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Chile Classification NMO Juniors
2015 Chile Classification NMO Juniors
2015 Chile Classification NMO Juniors
Part of
Chile Classification NMO Juniors
Subcontests
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2015 Chile Classification / Qualifying NMO Juniors XXVII
p1. Consider a grid board of
16
×
16
16\times 16
16
×
16
cells. What is the maximum number of lockers that can be painted black so that are there no more than
8
8
8
black boxes in each row, each column and each diagonal of the board? p2. On a blackboard,
4
4
4
points
A
A
A
,
B
B
B
,
C
C
C
and
D
D
D
are still drawn so that they form an convex quadrilateral. Find a point
E
E
E
inside the quadrilateral of so that the sum of the distances to the
4
4
4
vertices is the smallest possible. p3. Find all the prime numbers
p
p
p
such that
2
p
+
p
2
2^p + p^2
2
p
+
p
2
is a prime number. p4. Determine if the number
1
!
+
2
!
+
.
.
.
+
2015
!
1! + 2! + ... + 2015!
1
!
+
2
!
+
...
+
2015
!
is a perfect square. p5. Consider a triangle
△
A
B
C
\vartriangle ABC
△
A
BC
, and a point
P
P
P
inside it. When drawing the lines
A
P
AP
A
P
,
B
P
BP
BP
and
C
P
CP
CP
, the intersection points
D
D
D
,
E
E
E
and
F
F
F
are determined on the sides
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
respectively. The triangle is divided into 6 triangles (
△
A
F
P
\vartriangle AFP
△
A
FP
,
△
F
P
B
\vartriangle FPB
△
FPB
,
△
B
D
P
\vartriangle BDP
△
B
D
P
,
△
D
P
C
\vartriangle DPC
△
D
PC
,
△
C
P
E
\vartriangle CPE
△
CPE
,
△
E
P
A
\vartriangle EPA
△
EP
A
). Show that if
4
4
4
of these triangles have the same area then points
D
D
D
,
E
E
E
,
F
F
F
are the midpoints of the respective sides. p6. Sebastian and Fernando are preparing to play the following game: at a table there are
2015
2015
2015
tokens, which are red on one side and black on the other. Initially the tokens are randomly flipped and played alternately in turns. On each shift, it is allowed remove any non-zero quantity of tokens of the same color or flip any quantity not null of sheets of the same color. Whoever removes the last token wins. If Sebastian plays first, who has a winning strategy?PS. Juniors P3, P5 were also proposed as [url=https://artofproblemsolving.com/community/c4h2693883p23392778]Seniors P1, P4.