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Problems
Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
2015 Argentina National Olympiad
2015 Argentina National Olympiad
Part of
Argentina National Olympiad
Subcontests
(6)
6
1
Hide problems
list of subsets of 1-1001
Let
S
S
S
the set of natural numbers from
1
1
1
up to
1001
1001
1001
,
S
=
{
1
,
2
,
.
.
.
,
1001
}
S=\{1,2,...,1001\}
S
=
{
1
,
2
,
...
,
1001
}
. Lisandro thinks of a number
N
N
N
of
S
S
S
, and Carla has to find out that number with the following procedure. She gives Lisandro a list of subsets of
S
S
S
, Lisandro reads it and tells Carla how many subsets of her list contain
N
N
N
. If Carla wishes, she can repeat the same thing with a second list, and then with a third, but no more than
3
3
3
are allowed. What is the smallest total number of subsets that allow Carla to find
N
N
N
for sure?
5
1
Hide problems
p^3-4p+9 is a perfect square
Find all prime numbers
p
p
p
such that
p
3
−
4
p
+
9
p^3-4p+9
p
3
−
4
p
+
9
is a perfect square.
4
1
Hide problems
segment S of length 50 is covered by several segments of length 1
An segment
S
S
S
of length
50
50
50
is covered by several segments of length
1
1
1
, all of them contained in
S
S
S
. If any of these unit segments were removed,
S
S
S
would no longer be completely covered. Find the maximum number of unit segments with this property.Clarification: Assume that the segments include their endpoints.
2
1
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a+b and ab+1 are powers of 2
Find all pairs of natural numbers
a
,
b
a,b
a
,
b
, with
a
≠
b
a\ne b
a
=
b
, such that
a
+
b
a+b
a
+
b
and
a
b
+
1
ab+1
ab
+
1
are powers of
2
2
2
.
1
1
Hide problems
sum of k(3k+1 ) / {(3k-1)(3k+2) as irreducible fraction
Express the sum of
99
99
99
terms
1
⋅
4
2
⋅
5
+
2
⋅
7
5
⋅
8
+
…
+
k
(
3
k
+
1
)
(
3
k
−
1
)
(
3
k
+
2
)
+
…
+
99
⋅
298
296
⋅
299
\frac{1\cdot 4}{2\cdot 5}+\frac{2\cdot 7}{5\cdot 8}+\ldots +\frac{k(3k+1 )}{(3k-1)(3k+2)}+\ldots +\frac{99\cdot 298}{296\cdot 299}
2
⋅
5
1
⋅
4
+
5
⋅
8
2
⋅
7
+
…
+
(
3
k
−
1
)
(
3
k
+
2
)
k
(
3
k
+
1
)
+
…
+
296
⋅
299
99
⋅
298
as an irreducible fraction.
3
1
Hide problems
max value of the y -coordinate of P(x,y) after triangle rotation
Consider the points
O
=
(
0
,
0
)
,
A
=
(
−
2
,
0
)
O = (0,0), A = (- 2,0)
O
=
(
0
,
0
)
,
A
=
(
−
2
,
0
)
and
B
=
(
0
,
2
)
B = (0,2)
B
=
(
0
,
2
)
in the coordinate plane. Let
E
E
E
and
F
F
F
be the midpoints of
O
A
OA
O
A
and
O
B
OB
OB
respectively. We rotate the triangle
O
E
F
OEF
OEF
with a center in
O
O
O
clockwise until we obtain the triangle
O
E
′
F
′
OE'F'
O
E
′
F
′
and, for each rotated position, let
P
=
(
x
,
y
)
P = (x, y)
P
=
(
x
,
y
)
be the intersection of the lines
A
E
′
AE'
A
E
′
and
B
F
′
BF'
B
F
′
. Find the maximum possible value of the
y
y
y
-coordinate of
P
P
P
.