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Problems
Contests
National and Regional Contests
Peru Contests
Peru Cono Sur TST
2006 Team Selection Test For CSMO
2006 Team Selection Test For CSMO
Part of
Peru Cono Sur TST
Subcontests
(4)
4
1
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Xvii cono sur - peru tst 2006.
All the squares of a board of
(
n
+
1
)
×
(
n
−
1
)
(n+1)\times(n-1)
(
n
+
1
)
×
(
n
−
1
)
squares are painted with three colors such that, for any two different columns and any two different rows, the 4 squares in their intersections they don't have all the same color. Find the greatest possible value of
n
n
n
.
3
1
Hide problems
Xvii cono sur - peru tst 2006.
The set
M
=
{
1
;
2
;
3
;
…
;
29
;
30
}
M= \{1;2;3;\ldots ; 29;30\}
M
=
{
1
;
2
;
3
;
…
;
29
;
30
}
is divided in
k
k
k
subsets such that if
a
+
b
=
n
2
,
(
a
,
b
∈
M
,
a
≠
b
,
n
a+b=n^2, (a,b \in M, a\neq b, n
a
+
b
=
n
2
,
(
a
,
b
∈
M
,
a
=
b
,
n
is an integer number
)
)
)
, then
a
a
a
and
b
b
b
belong different subsets. Determine the minimum value of
k
k
k
.
2
1
Hide problems
Xvii cono sur - peru tst 2006.
Let
A
A
1
AA_1
A
A
1
and
B
B
1
BB_1
B
B
1
be the altitudes of an acute-angled, non-isosceles triangle
A
B
C
ABC
A
BC
. Also, let
A
0
A_0
A
0
and
B
0
B_0
B
0
be the midpoints of its sides
B
C
BC
BC
and
C
A
CA
C
A
, respectively. The line
A
1
B
1
A_1B_1
A
1
B
1
intersects the line
A
0
B
0
A_0B_0
A
0
B
0
at a point
C
′
C'
C
′
. Prove that the line
C
C
′
CC'
C
C
′
is perpendicular to the Euler line of the triangle
A
B
C
ABC
A
BC
(this is the line that joins the orthocenter and the circumcenter of the triangle
A
B
C
ABC
A
BC
).
1
1
Hide problems
Xvii cono sur - peru tst 2006.
Find all the pairs of positive numbers such that the last digit of their sum is 3, their difference is a primer number and their product is a perfect square.