MathDB
Problems
Contests
International Contests
Cono Sur Olympiad
2008 Cono Sur Olympiad
2008 Cono Sur Olympiad
Part of
Cono Sur Olympiad
Subcontests
(6)
6
1
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Palindromes
A palindrome is a number that is the same when its digits are reversed. Find all numbers that have at least one multiple that is a palindrome.
5
1
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Isosceles triangle with inscribed semicircle
Let
A
B
C
ABC
A
BC
be an isosceles triangle with base
A
B
AB
A
B
. A semicircle
Γ
\Gamma
Γ
is constructed with its center on the segment AB and which is tangent to the two legs,
A
C
AC
A
C
and
B
C
BC
BC
. Consider a line tangent to
Γ
\Gamma
Γ
which cuts the segments
A
C
AC
A
C
and
B
C
BC
BC
at
D
D
D
and
E
E
E
, respectively. The line perpendicular to
A
C
AC
A
C
at
D
D
D
and the line perpendicular to
B
C
BC
BC
at
E
E
E
intersect each other at
P
P
P
. Let
Q
Q
Q
be the foot of the perpendicular from
P
P
P
to
A
B
AB
A
B
. Show that
P
Q
C
P
=
1
2
A
B
A
C
\frac{PQ}{CP}=\frac{1}{2}\frac{AB}{AC}
CP
PQ
=
2
1
A
C
A
B
.
4
1
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Coloring cells
What is the largest number of cells that can be colored in a
7
×
7
7\times7
7
×
7
table in such a way that any
2
×
2
2\times2
2
×
2
subtable has at most 2 colored cells?
3
1
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Two friends must guess the other's number
Two friends
A
A
A
and
B
B
B
must solve the following puzzle. Each of them receives a number from the set
{
1
,
2
,
…
,
250
}
\{1,2,…,250\}
{
1
,
2
,
…
,
250
}
, but they don’t see the number that the other received. The objective of each friend is to discover the other friend’s number. The procedure is as follows: each friend, by turns, announces various not necessarily distinct positive integers: first
A
A
A
says a number, then
B
B
B
says one,
A
A
A
says a number again, etc., in such a way that the sum of all the numbers said is
20
20
20
. Demonstrate that there exists a strategy that
A
A
A
and
B
B
B
have previously agreed on such that they can reach the objective, no matter which number each one received at the beginning of the puzzle.
2
1
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Point on the interior of a triangle
Let
P
P
P
be a point in the interior of triangle
A
B
C
ABC
A
BC
. Let
X
X
X
,
Y
Y
Y
, and
Z
Z
Z
be points on sides
B
C
BC
BC
,
A
C
AC
A
C
, and
A
B
AB
A
B
respectively, such that
<
P
X
C
=
<
P
Y
A
=
<
P
Z
B
<PXC=<PYA=<PZB
<
PXC
=<
P
Y
A
=<
PZB
. Let
U
U
U
,
V
V
V
, and
W
W
W
be points on sides
B
C
BC
BC
,
A
C
AC
A
C
, and
A
B
AB
A
B
, respectively, or on their extensions if necessary, with
X
X
X
in between
B
B
B
and
U
U
U
,
Y
Y
Y
in between
C
C
C
and
V
V
V
, and
Z
Z
Z
in between
A
A
A
and
W
W
W
, such that
P
U
=
2
P
X
PU=2PX
P
U
=
2
PX
,
P
V
=
2
P
Y
PV=2PY
P
V
=
2
P
Y
, and
P
W
=
2
P
Z
PW=2PZ
P
W
=
2
PZ
. If the area of triangle
X
Y
Z
XYZ
X
Y
Z
is
1
1
1
, find the area of triangle
U
V
W
UVW
U
VW
.
1
1
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Reversing the digits
We define
I
(
n
)
I(n)
I
(
n
)
as the result when the digits of
n
n
n
are reversed. For example,
I
(
123
)
=
321
I(123)=321
I
(
123
)
=
321
,
I
(
2008
)
=
8002
I(2008)=8002
I
(
2008
)
=
8002
. Find all integers
n
n
n
,
1
≤
n
≤
10000
1\leq{n}\leq10000
1
≤
n
≤
10000
for which
I
(
n
)
=
⌈
n
2
⌉
I(n)=\lceil{\frac{n}{2}}\rceil
I
(
n
)
=
⌈
2
n
⌉
. Note:
⌈
x
⌉
\lceil{x}\rceil
⌈
x
⌉
denotes the smallest integer greater than or equal to
x
x
x
. For example,
⌈
2.1
⌉
=
3
\lceil{2.1}\rceil=3
⌈
2.1
⌉
=
3
,
⌈
3.9
⌉
=
4
\lceil{3.9}\rceil=4
⌈
3.9
⌉
=
4
,
⌈
7
⌉
=
7
\lceil{7}\rceil=7
⌈
7
⌉
=
7
.