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Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
2021 Argentina National Olympiad
2021 Argentina National Olympiad
Part of
Argentina National Olympiad
Subcontests
(6)
6
3
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Argentina MO 2021 National Level 1 P6
Milly chooses a positive integer
n
n
n
and then Uriel colors each integer between
1
1
1
and
n
n
n
inclusive red or blue. Then Milly chooses four numbers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
of the same color (there may be repeated numbers). If
a
+
b
+
c
=
d
a+b+c= d
a
+
b
+
c
=
d
then Milly wins. Determine the smallest
n
n
n
Milly can choose to ensure victory, no matter how Uriel colors.
Argentina MO 2021 National Level 2 P6
Decide if it is possible to choose
330
330
330
points in the plane so that among all the distances that are formed between two of them there are at least
1700
1700
1700
that are equal.
Argentina MO 2021 National Level 3 P6
We say that a positive integer
k
k
k
is tricubic if there are three positive integers
a
,
b
,
c
,
a, b, c,
a
,
b
,
c
,
not necessarily different, such that
k
=
a
3
+
b
3
+
c
3
.
k=a^3+b^3+c^3.
k
=
a
3
+
b
3
+
c
3
.
a) Prove that there are infinitely many positive integers
n
n
n
that satisfy the following condition: exactly one of the three numbers
n
,
n
+
2
n, n+2
n
,
n
+
2
and
n
+
28
n+28
n
+
28
is tricubic. b) Prove that there are infinitely many positive integers
n
n
n
that satisfy the following condition: exactly two of the three numbers
n
,
n
+
2
n, n+2
n
,
n
+
2
and
n
+
28
n+28
n
+
28
are tricubic. c) Prove that there are infinitely many positive integers
n
n
n
that satisfy the following condition: the three numbers
n
,
n
+
2
n, n+2
n
,
n
+
2
and
n
+
28
n+28
n
+
28
are tricubic.
5
3
Hide problems
Argentina MO 2021 National Level 1 P5
Mica wrote a list of numbers using the following procedure. The first number is
1
1
1
, and then, at each step, he wrote the result of adding the previous number plus
3
3
3
. The first numbers on Mica's list are
1
,
4
,
7
,
10
,
13
,
16
,
…
.
1, 4, 7, 10, 13, 16,\dots.
1
,
4
,
7
,
10
,
13
,
16
,
…
.
Next, Facu underlined all the numbers in Mica's list that are greater than
10
10
10
and less than
100000
,
100000,
100000
,
and that have all their digits the same. What are the numbers that Facu underlined?
Argentina MO 2021 National Level 2 P5
Determine all positive integers
n
n
n
such that
n
⋅
2
n
−
1
+
1
n\cdot 2^{n-1}+1
n
⋅
2
n
−
1
+
1
is a perfect square.
Argentina MO 2021 National Level 3 P5
The sequence
a
n
(
n
≥
1
)
a_n (n\geq 1)
a
n
(
n
≥
1
)
of natural numbers is defined as
a
n
+
1
=
a
n
+
b
n
,
a_{n+1}=a_n+b_n,
a
n
+
1
=
a
n
+
b
n
,
where
b
n
b_n
b
n
is the number that has the same digits as
a
n
a_n
a
n
but in the opposite order (
b
n
b_n
b
n
can start with
0
0
0
). For example, if
a
1
=
180
,
a_1=180,
a
1
=
180
,
then
a
2
=
261
,
a
3
=
423.
a_2=261, a_3=423.
a
2
=
261
,
a
3
=
423.
a) Decide if
a
1
a_1
a
1
can be chosen so that
a
7
a_7
a
7
is prime. b) Decide if
a
1
a_1
a
1
can be chosen so that
a
5
a_5
a
5
is prime.
4
3
Hide problems
Argentina MO 2021 National Level 1 P4
Martu wants to build a set of cards with the following properties:• Each card has a positive integer on it. • The number on each card is equal to one of
5
5
5
possible numbers. • If any two cards are taken and added together, it is always possible to find two other cards in the set such that the sum is the same.Determine the fewest number of cards Martu's set can have and give an example for that number.
Argentina MO 2021 National Level 2 P4
The sum of several positive integers, not necessarily different, all of them less than or equal to
10
10
10
, is equal to
S
S
S
. We want to distribute all these numbers into two groups such that the sum of the numbers in each group is less than or equal to
80.
80.
80.
Determine all values of
S
S
S
for which this is possible.
Argentina MO 2021 National Level 3 P4
Find the real numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
such that,
1
x
+
1
y
+
z
=
1
2
,
1
y
+
1
z
+
x
=
1
3
,
1
z
+
1
x
+
y
=
1
4
.
