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National and Regional Contests
Peru Contests
Peru MO (ONEM)
2021 Peru MO (ONEM)
2021 Peru MO (ONEM)
Part of
Peru MO (ONEM)
Subcontests
(4)
4
1
Hide problems
n-sided poly with some operations
Let
n
≥
3
n\geq 3
n
≥
3
be a positive integer and a circle
ω
\omega
ω
is given. A regular polygon(with
n
n
n
sides)
P
P
P
is drawn and your vertices are in the circle
ω
\omega
ω
and these vertices are red. One operation is choose three red points
A
,
B
,
C
A,B,C
A
,
B
,
C
, such that
A
B
=
B
C
AB=BC
A
B
=
BC
and delete the point
B
B
B
. Prove that one can do some operations, such that only two red points remain in the circle.
2
1
Hide problems
Ana can or not can avoid
The numbers
1
1
1
to
25
25
25
will be written in a table
5
×
5
5 \times 5
5
×
5
. First, Ana chooses
k
k
k
of these numbers(
1
1
1
to
25
25
25
), and write in some cells. Then, Enrique writes the remaining numbers with the following goal: The product of the numbers in some column/row is a perfect square. a) Prove that if
k
=
5
k=5
k
=
5
, Ana can avoid Enrique to reach his goal. b) Prove that if
k
=
4
k=4
k
=
4
, Enrique can reach his goal.
3
1
Hide problems
3 incircles with radius r
Let
M
,
N
,
P
M,N,P
M
,
N
,
P
be points in the sides
B
C
,
A
C
,
A
B
BC,AC,AB
BC
,
A
C
,
A
B
of
△
A
B
C
\triangle ABC
△
A
BC
respectively. The quadrilateral
M
C
N
P
MCNP
MCNP
has an incircle of radius
r
r
r
, if the incircles of
△
B
P
M
\triangle BPM
△
BPM
and
△
A
N
P
\triangle ANP
△
A
NP
also have the radius
r
r
r
. Prove that
A
P
⋅
M
P
=
B
P
⋅
N
P
AP\cdot MP=BP\cdot NP
A
P
⋅
MP
=
BP
⋅
NP
1
1
Hide problems
6/8 rational(s) is a product
a) Determine if it's possible write
6
6
6
positive rational numbers, pairwise distinct, in a circle such that each one is equal to the product of your neighbor numbers. b) Determine if it's possible write
8
8
8
positive rational numbers, pairwise distinct, in a circle such that each one is equal to the product of your neighbor numbers.