MathDB
Problems
Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
1995 Argentina National Olympiad
1995 Argentina National Olympiad
Part of
Argentina National Olympiad
Subcontests
(6)
6
1
Hide problems
max length of ant's path , 27 points in 3D
The
27
27
27
points
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of the space are marked such that
a
a
a
,
b
b
b
and
c
c
c
take the values
0
0
0
,
1
1
1
or
2
2
2
. We will call these points "junctures". Using
54
54
54
rods of length
1
1
1
, all the joints that are at a distance of
1
1
1
are joined together. A cubic structure of
2
×
2
×
2
2\times 2\times 2
2
×
2
×
2
is thus formed. An ant starts from a juncture
A
A
A
and moves along the rods; When it reaches a juncture it turns
9
0
∘
90^\circ
9
0
∘
and changes rod. If the ant returns to
A
A
A
and has not visited any juncture more than once except
A
A
A
, which it visited
2
2
2
times, at the beginning of the walk and at the end of it, what is the greatest length that the path of the ant can have?
5
1
Hide problems
x^3+\sqrt{3}(a-1)x^2-6ax+b=0
Let
a
,
b
a,b
a
,
b
be real numbers such that the equation
x
3
+
3
(
a
−
1
)
x
2
−
6
a
x
+
b
=
0
x^3+\sqrt{3}(a-1)x^2-6ax+b=0
x
3
+
3
(
a
−
1
)
x
2
−
6
a
x
+
b
=
0
has three real roots. Prove that
∣
b
∣
≤
∣
a
+
1
∣
3
|b|\leq |a+1|^3
∣
b
∣
≤
∣
a
+
1
∣
3
.>Clarification:
∣
x
∣
|x|
∣
x
∣
indicates the absolute value of
x
x
x
. For example,
∣
5
∣
=
5
|5|=5
∣5∣
=
5
;
∣
−
1.23
∣
=
1.23
|-1.23|=1.23
∣
−
1.23∣
=
1.23
; etc
4
1
Hide problems
min n= sum of 9 consecutives, 10 consecutices, 11 consecutives
Find the smallest natural number that is the sum of
9
9
9
consecutive natural numbers, is the sum of
10
10
10
consecutive natural numbers and is also the sum of
11
11
11
consecutive natural numbers.
2
1
Hide problems
1/x+1/y=1/n
For each positive integer
n
n
n
let
p
(
n
)
p(n)
p
(
n
)
be the number of ordered pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
of positive integers such that
1
x
+
1
y
=
1
n
.
\dfrac{1}{x}+\dfrac{1}{y} =\dfrac{1}{n}.
x
1
+
y
1
=
n
1
.
For example, for
n
=
2
n=2
n
=
2
the pairs are
(
3
,
6
)
,
(
4
,
4
)
,
(
6
,
3
)
(3,6),(4,4),(6,3)
(
3
,
6
)
,
(
4
,
4
)
,
(
6
,
3
)
. Therefore
p
(
2
)
=
3
p(2)=3
p
(
2
)
=
3
. a) Determine
p
(
n
)
p(n)
p
(
n
)
for all
n
n
n
and calculate
p
(
1995
)
p(1995)
p
(
1995
)
. b) Determine all pairs
n
n
n
such that
p
(
n
)
=
3
p(n)=3
p
(
n
)
=
3
.
1
1
Hide problems
stones at vertices of regular polygon
A
0
A
1
…
A
n
A_0A_1\ldots A_n
A
0
A
1
…
A
n
is a regular polygon with
n
+
1
n+1
n
+
1
vertices (n>2). Initially
n
n
n
stones are placed at vertex
A
0
A_0
A
0
. In each allowed operation,
2
2
2
stones are moved simultaneously, at the player's choice: each stone is moved from the vertex where it is located to one of the adjacent
2
2
2
vertices. Find all the values of
n
n
n
for which it is possible to have, after a succession of permitted operations, a stone at each of the vertices
A
1
,
A
2
,
…
,
A
n
A_1,A_2,\ldots ,A_n
A
1
,
A
2
,
…
,
A
n
.Clarification: The two stones that move in an allowed operation can be at the same vertex or at different vertices.
3
1
Hide problems
Parallelogram
Let ABCD be a parallelogram, and P a point such that
2
P
D
A
=
A
B
P
2 PDA=ABP
2
P
D
A
=
A
BP
and
2
P
A
D
=
P
C
D
2 PAD=PCD
2
P
A
D
=
PC
D
Show that
A
B
=
B
P
=
C
P
AB=BP=CP
A
B
=
BP
=
CP