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Chile Classification NMO Juniors
2019 Chile Classification NMO Juniors
2019 Chile Classification NMO Juniors
Part of
Chile Classification NMO Juniors
Subcontests
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2019 Chile Classification / Qualifying NMO Juniors XXXI
p1. Consider the sequence of positive integers
2
,
3
,
5
,
6
,
7
,
8
,
10
,
11...
2, 3, 5, 6, 7, 8, 10, 11 ...
2
,
3
,
5
,
6
,
7
,
8
,
10
,
11...
. which are not perfect squares. Calculate the
2019
2019
2019
-th term of the sequence. p2. In a triangle
A
B
C
ABC
A
BC
, let
D
D
D
be the midpoint of side
B
C
BC
BC
and
E
E
E
be the midpoint of segment
A
D
AD
A
D
. Lines
A
C
AC
A
C
and
B
E
BE
BE
intersect at
F
F
F
. Show that
3
A
F
=
A
C
3AF = AC
3
A
F
=
A
C
. p3. Find all positive integers
n
n
n
such that
n
!
+
2019
n! + 2019
n
!
+
2019
is a square perfect. p4. In a party, there is a certain group of people, none of whom has more than
3
3
3
friends in this. However, if two people are not friends at least they have a friend in this party. What is the largest possible number of people in the party?