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Problems
Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
1989 Chile National Olympiad
1989 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(7)
3
1
Hide problems
sum of areas among semicircles, with diameters sides of right triangle
In a right triangle with legs
a
a
a
,
b
b
b
and hypotenuse
c
c
c
, draw semicircles with diameters on the sides of the triangle as indicated in the figure. The purple areas have values
X
,
Y
X,Y
X
,
Y
. Calculate
X
+
Y
X + Y
X
+
Y
. https://cdn.artofproblemsolving.com/attachments/1/a/5086dc7172516b0a986ef1af192c15eba4d6fc.png
7
1
Hide problems
5 hats, 3 white and 2 black, guess the color (Chile NMO 1989 P7)
Three wise men live in an old region. As they do not always agree on their advice to the king, he decided to stay with the wisest of the three, killing the others. To decide which of them was saved, performed the following test: He put each sage a hat, without him seeing its color, then locked them in a common room and told them:
∙
\bullet
∙
Only the first to guess the color of his own hat will save his life.
∙
\bullet
∙
In total there are five hats, three are white and two are black.
∙
\bullet
∙
You cannot communicate with each other, but you can look at each other. After a long time, one of the wise men says: "I know the color of my hat." What color did they have? How did you figure it out? What color were the other hats used?
6
1
Hide problems
(f(n+1) -1)^2 +(f (n)-1)^2 = 2f(n) f(n+1) + 4 (Chile NMO 1989 P6)
The function
f
f
f
, with domain on the set of non-negative integers, is defined by the following :
∙
\bullet
∙
f
(
0
)
=
2
f (0) = 2
f
(
0
)
=
2
∙
\bullet
∙
(
f
(
n
+
1
)
−
1
)
2
+
(
f
(
n
)
−
1
)
2
=
2
f
(
n
)
f
(
n
+
1
)
+
4
(f (n + 1) -1)^2 + (f (n)-1) ^2 = 2f (n) f (n + 1) + 4
(
f
(
n
+
1
)
−
1
)
2
+
(
f
(
n
)
−
1
)
2
=
2
f
(
n
)
f
(
n
+
1
)
+
4
, taking
f
(
n
)
f (n)
f
(
n
)
the largest possible value. Determine
f
(
n
)
f (n)
f
(
n
)
.
4
1
Hide problems
N locks, 5 with some keys, any 3 may open it, no 2 can (Chile NMO 1989 P4)
The vault of a bank has
N
N
N
locks. To open it, they must be operated simultaneously. Five executives have some of the keys, so any trio can open the vault, but no pair can do it. Determine
N
N
N
.
2
1
Hide problems
no of squares in mxn rectangle, crossed by diagonal (Chile NMO 1989 P2)
We have a rectangle with integer sides
m
,
n
m, n
m
,
n
that is subdivided into
m
n
mn
mn
squares of side
1
1
1
. Find the number of little squares that are crossed by the diagonal (without counting those that are touched only in one vertex)
1
1
Hide problems
1998 has same sum of digits in base b and 10 (Chile NMO 1989 P1)
Writing
1989
1989
1989
in base
b
b
b
, we obtain a three-digit number:
x
y
z
xyz
x
yz
. It is known that the sum of the digits is the same in base
10
10
10
and in base
b
b
b
, that is,
1
+
9
+
8
+
9
=
x
+
y
+
z
1 + 9 + 8 + 9 = x + y + z
1
+
9
+
8
+
9
=
x
+
y
+
z
. Determine
x
,
y
,
z
,
b
.
x,y,z,b.
x
,
y
,
z
,
b
.
5
1
Hide problems
altitude in a triangle with rational sides, defines rational segments
The lengths of the three sides of a
△
A
B
C
\triangle ABC
△
A
BC
are rational. The altitude
C
D
CD
C
D
determines on the side
A
B
AB
A
B
two segments
A
D
AD
A
D
and
D
B
DB
D
B
. Prove that
A
D
,
D
B
AD, DB
A
D
,
D
B
are rational.