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Problems
Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
1995 Chile National Olympiad
1995 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(7)
6
1
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compare (1995^{1994} + 1)/(1995^{1995}+1) (1995 Chile NMO P6)
Which of the following rationals is greater ,
199
5
1994
+
1
199
5
1995
+
1
\frac{1995^{1994} + 1}{1995^{1995} + 1}
199
5
1995
+
1
199
5
1994
+
1
or
199
5
1995
+
1
199
5
1996
+
1
\frac{1995^{1995} + 1}{ 1995^{1996} +1}
199
5
1996
+
1
199
5
1995
+
1
?
5
1
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5 lions and 4 tigers on a line (1995 Chile NMO P5)
A tamer wants to line up five lions and four tigers. We know that a tiger cannot go after another. How many ways can the beasts be distributed? The tamer cannot distinguish two animals of the same species.
4
1
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3digit numbers at vertices of cube (1995 Chile NMO P4)
It is possible to write the numbers
111
111
111
,
112
112
112
,
121
121
121
,
122
122
122
,
211
211
211
,
212
212
212
,
221
221
221
and
222
222
222
at the vertices of a cube, so that the numbers written in adjacent vertices match at most in one digit?
3
1
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p(a)=b, p(b)=c, p(c)=a not possible for integer cubic (1995 Chile NMO P3)
If
p
(
x
)
=
c
0
+
c
1
x
+
c
2
x
2
+
c
3
x
3
p (x) = c_0 + c_1x + c_2x^2 + c_3x^3
p
(
x
)
=
c
0
+
c
1
x
+
c
2
x
2
+
c
3
x
3
is a polynomial with integer coefficients with
a
,
b
,
c
a, b,c
a
,
b
,
c
integers and different from each other, prove that it cannot happen simultaneously that
p
(
a
)
=
b
p (a) = b
p
(
a
)
=
b
,
p
(
b
)
=
c
p (b) = c
p
(
b
)
=
c
and
p
(
c
)
=
a
p (c) = a
p
(
c
)
=
a
.
1
1
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12 divides (a-b) (a-c) (a-d) (b- c) (b-d) (c-d) (1995 Chile NMO P1)
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be integers. Prove that
12
12
12
divides
(
a
−
b
)
(
a
−
c
)
(
a
−
d
)
(
b
−
c
)
(
b
−
d
)
(
c
−
d
)
(a-b) (a-c) (a-d) (b- c) (b-d) (c-d)
(
a
−
b
)
(
a
−
c
)
(
a
−
d
)
(
b
−
c
)
(
b
−
d
)
(
c
−
d
)
.
7
1
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3 circles inscribed in a semicirlce, radii inequalities (1995 Chile Level 2 P7)
In a semicircle of radius
4
4
4
three circles are inscribed, as indicated in the figure. Larger circles have radii
R
1
R_1
R
1
and
R
2
R_2
R
2
, and the larger circle has radius
r
r
r
. a) Prove that
1
r
=
1
R
1
+
1
R
2
\dfrac {1} {\sqrt{r}} = \dfrac {1} {\sqrt{R_1}} + \dfrac {1} {\sqrt{R_2}}
r
1
=
R
1
1
+
R
2
1
b) Prove that
R
1
+
R
2
≤
8
(
2
−
1
)
R_1 + R_2 \le 8 (\sqrt{2} -1)
R
1
+
R
2
≤
8
(
2
−
1
)
c) Prove that
r
≤
2
−
1
r \le \sqrt{2} -1
r
≤
2
−
1
https://cdn.artofproblemsolving.com/attachments/0/9/aaaa65d1f4da4883973751e1363df804b9944c.jpg
2
1
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black area by 6 arcs of radius = circle radius (1995 Chile Level 2 P2)
In a circle of radius
1
1
1
, six arcs of radius
1
1
1
are drawn, which cut the circle as in the figure. Determine the black area. https://cdn.artofproblemsolving.com/attachments/8/9/0323935be8406ea0c452b3c8417a8148c977e3.jpg