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Problems
Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
2022 Argentina National Olympiad
2022 Argentina National Olympiad
Part of
Argentina National Olympiad
Subcontests
(6)
6
1
Hide problems
P(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x+a_0
For every positive integer
n
n
n
, we consider the polynomial of real coefficients, of
2
n
+
1
2n+1
2
n
+
1
terms,
P
(
x
)
=
a
2
n
x
2
n
+
a
2
n
−
1
x
2
n
−
1
+
.
.
.
+
a
1
x
+
a
0
P(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x+a_0
P
(
x
)
=
a
2
n
x
2
n
+
a
2
n
−
1
x
2
n
−
1
+
...
+
a
1
x
+
a
0
where all coefficients are real numbers satisfying
100
≤
a
i
≤
101
100 \le a_i \le 101
100
≤
a
i
≤
101
for
0
≤
i
≤
2
n
0 \le i \le 2n
0
≤
i
≤
2
n
. Find the smallest possible value of
n
n
n
such that the polynomial can have at least one real root.
5
1
Hide problems
x^3+y^3=4(x^2y+xy^2-5)
Find all pairs of positive integers
x
,
y
x,y
x
,
y
such that
x
3
+
y
3
=
4
(
x
2
y
+
x
y
2
−
5
)
.
x^3+y^3=4(x^2y+xy^2-5).
x
3
+
y
3
=
4
(
x
2
y
+
x
y
2
−
5
)
.
4
1
Hide problems
max no of pieces that threaten in 19x19 square, inside 10^3x10^3 board
We consider a square board of
1000
×
1000
1000\times 1000
1000
×
1000
with
1000000
1000000
1000000
squares
1
×
1
1\times 1
1
×
1
. A piece placed on a square threatens all squares on the board that are inside a
19
×
19
19\times 19
19
×
19
square. with a center in the square where the piece is placed, and with sides parallel to those of the board, except for the squares in the same row and those in the same column. Determine the maximum number of pieces that can be placed on the board so that no two pieces threaten each other.
2
1
Hide problems
x_1+x_2+...+x_k is divided by k for all k with 1<=k<=n
Determine all positive integers
n
n
n
such that numbers from
1
1
1
to
n
n
n
can be sorted in some order
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
with the property that the number
x
1
+
x
2
+
.
.
.
+
x
k
x_1+x_2+...+x_k
x
1
+
x
2
+
...
+
x
k
is divisible by
k
k
k
, for all
1
≤
k
≤
n
1\le k\le n
1
≤
k
≤
n
., that is
1
1
1
is divides
x
1
x_1
x
1
,
2
2
2
divides
x
1
+
x
2
x_1+x_2
x
1
+
x
2
,
3
3
3
divides
x
1
+
x
2
+
x
3
x_1+x_2+x_3
x
1
+
x
2
+
x
3
, and so on until
n
n
n
divides
x
1
+
x
2
+
.
.
.
+
x
n
x_1+x_2+...+x_n
x
1
+
x
2
+
...
+
x
n
.
1
1
Hide problems
sum of max prime divisors of n such that P^k <=n
For every positive integer
n
n
n
,
P
(
n
)
P(n)
P
(
n
)
is defined as follows: For each prime divisor
p
p
p
of
n
n
n
is considered the largest integer
k
k
k
such that
p
k
≤
n
p^k\le n
p
k
≤
n
and all the
p
k
p^k
p
k
are added. For example, for
n
=
100
=
2
2
⋅
5
2
n=100=2^2 \cdot 5^2
n
=
100
=
2
2
⋅
5
2
, as
2
6
<
100
<
2
7
2^6<100<2^7
2
6
<
100
<
2
7
and
5
2
<
100
<
5
3
5^2<100<5^3
5
2
<
100
<
5
3
, it turns out that
P
(
100
)
=
2
6
+
5
2
=
89
P(100)=2^6+5^2=89
P
(
100
)
=
2
6
+
5
2
=
89
Prove that there are infinitely many positive integers
n
n
n
such that
P
(
n
)
>
n
P(n)>n
P
(
n
)
>
n
..
3
1
Hide problems
locus of the midpoints of the sides KL for all equilaterals KLM
Given a square
A
B
C
D
ABCD
A
BC
D
, let us consider an equilateral triangle
K
L
M
KLM
K
L
M
, whose vertices
K
K
K
,
L
L
L
and
M
M
M
belong to the sides
A
B
AB
A
B
,
B
C
BC
BC
and
C
D
CD
C
D
respectively. Find the locus of the midpoints of the sides
K
L
KL
K
L
for all possible equilateral triangles
K
L
M
KLM
K
L
M
. Note: The set of points that satisfy a property is called a locus.