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Problems
Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
1998 Argentina National Olympiad
1998 Argentina National Olympiad
Part of
Argentina National Olympiad
Subcontests
(6)
6
1
Hide problems
sum x_i<=3, sum x_i^2 >=1, 3 may be chosen with sum >=1
Given
n
n
n
non-negative real numbers,
n
≥
3
n\geq 3
n
≥
3
, such that the sum of the
n
n
n
numbers is less than or equal to
3
3
3
and the sum of the squares of the
n
n
n
numbers is greater than or equal to
1
1
1
, prove that among the
n
n
n
numbers three can be chosen whose sum is greater than or equal to
1
1
1
.
4
1
Hide problems
x -[x/2] -[x/3] -[x/6]
Determine all possible values of the expression
x
−
[
x
2
]
−
[
x
3
]
−
[
x
6
]
x-\left [\frac{x}{2}\right ]-\left [\frac{x}{3}\right ]-\left [\frac{x} {6}\right ]
x
−
[
2
x
]
−
[
3
x
]
−
[
6
x
]
by varying
x
x
x
in the real numbers.Clarification: The brackets indicate the integer part of the number they enclose.
3
1
Hide problems
no of sequences of type 1 is equal to number of sequences of type 2
Given two integers
m
≥
2
m\geq 2
m
≥
2
and
n
≥
2
n\geq 2
n
≥
2
we consider two types of sequences of length
m
⋅
n
m\cdot n
m
⋅
n
formed exclusively by
0
0
0
and
1
1
1
TYPE 1 sequences are all those that verify the following two conditions:
∙
\bullet
∙
a
k
a
k
+
m
=
0
a_ka_{k+m} = 0
a
k
a
k
+
m
=
0
for all
k
=
1
,
2
,
3
,
.
.
.
k = 1, 2, 3, ...
k
=
1
,
2
,
3
,
...
∙
\bullet
∙
If
a
k
a
k
+
1
=
1
a_ka_{k+1} = 1
a
k
a
k
+
1
=
1
, then
k
k
k
is a multiple of
m
m
m
. TYPE 2 sequences are all those that verify the following two conditions:
∙
\bullet
∙
a
k
a
k
+
n
=
0
a_ka_{k+n} = 0
a
k
a
k
+
n
=
0
for all
k
=
1
,
2
,
3
,
.
.
.
k = 1, 2, 3, ...
k
=
1
,
2
,
3
,
...
∙
\bullet
∙
If
a
k
a
k
+
1
=
1
a_ka_{k+1} = 1
a
k
a
k
+
1
=
1
, then
k
k
k
is a multiple of
n
n
n
.Prove that the number of sequences of type 1 is equal to the number of sequences of type 2.
1
1
Hide problems
list with an even number of integers
Jorge writes a list with an even number of integers, not all equal to
0
0
0
(there may be repeated numbers). Show that Martin can cross out a number from the list, of his choice, so that it is impossible for Jorge to separate the remaining numbers into two groups in such a way that the sum of all the numbers in one group is equal to the sum of all the others. numbers from the other group.
5
1
Hide problems
right isosceles triangle with min area wanted inscribed in right isosceles
Let
A
B
C
ABC
A
BC
a right isosceles triangle with hypotenuse
A
B
=
2
AB=\sqrt2
A
B
=
2
. Determine the positions of the points
X
,
Y
,
Z
X,Y,Z
X
,
Y
,
Z
on the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
respectively so that the triangle
X
Y
Z
XYZ
X
Y
Z
is isosceles, right, and with minimum area.
2
1
Hide problems
quadrilateral of orthocenters is a parallelogram
Let a quadrilateral
A
B
C
D
ABCD
A
BC
D
have an inscribed circle and let
K
,
L
,
M
,
N
K, L, M, N
K
,
L
,
M
,
N
be the tangency points of the sides
A
B
,
B
C
,
C
D
AB, BC, CD
A
B
,
BC
,
C
D
and
D
A
DA
D
A
, respectively. Consider the orthocenters of each of the triangles
△
A
K
N
,
△
B
L
K
,
△
C
M
L
\vartriangle AKN, \vartriangle BLK, \vartriangle CML
△
A
K
N
,
△
B
L
K
,
△
CM
L
and
△
D
N
M
\vartriangle DNM
△
D
NM
. Prove that these four points are the vertices of a parallelogram.