MathDB
Problems
Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
1998 Chile National Olympiad
1998 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(7)
7
1
Hide problems
2 dice with same sum as 2 ordinary ones (Chile NMO 1998 P7)
When rolling two normal dice, the set of possible outcomes of the sum of the points is
2
,
3
,
3
,
4
,
4
,
4
,
.
.
.
,
11
,
11
,
12
2, 3, 3, 4,4, 4,..., 11, 11,12
2
,
3
,
3
,
4
,
4
,
4
,
...
,
11
,
11
,
12
. Notice that this sequence can be obtained from the identity
(
x
+
x
2
+
x
3
+
x
4
+
x
5
+
x
6
)
(
x
+
x
2
+
x
3
+
x
4
+
x
5
+
x
6
)
=
x
2
+
2
x
3
+
3
x
4
+
.
.
.
+
2
x
11
+
x
12
.
(x + x^2 + x^3 + x^4 + x^5 + x^6) (x + x^2 + x^3 + x^4 + x^5 + x^6) = x^2 + 2x^3 + 3x^4 +... + 2x^{11} + x^{12}.
(
x
+
x
2
+
x
3
+
x
4
+
x
5
+
x
6
)
(
x
+
x
2
+
x
3
+
x
4
+
x
5
+
x
6
)
=
x
2
+
2
x
3
+
3
x
4
+
...
+
2
x
11
+
x
12
.
Design a crazy pair of dice, that is, two other cubes, not necessarily the same, with a natural number indicated on each face, such that the set of possible results of the sum of its points is equal to of two normal dice.
6
1
Hide problems
create square by 4 pieces cut from an equilateral (Chile NMO 1998 P6)
Given an equilateral triangle, cut it into four polygonal shapes so that, reassembled appropriately, these figures form a square.
5
1
Hide problems
3 =a^3+b^3+c^3+d^3 (Chile NMO 1998 P5)
Show that the number
3
3
3
can be written in a infinite number of different ways as the sum of the cubes of four integers.
4
1
Hide problems
x^{3/2} + 6x^{5/4} + 8x^{3/4}\ge 15x (Chile NMO 1998 P4)
a) Prove that for any nonnegative real
x
x
x
, holds
x
3
2
+
6
x
5
4
+
8
x
3
4
≥
15
x
.
x^{\frac32} + 6x^{\frac54} + 8x^{\frac34}\ge 15x.
x
2
3
+
6
x
4
5
+
8
x
4
3
≥
15
x
.
b) Determine all x for which the equality holds
3
1
Hide problems
nested \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+...}}}} (Chile NMO 1998 P3)
Evaluate
1
+
2
1
+
3
1
+
4
1
+
.
.
.
\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+...}}}}
1
+
2
1
+
3
1
+
4
1
+
...
.
1
1
Hide problems
sum x_i =1998, a<x_i<b (Chile NMO 1998 P1)
Find all pairs of naturals
a
,
b
a,b
a
,
b
with
a
<
b
a <b
a
<
b
, such that the sum of the naturals greater than
a
a
a
and less than
b
b
b
equals
1998
1998
1998
.
2
1
Hide problems
locus wanted, semicircle, constant length (1998 Chile Level 2 P2)
Given a semicircle of diameter
A
B
AB
A
B
, with
A
B
=
2
r
AB = 2r
A
B
=
2
r
, be
C
D
CD
C
D
a variable string, but of fixed length
c
c
c
. Let
E
E
E
be the intersection point of lines
A
C
AC
A
C
and
B
D
BD
B
D
, and let
F
F
F
be the intersection point of lines
A
D
AD
A
D
and
B
C
BC
BC
. a) Prove that the lines
E
F
EF
EF
and
A
B
AB
A
B
are perpendicular. b) Determine the locus of the point
E
E
E
. c) Prove that
E
F
EF
EF
has a constant measure, and determine it based on
c
c
c
and
r
r
r
.