\frac{1}{x}+\frac{1}{y+z}=\frac{1}{2}, \frac{1}{y}+\frac{1}{z+x}=\frac{1}{3}, \frac{1}{z}+\frac{1}{x+y}=\frac{1}{4}.
x
1
+
y
+
z
1
=
2
1
,
y
1
+
z
+
x
1
=
3
1
,
z
1
+
x
+
y
1
=
4
1
.
3
3
Hide problems
Argentina MO 2021 National Level 1 P3
Let
A
B
C
ABC
A
BC
be an isosceles right triangle at
A
A
A
with
A
B
=
A
C
AB=AC
A
B
=
A
C
. Let
M
M
M
and
N
N
N
be on side
B
C
BC
BC
, with
M
M
M
between
B
B
B
and
N
,
N,
N
,
such that
B
M
2
+
N
C
2
=
M
N
2
.
BM^2+ NC^2= MN^2.
B
M
2
+
N
C
2
=
M
N
2
.
Determine the measure of the angle
∠
M
A
N
.
\angle MAN.
∠
M
A
N
.
Argentina MO 2021 National Level 2 P3
A circle is divided into
2
n
2n
2
n
equal arcs by
2
n
2n
2
n
points. Find all
n
>
1
n>1
n
>
1
such that these points can be joined in pairs using
n
n
n
segments, all of different lengths and such that each point is the endpoint of exactly one segment.
Argentina MO 2021 National Level 3 P3
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral inscribed in a circle such that
∠
A
B
C
=
6
0
∘
.
\angle ABC=60^{\circ}.
∠
A
BC
=
6
0
∘
.
a) Prove that if
B
C
=
C
D
BC=CD
BC
=
C
D
then
A
B
=
C
D
+
D
A
.
AB= CD+DA.
A
B
=
C
D
+
D
A
.
b) Is it true that if
A
B
=
C
D
+
D
A
AB= CD+DA
A
B
=
C
D
+
D
A
then
B
C
=
C
D
BC=CD
BC
=
C
D
?
2
3
Hide problems
Argentina MO 2021 National Level 1 P2
On each OMA lottery ticket there is a
9
9
9
-digit number that only uses the digits
1
,
2
1, 2
1
,
2
and
3
3
3
(not necessarily all three). Each ticket has one of the three colors red, blue or green. It is known that if two banknotes do not match in any of the
9
9
9
figures, then they are of different colors. Bill
122222222
122222222
122222222
is red,
222222222
222222222
222222222
is green, what color is bill
123123123
123123123
123123123
?
Argentina MO 2021 National Level 2 P2
In a semicircle with center
O
O
O
, let
C
C
C
be a point on the diameter
A
B
AB
A
B
different from
A
,
B
A, B
A
,
B
and
O
.
O.
O
.
Draw through
C
C
C
two rays such that the angles that these rays form with the diameter
A
B
AB
A
B
are equal and that they intersect at the semicircle at
D
D
D
and at
E
E
E
. The line perpendicular to
C
D
CD
C
D
through
D
D
D
intersects the semicircle at
K
.
K.
K
.
Prove that if
D
≠
E
,
D\neq E,
D
=
E
,
then
K
E
KE
K
E
is parallel to
A
B
.
AB.
A
B
.
Argentina MO 2021 National Level 3 P2
Let
m
m
m
be a positive integer for which there exists a positive integer
n
n
n
such that the multiplication
m
n
mn
mn
is a perfect square and
m
−
n
m- n
m
−
n
is prime. Find all
m
m
m
for
1000
≤
m
≤
2021.
1000\leq m \leq 2021.
1000
≤
m
≤
2021.
1
3
Hide problems
Argentina MO 2021 National Level 1 P1
Determine all pairs of prime numbers
p
p
p
and
q
q
q
greater than
1
1
1
and less than
100
100
100
, such that the following five numbers:
p
+
6
,
p
+
10
,
q
+
4
,
q
+
10
,
p
+
q
+
1
,
p+6,p+10,q+4,q+10,p+q+1,
p
+
6
,
p
+
10
,
q
+
4
,
q
+
10
,
p
+
q
+
1
,
are all prime numbers.
Argentina MO 2021 National Level 2 P1
You have two blackboards
A
A
A
and
B
B
B
. You have to write on them some of the integers greater than or equal to
2
2
2
and less than or equal to
20
20
20
in such a way that each number on blackboard
A
A
A
is co-prime with each number on blackboard
B
.
B.
B
.
Determine the maximum possible value of multiplying the number of numbers written in
A
A
A
by the number of numbers written in
B
B
B
.
Argentina MO 2021 National Level 3 P1
An infinite sequence of digits
1
1
1
and
2
2
2
is determined by the following two properties:i) The sequence is built by writing, in some order, blocks
12
12
12
and blocks
112.
112.
112.
ii) If each block
12
12
12
is replaced by
1
1
1
and each block
112
112
112
by
2
2
2
, the same sequence is again obtained.In which position is the hundredth digit
1
1
1
? What is the thousandth digit of the sequence?