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Problems
Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
1996 Chile National Olympiad
1996 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(7)
7
1
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sum 1/1x4+1/4x7+1/7x10+1/1995x1996 , 1996 Chile NMO p7
(a) Let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
be integers such that
a
d
≠
b
c
ad\ne bc
a
d
=
b
c
. Show that is always possible to write the fraction
1
(
a
x
+
b
)
(
c
x
+
d
)
\frac{1}{(ax+b)(cx+d)}
(
a
x
+
b
)
(
c
x
+
d
)
1
in the form
r
a
x
+
b
+
s
c
x
+
d
\frac{r}{ax+b}+\frac{s}{cx+d}
a
x
+
b
r
+
c
x
+
d
s
(b) Find the sum
1
1
⋅
4
+
1
4
⋅
7
+
1
7
⋅
10
+
.
.
.
+
1
1995
⋅
1996
\frac{1}{1 \cdot 4}+\frac{1}{4 \cdot 7}+\frac{1}{7 \cdot 10}+...+\frac{1}{1995 \cdot 1996}
1
⋅
4
1
+
4
⋅
7
1
+
7
⋅
10
1
+
...
+
1995
⋅
1996
1
6
1
Hide problems
equal arcs wanted, 2 intersecting circles (1996 Chile NMO P6)
Two circles,
C
C
C
and
K
K
K
, are secant at
A
A
A
and
B
B
B
. Let
P
P
P
be a point on the arc
A
B
AB
A
B
of
C
C
C
. Lines
P
A
PA
P
A
and
P
B
PB
PB
intersect
K
K
K
again at
R
R
R
and
S
S
S
respectively. Let
P
′
P'
P
′
be another point at same arc as
P
P
P
, so that lines
P
′
A
P'A
P
′
A
and
P
′
B
P'B
P
′
B
again intersect
K
K
K
at
R
′
R'
R
′
and
S
′
S'
S
′
, respectively. Prove that the arcs
R
S
RS
RS
and
R
′
S
′
R'S'
R
′
S
′
have equal measures. https://cdn.artofproblemsolving.com/attachments/2/4/88693c36159179fb2b098b671a2f8281b37aae.png
5
1
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1,048,576 different ways to choose 2 cakes (1996 Chile NMO P5)
Some time ago, on a radio program, a baker announced a special promotion in the purchase of two stuffed cakes. Each cake could contain up to five fillings of which had in the pastry. On the show, a lady said there were
1
,
048
,
576
1,048,576
1
,
048
,
576
different possibilities to choose the two stuffed cakes. How many different fillings did the pastry chef have?
4
1
Hide problems
a,b>=5 if ax^2-bx + c = 0 has two roots in [0, 1] (1996 Chile NMO P4)
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be naturals. The equation
a
x
2
−
b
x
+
c
=
0
ax^2-bx + c = 0
a
x
2
−
b
x
+
c
=
0
has two roots at
[
0
,
1
]
[0, 1]
[
0
,
1
]
. Prove that
a
≥
5
a\ge 5
a
≥
5
and
b
≥
5
b\ge 5
b
≥
5
.
1
1
Hide problems
buy a pair of shoes without paying (1996 Chile NMO P1)
A shoe brand proposes: Buy a pair of shoes without paying. It's about this: you go to the factory and pay
20
,
000
$
20,000 \$
20
,
000$
for a pair of shoes, get the shoes and ten stamps, with a unit cost of each stamp
2000
$
2000 \$
2000$
. By selling these stamps you will get your money back. The ones who buy these stamps go to the factory, delivers them and for
18
,
000
$
18,000 \$
18
,
000$
they receive their pair of shoes and the ten stamps, thus continuing the cycle.
∙
\bullet
∙
How much does the factory receive for each pair of shoes?
∙
\bullet
∙
Can this operation be repeated a hundred times, assuming that no one repeats itself?[hide=original wording]Una marca de zapatos propone: Compre un par de zapatos sin pagar. Se trata de lo siguiente: usted va a la fabrica y paga \$ 20000 por un par de zapatos; recibe los zapatos y diez estampillas, con un costo unitario de ]\$ 2000. Al vender estas estampillas recuperara su dinero. Quienes compren estas estampillas van a la fabrica, la entregan y por \$ 18000 reciben su par de zapatos y las diez estampillas, continuando as el ciclo. - Cuanto recibe la fabrica por cada par de zapatos? - Se puede repetir esta operacion cien veces, suponiendo que nadie se repite?
3
1
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max no of lines among 2n triangles no forming triangle (1996 Chile NMO P3)
Let
n
>
2
n> 2
n
>
2
be a natural. Given
2
n
2n
2
n
points in the plane, no
3
3
3
are collinear. What is the maximum number of lines that can be drawn between them, without forming a triangle?[hide=original wording]Sea n > 2 un natural. Dados 2n puntos en el plano, tres a tres no colineales, Cual es el numero maximo de trazos que pueden dibujarse entre ellos, sin formar un triangulo?
2
1
Hide problems
trangle construction, intersections with circumcircle (Chile 1996 P2)
Construct the
△
A
B
C
\triangle ABC
△
A
BC
, with
A
C
<
B
C
AC <BC
A
C
<
BC
, if the circumcircle is known, and the points
D
,
E
,
F
D, E, F
D
,
E
,
F
in it, where they intersect, respectively, the altitude, the median and the angle bisector that they start from the vertex
C
C
C
.