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Problems
Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
2012 Chile National Olympiad
2012 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(4)
3
1
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sequence of consecutive days where made exactly 14 friends in FB
A person enters the social network facebook. He befriends at least one person a day for the first
30
30
30
days. At the end of those
30
30
30
days, it has been exactly
45
45
45
friends. Prove that there is a sequence of consecutive days where made exactly
14
14
14
friends.
2
1
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sum 1/a_i <=80
Let
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
be all positive integers with
2012
2012
2012
digits or less, none of which is a
9
9
9
. Prove that
1
a
1
+
1
a
2
+
.
.
.
+
1
a
n
≤
80.
\frac{1}{a_1}+\frac{1}{a_2}+ ... +\frac{1}{a_{n}}\le 80.
a
1
1
+
a
2
1
+
...
+
a
n
1
≤
80.
1
1
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min no of movements in chess for horse
What is the minimum number of movements that a horse must carry out on chess, on an
8
×
8
8\times 8
8
×
8
board, to reach the upper right square starting at the lower left? Remember that the horse moves in the usual
L
L
L
-shaped manner.
4
1
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ratio in an isosceles triangle wanted, angles data given Chile 2008 L2 P2
Consider an isosceles triangle
A
B
C
ABC
A
BC
, where
A
B
=
A
C
AB = AC
A
B
=
A
C
.
D
D
D
is a point on the
A
C
AC
A
C
side and
P
P
P
a point on the segment
B
D
BD
B
D
so that the angle
∠
A
P
C
=
9
0
o
\angle APC = 90^o
∠
A
PC
=
9
0
o
and
∠
A
B
P
=
∠
B
C
P
\angle ABP = \angle BCP
∠
A
BP
=
∠
BCP
. Determine the ratio
A
D
:
D
C
AD: DC
A
D
:
D
C
.