C
AC \perp BC
A
C
⊥
BC
. Consider a point
D
D
D
on
A
B
AB
A
B
such that
C
D
=
k
CD=k
C
D
=
k
, and the radius of the inscribe circles on
△
A
D
C
\triangle {ADC}
△
A
D
C
and
△
C
D
B
\triangle {CDB}
△
C
D
B
are equals. Prove that the area of
△
A
B
C
\triangle {ABC}
△
A
BC
is equal to
k
2
k^2
k
2
.
1
2
Hide problems
Not a perfect square
The positive integrer number
n
n
n
has
1994
1994
1994
digits.
14
14
14
of its digits are
0
0
0
's and the number of times that the other digits:
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
1, 2, 3, 4, 5, 6, 7, 8, 9
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
appear are in proportion
1
:
2
:
3
:
4
:
5
:
6
:
7
:
8
:
9
1: 2: 3: 4: 5: 6: 7: 8: 9
1
:
2
:
3
:
4
:
5
:
6
:
7
:
8
:
9
, respectively. Prove that
n
n
n
is not a perfect square.
Can she always win?
Pedro and Cecilia play the following game: Pedro chooses a positive integer number
a
a
a
and Cecilia wins if she finds a positive integrer number
b
b
b
, prime with
a
a
a
, such that, in the factorization of
a
3
+
b
3
a^3+b^3
a
3
+
b
3
will appear three different prime numbers. Prove that Cecilia can always win.
2
2
Hide problems
Proof in a circle
Consider a circle
C
C
C
with diameter
A
B
=
1
AB=1
A
B
=
1
. A point
P
0
P_0
P
0
is chosen on
C
C
C
,
P
0
≠
A
P_0 \ne A
P
0
=
A
, and starting in
P
0
P_0
P
0
a sequence of points
P
1
,
P
2
,
…
,
P
n
,
…
P_1, P_2, \dots, P_n, \dots
P
1
,
P
2
,
…
,
P
n
,
…
is constructed on
C
C
C
, in the following way:
Q
n
Q_n
Q
n
is the symmetrical point of
A
A
A
with respect of
P
n
P_n
P
n
and the straight line that joins
B
B
B
and
Q
n
Q_n
Q
n
cuts
C
C
C
at
B
B
B
and
P
n
+
1
P_{n+1}
P
n
+
1
(not necessary different). Prove that it is possible to choose
P
0
P_0
P
0
such that: i
∠
P
0
A
B
<
1
\angle {P_0AB} < 1
∠
P
0
A
B
<
1
. ii In the sequence that starts with
P
0
P_0
P
0
there are
2
2
2
points,
P
k
P_k
P
k
and
P
j
P_j
P
j
, such that
△
A
P
k
P
j
\triangle {AP_kP_j}
△
A
P
k
P
j
is equilateral.
Quadratic eqution
Solve the following equation in integers with gcd (x, y) = 1
x
2
+
y
2
=
2
z
2
x^2 + y^2 = 2 z^2
x
2
+
y
2
=
2
z
2