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Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
1997 Argentina National Olympiad
1997 Argentina National Olympiad
Part of
Argentina National Olympiad
Subcontests
(6)
1
1
Hide problems
220 triangles with k +s points on 2 parallel lines
Let
s
s
s
and
t
t
t
be two parallel lines. We have marked
k
k
k
points on line
s
s
s
and
n
n
n
points on line
t
t
t
(
k
≥
n
k\geq n
k
≥
n
). If it is known that the total number of triangles that have their three vertices at marked points is
220
220
220
, find all possible values of
k
k
k
and
n
n
n
.
6
1
Hide problems
10 naturals of arithmetic progression. wanted
Decide if there are ten natural and distinct numbers
a
1
,
a
2
,
…
,
a
10
a_1,a_2,\ldots ,a_{10}
a
1
,
a
2
,
…
,
a
10
such that:
∙
\bullet
∙
Each of them is a power of a natural number with a natural exponent and greater than
1
1
1
.
∙
\bullet
∙
The numbers
a
1
,
a
2
,
…
,
a
10
a_1,a_2,\ldots ,a_{10}
a
1
,
a
2
,
…
,
a
10
form an arithmetic progression.
4
1
Hide problems
first 1997 natural numbers written on blackboard
The first
1997
1997
1997
natural numbers are written on the blackboard:
1
,
2
,
3
,
…
,
1997
1,2,3,\ldots ,1997
1
,
2
,
3
,
…
,
1997
. In front of each number, a "
+
+
+
" sign or a "
−
-
−
" sign will be written in order, from left to right. To decide each sign, a coin is tossed; If it comes up heads, you write "
+
+
+
", if it comes up tails, you write "
−
-
−
". Once the
1997
1997
1997
signs are written, the algebraic sum of the expression on the blackboard is carried out and the result is
S
S
S
. What is the probability that
S
S
S
is greater than
0
0
0
?Clarification: The probability of an event is equal to the number of favorable cases/number of possible cases.
3
1
Hide problems
max S=x_1(1-x_2)+x_2(1-x_3)+x_3(1-x_4)+\cdots +x_{99}(1-x_{100})+x_ {100}(1-x_1)
Let
x
1
,
x
2
,
x
3
,
…
,
x
100
x_1,x_2,x_3,\ldots ,x_{100}
x
1
,
x
2
,
x
3
,
…
,
x
100
be one hundred real numbers greater than or equal to
0
0
0
and less than or equal to
1
1
1
. Find the maximum possible value of the sum
S
=
x
1
(
1
−
x
2
)
+
x
2
(
1
−
x
3
)
+
x
3
(
1
−
x
4
)
+
⋯
+
x
99
(
1
−
x
100
)
+
x
100
(
1
−
x
1
)
.
S=x_1(1-x_2)+x_2(1-x_3)+x_3(1-x_4)+\cdots +x_{99}(1-x_{100})+x_ {100}(1-x_1).
S
=
x
1
(
1
−
x
2
)
+
x
2
(
1
−
x
3
)
+
x
3
(
1
−
x
4
)
+
⋯
+
x
99
(
1
−
x
100
)
+
x
100
(
1
−
x
1
)
.
5
1
Hide problems
locus of points P with equal triangle areas (ABP)=(CDP)
Given two non-parallel segments
A
B
AB
A
B
and
C
D
CD
C
D
on the plane, find the locus of points
P
P
P
on the plane such that the area of triangle
A
B
P
ABP
A
BP
is equal to the area of triangle
C
D
P
CDP
C
D
P
.
2
1
Hide problems
ABC$ is isosceles or right when <CAM +<MCB = 90^o, M midpoint of ABC
Let
A
B
C
ABC
A
BC
be a triangle and
M
M
M
be the midpoint of
A
B
AB
A
B
. If it is known that
∠
C
A
M
+
∠
M
C
B
=
9
0
o
\angle CAM + \angle MCB = 90^o
∠
C
A
M
+
∠
MCB
=
9
0
o
, show that triangle
A
B
C
ABC
A
BC
is isosceles or right.