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Problems
Contests
International Contests
Cono Sur Olympiad
1993 Cono Sur Olympiad
1993 Cono Sur Olympiad
Part of
Cono Sur Olympiad
Subcontests
(3)
3
2
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A set b
Find the number of elements that a set
B
B
B
can have, contained in
(
1
,
2
,
.
.
.
,
n
)
(1, 2, ... , n)
(
1
,
2
,
...
,
n
)
, according to the following property: For any elements
a
a
a
and
b
b
b
on
B
B
B
(
a
≠
b
a \ne b
a
=
b
),
(
a
−
b
)
∤
(
a
+
b
)
(a-b) \not| (a+b)
(
a
−
b
)
∣
(
a
+
b
)
.
Proof with points
Prove that, given a positive integrer
n
n
n
, there exists a positive integrer
k
n
k_n
k
n
with the following property: Given any
k
n
k_n
k
n
points in the space,
4
4
4
by
4
4
4
non-coplanar, and associated integrer numbers between
1
1
1
and
n
n
n
to each sharp edge that meets
2
2
2
of this points, there's necessairly a triangle determined by
3
3
3
of them, whose sharp edges have associated the same number.
2
2
Hide problems
Proof in a circle
Consider a circle with centre
O
O
O
, and
3
3
3
points on it,
A
,
B
A,B
A
,
B
and
C
C
C
, such that
∠
A
O
B
<
∠
B
O
C
\angle {AOB}< \angle {BOC}
∠
A
OB
<
∠
BOC
. Let
D
D
D
be the midpoint on the arc
A
C
AC
A
C
that contains the point
B
B
B
. Consider a point
K
K
K
on
B
C
BC
BC
such that
D
K
⊥
B
C
DK \perp BC
DK
⊥
BC
. Prove that
A
B
+
B
K
=
K
C
AB+BK=KC
A
B
+
B
K
=
K
C
.
Prove succession
Prove that there exists a succession
a
1
,
a
2
,
.
.
.
,
a
k
,
.
.
.
a_1, a_2, ... , a_k, ...
a
1
,
a
2
,
...
,
a
k
,
...
, where each
a
i
a_i
a
i
is a digit (
a
i
∈
(
0
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
)
a_i \in (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
a
i
∈
(
0
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
)
) and
a
0
=
6
a_0=6
a
0
=
6
, such that, for each positive integrer
n
n
n
, the number
x
n
=
a
0
+
10
a
1
+
100
a
2
+
.
.
.
+
1
0
n
−
1
a
n
−
1
x_n=a_0+10a_1+100a_2+...+10^{n-1}a_{n-1}
x
n
=
a
0
+
10
a
1
+
100
a
2
+
...
+
1
0
n
−
1
a
n
−
1
verify that
x
n
2
−
x
n
x_n^2-x_n
x
n
2
−
x
n
is divisible by
1
0
n
10^n
1
0
n
.
1
2
Hide problems
On a table there is a pile with 2001 tokens
On a table there is a pile with
T
T
T
tokens which incrementally shall be converted into piles with three tokens each. Each step is constituted of selecting one pile removing one of its tokens. And then the remaining pile is separated into two piles. Is there a sequence of steps that can accomplish this process? a.) T \equal{} 1000 (Cono Sur) b.) T \equal{} 2001 (BWM)
Chess board
On a chess board (
8
∗
8
8*8
8
∗
8
) there are written the numbers
1
1
1
to
64
64
64
: on the first line, from left to right, there are the numbers
1
,
2
,
3
,
.
.
.
,
8
1, 2, 3, ... , 8
1
,
2
,
3
,
...
,
8
; on the second line, from left to right, there are the numbers
9
,
10
,
11
,
.
.
.
,
16
9, 10, 11, ... , 16
9
,
10
,
11
,
...
,
16
;etc. The \"+\" and \"-\" signs are put to each number such that, in each line and in each column, there are
4
4
4
\"+\" signs and
4
4
4
\"-\" signs. Then, the
64
64
64
numbers are added. Find all the possible values of this sum.