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Problems
Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
1993 Chile National Olympiad
1993 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(7)
7
1
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6 exchange their vests of different colors (Chile NMO 1993 P7)
Six young people - Antonio, Bernardo, Carlos, Diego, Eduardo, and Francisco, attended a meeting in vests of different colors. After the meeting, they decided to exchange the vests as souvenir.
1
)
1)
1
)
. Each of them came out of the meeting room, wearing a vest with color different from the one with which they went into the meeting room.
2
)
2)
2
)
. The vest with which Antonio came out of the meeting room was belong to the young man who came out with Bernardo's vest.
3
)
3)
3
)
. The owner of the vest with which Carlos came out of the meeting room, came out with the vest that was belong to the young man who came out with Diego's vest.
4
)
4)
4
)
. The one who came out of the meeting room with Eduardo's vest was not the owner of the vest with which Francisco came out. Determine who came out of the meeting room with Antonio's vest, and who owns the vest with which Antonio came out.[hide=original wording]Seis jovenes que asistieron a una reunion vistiendo chalecos de distintos colores, decidieron intercambiarlos y salieron vistiendo todos de color diferente a aquel con que llegaron. El chaleco con que salio Antonio perteneca al joven que salio con el chaleco de Bernardo. El dueno del chaleco con que salio Carlos, salio con el chaleco que perteneca al joven que se llevo el de Diego. Quien se llevo el chaleco de Eduardo no era el dueno del que se llevo Francisco. Determine quien salio con el chaleco de Antonio, y quien es el dueno del chaleco que se llevo Antonio.
5
1
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a (1- b)>1/4 , b (1-c)>1/4 , c (1-a)>1/4 (Chile NMO 1993 P5)
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
three positive numbers less than
1
1
1
. Prove that cannot occur simultaneously these three inequalities:
a
(
1
−
b
)
>
1
4
a (1- b)>\frac14
a
(
1
−
b
)
>
4
1
b
(
1
−
c
)
>
1
4
b (1-c)>\frac14
b
(
1
−
c
)
>
4
1
c
(
1
−
a
)
>
1
4
c (1-a)>\frac14
c
(
1
−
a
)
>
4
1
4
1
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members in 4 commissions in a club (Chile NMO 1993 P4)
In some club, each member is on two commissions. Furthermore, it is known that two any commissions always have exactly one member in common. Knowing there are five commissions. How many members does the club have?
3
1
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(8r + 21)/(3r + 8) approximation of \sqrt7 that r (Chile NMO 1993 P3)
Let
r
r
r
be a positive rational. Prove that
8
r
+
21
3
r
+
8
\frac{8r + 21}{3r + 8}
3
r
+
8
8
r
+
21
is a better approximation to
7
\sqrt7
7
that
r
r
r
.
6
1
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sum of distances of interior point of rectangles >= 2 area (Chile 1993 P6)
Let
A
B
C
D
ABCD
A
BC
D
be a rectangle of area
S
S
S
, and
P
P
P
be a point inside it. We denote by
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
the distances from
P
P
P
to the vertices
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
respectively. Prove that
a
2
+
b
2
+
c
2
+
d
2
≥
2
S
a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2\ge 2S
a
2
+
b
2
+
c
2
+
d
2
≥
2
S
. When there is equality?
1
1
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shortest path through 4 vertices of a square (Chile 1993 P1)
There are four houses, located on the vertices of a square. You want to draw a road network, so that you can go from any house to any other. Prove that the network formed by the diagonals is not the shortest. Find a shorter network.
2
1
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given a rectangle, circumscribe a rectangle of maximum area (Chile 1993 P2)
Given a rectangle, circumscribe a rectangle of maximum area.