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Chile Junior Math Olympiad
2019 Chile Junior Math Olympiad
2019 Chile Junior Math Olympiad
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Chile Junior Math Olympiad
Subcontests
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2019 Chile NMO Juniors XXXI
p1. In how many ways can
2019
2019
2019
be written as the sum of consecutive positive integers? p2. On a table there are
2019
2019
2019
chips. Two players take turns drawing chips. The first player to play can draw any odd number of chips between
1
1
1
and
99
99
99
. The other player can draw any even number of chips between
2
2
2
and
100
100
100
. The player who can not continue playing loses. Determine if any of the players has a winning strategy. [url=https://artofproblemsolving.com/community/c4h2454973p20439401]p3. Consider a rectangle
A
B
C
D
ABCD
A
BC
D
with
∣
A
B
∣
>
∣
B
C
∣
| AB | > | BC |
∣
A
B
∣
>
∣
BC
∣
and let
E
E
E
be the midpoint of
C
D
CD
C
D
side.
F
F
F
is chosen in
C
D
CD
C
D
such that
∣
C
F
∣
=
∣
B
C
∣
| CF | = | BC |
∣
CF
∣
=
∣
BC
∣
. Suppose
A
C
⊥
B
E
AC \perp BE
A
C
⊥
BE
. Prove that
∣
A
B
∣
=
∣
B
F
∣
| AB | = | BF |
∣
A
B
∣
=
∣
BF
∣
. p4. Each face of a cube of dimensions
1000
×
1000
×
1000
1000\times 1000\times 1000
1000
×
1000
×
1000
is divided into
100
0
2
1000^2
100
0
2
unit squares of
1
×
1
1\times 1
1
×
1
. Determine the largest number of little squares units that can be marked on the cube in such a way that no pair of them share a side in common.