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Chile Classification NMO
1994 Chile Classification NMO
1994 Chile Classification NMO
Part of
Chile Classification NMO
Subcontests
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1994 Chile Classification / Qualifying NMO VI
p1. The Fernández family, made up of father, mother and their three children: Pablo, Claudio and Carolina, subscribed for the year 1994 to the city newspaper. Subscription started on Saturday January
1
1
1
st. The family agreed that each member would get up early to pick up the newspaper, alternately starting with paper and in the order indicated. How many days on Sunday it will be time for each family member? p2. Given three lines in the plane, concurrent at a point
O
O
O
, consider three consecutive angles that are formed between them (obviously they add up to
18
0
o
180^o
18
0
o
). Take a point
P
P
P
in the plane, out of those lines, and let
A
A
A
,
B
B
B
,
C
C
C
be their orthogonal projections on the three lines. Prove that the
△
A
B
C
\vartriangle ABC
△
A
BC
has the same angles as those that the lines form to each other. p3. Consider ten positive, not necessarily distinct, integers that add up to
95
95
95
. Find the smallest possible value of the sum of its squares. p4. In
△
A
B
C
\vartriangle ABC
△
A
BC
, with right angle at
C
C
C
, the altitude
h
c
=
C
D
h_c = CD
h
c
=
C
D
is drawn. Let
r
,
r
1
,
r
2
r,r_1,r_2
r
,
r
1
,
r
2
be the inradii, of the triangles
△
A
B
C
\vartriangle ABC
△
A
BC
,
△
A
D
C
\vartriangle ADC
△
A
D
C
,
△
B
C
D
\vartriangle BCD
△
BC
D
, respectively. Prove that
r
+
r
1
+
r
2
=
h
c
r + r_1 + r_2 = h_c
r
+
r
1
+
r
2
=
h
c
p5. Consider
8
n
−
4
8n- 4
8
n
−
4
points on a grid, arranged in the shape of a cross, as shown in the figure. Determine the maximum number of squares with vertices at the
8
n
−
4
8n-4
8
n
−
4
points. In the figure, they have outlined some cases. https://cdn.artofproblemsolving.com/attachments/c/d/e7090c54eaf2bd3749cdee2ff77172ada7866a.jpg p6. Let
A
A
A
and
D
D
D
be two opposite vertices of a regular hexagon. A frog begins to jump to starting at
A
A
A
. From any vertex in the hexagon, other than
D
D
D
, the frog can jump to any vertex adjacent, but as soon as it reaches
D
D
D
, it stops jumping (stops there). Let
P
(
n
)
P (n)
P
(
n
)
be the number of different paths, with exactly
n
n
n
jumps, from
A
A
A
to
D
D
D
. Prove that:
∙
\bullet
∙
P
(
n
)
=
0
P (n) = 0
P
(
n
)
=
0
if
n
n
n
is even
∙
\bullet
∙
P
(
2
n
+
1
)
=
2
⋅
3
n
−
1
P (2n + 1) = 2\cdot 3^{n- 1}
P
(
2
n
+
1
)
=
2
⋅
3
n
−
1
p7. Consider the product of all positive integers multiples of
6
6
6
, which are less than thousand. Find the number of zeros this product ends with.