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Problems
Contests
International Contests
Cono Sur Olympiad
1989 Cono Sur Olympiad
1989 Cono Sur Olympiad
Part of
Cono Sur Olympiad
Subcontests
(3)
3
2
Hide problems
F(1998)
A number
p
p
p
is
p
e
r
f
e
c
t
perfect
p
er
f
ec
t
if the sum of its divisors, except
p
p
p
is
p
p
p
. Let
f
f
f
be a function such that:
f
(
n
)
=
0
f(n)=0
f
(
n
)
=
0
, if n is perfect
f
(
n
)
=
0
f(n)=0
f
(
n
)
=
0
, if the last digit of n is 4
f
(
a
.
b
)
=
f
(
a
)
+
f
(
b
)
f(a.b)=f(a)+f(b)
f
(
a
.
b
)
=
f
(
a
)
+
f
(
b
)
Find
f
(
1998
)
f(1998)
f
(
1998
)
Cuboid
Show that reducing the dimensions of a cuboid we can't get another cuboid with half the volume and half the surface.
2
2
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Finf the sum
Find the sum
1
+
11
+
111
+
⋯
+
111
…
111
⏟
n
digits
.
1+11+111+\cdots+\underbrace{111\ldots111}_{n\text{ digits}}.
1
+
11
+
111
+
⋯
+
n
digits
111
…
111
.
Constant sum
Let
A
B
C
D
ABCD
A
BC
D
be a square with diagonals
A
C
AC
A
C
and
B
D
BD
B
D
, and
P
P
P
a point in one of the sides of the square. Show that the sum of the distances from P to the diagonals is constant.
1
2
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Find x in Triangle
Two isosceles triangles with sidelengths
x
,
x
,
a
x,x,a
x
,
x
,
a
and
x
,
x
,
b
x,x,b
x
,
x
,
b
(
a
≠
b
a \neq b
a
=
b
) have equal areas. Find
x
x
x
.
N is a square
Let
n
n
n
be square with 4 digits, such that all its digits are less than 6. If we add 1 to each digit the resulting number is another square. Find
n
n
n