MathDB
Problems
Contests
International Contests
Cono Sur Olympiad
1996 Cono Sur Olympiad
1996 Cono Sur Olympiad
Part of
Cono Sur Olympiad
Subcontests
(6)
1
1
Hide problems
areas of a square divided in squares and rectangles
In the following figure, the largest square is divided into two squares and three rectangles, as shown:The area of each smaller square is equal to
a
a
a
and the area of each small rectangle is equal to
b
b
b
. If
a
+
b
=
24
a+b=24
a
+
b
=
24
and the root square of
a
a
a
is a natural number, find all possible values for the area of the largest square. https://cdn.artofproblemsolving.com/attachments/f/a/0b424d9c293889b24d9f31b1531bed5081081f.png
6
1
Hide problems
Points in the plane
Find all integers
n
≤
3
n \leq 3
n
≤
3
such that there is a set
S
n
S_n
S
n
formed by
n
n
n
points of the plane that satisfy the following two conditions: Any three points are not collinear. No point is found inside the circle whose diameter has ends at any two points of
S
n
S_n
S
n
.NOTE: The points on the circumference are not considered to be inside the circle.
2
1
Hide problems
An Easy Sequence Problem
Consider a sequence of real numbers defined by:
a
n
+
1
=
a
n
+
1
a
n
a_{n + 1} = a_n + \frac{1}{a_n}
a
n
+
1
=
a
n
+
a
n
1
for
n
=
0
,
1
,
2
,
.
.
.
n = 0, 1, 2, ...
n
=
0
,
1
,
2
,
...
Prove that, for any positive real number
a
0
a_0
a
0
, is true that
a
1996
a_{1996}
a
1996
is greater than
63
63
63
.
5
1
Hide problems
A good problem( very easy )
We want to cover totally a square(side is equal to
k
k
k
integer and
k
>
1
k>1
k
>
1
) with this rectangles:
1
1
1
rectangle (
1
×
1
1\times 1
1
×
1
),
2
2
2
rectangles (
2
×
1
2\times 1
2
×
1
),
4
4
4
rectangles (
3
×
1
3\times 1
3
×
1
),....,
2
n
2^n
2
n
rectangles (
n
+
1
×
1
n + 1 \times 1
n
+
1
×
1
), such that the rectangles can't overlap and don't exceed the limits of square. Find all
k
k
k
, such that this is possible and for each
k
k
k
found you have to draw a solution
4
1
Hide problems
Numbers in a sequence
The sequence
0
,
1
,
1
,
1
,
1
,
1
,
.
.
.
.
,
1
0, 1, 1, 1, 1, 1,....,1
0
,
1
,
1
,
1
,
1
,
1
,
....
,
1
where have
1
1
1
number zero and
1995
1995
1995
numbers one. If we choose two or more numbers in this sequence(but not the all
1996
1996
1996
numbers) and substitute one number by arithmetic mean of the numbers selected, we obtain a new sequence with
1996
1996
1996
numbers!!! Show that, we can repeat this operation until we have all
1996
1996
1996
numbers are equal Note: It's not necessary to choose the same quantity of numbers in each operation!!!
3
1
Hide problems
Bottles and price
A shop sells bottles with this capacity:
1
L
,
2
L
,
3
L
,
.
.
.
,
1996
L
1L, 2L, 3L,..., 1996L
1
L
,
2
L
,
3
L
,
...
,
1996
L
, the prices of bottles satifies this
2
2
2
conditions:
1
1
1
. Two bottles have the same price, if and only if, your capacities satifies
m
−
n
=
1000
m - n = 1000
m
−
n
=
1000
2
2
2
. The price of bottle
m
m
m
(
1001
>
m
>
0
1001>m>0
1001
>
m
>
0
) is
1996
−
m
1996 - m
1996
−
m
dollars. Find all pair(s)
m
m
m
and
n
n
n
such that: a)
m
+
n
=
1000
m + n = 1000
m
+
n
=
1000
b) the cost is smallest possible!!! c) with the pair, the shop can measure
k
k
k
liters, with
0
<
k
<
1996
0<k<1996
0
<
k
<
1996
(for all
k
k
k
integer) Note: The operations to measure are: i) To fill or empty any one of two bottles ii)Pass water of a bottle for other bottle We can measure
k
k
k
liters when the capacity of one bottle plus the capacity of other bottle is equal to
k
k
k