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Chile Classification NMO
1993 Chile Classification NMO
1993 Chile Classification NMO
Part of
Chile Classification NMO
Subcontests
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1993 Chile Classification / Qualifying NMO V
p1. Show that the difference between the cubes of two consecutive naturals is the sum of a square of one natural plus three times the square of another. p2. The area of a plane figure
F
F
F
is denoted by
A
F
A_F
A
F
. Given a circumference
C
C
C
, consider all pairs of triangles
(
T
,
S
)
(T, S)
(
T
,
S
)
, such that
C
C
C
is the incircle of
T
T
T
and the circumcircle of
S
S
S
. Between all these pairs, find a pair
(
T
0
,
S
0
)
(T_0,S_0)
(
T
0
,
S
0
)
such that the expression
A
T
0
A
S
0
\frac{A_{T_0}}{A_{S_0}}
A
S
0
A
T
0
has the minimum value. p3. Let
C
C
C
be the set of all naturals that can be expressed as
m
2
+
n
2
m^2 + n^2
m
2
+
n
2
, with
m
,
n
m, n
m
,
n
natural. Consider
s
,
t
∈
C
s, t \in C
s
,
t
∈
C
. Prove that:
∙
\bullet
∙
s
t
∈
C
st \in C
s
t
∈
C
.
∙
\bullet
∙
If
t
≠
0
t \ne 0
t
=
0
then there are rationals
x
,
y
x,y
x
,
y
such that
s
t
=
x
2
+
y
2
\frac{s}{t}= x^2 + y^2
t
s
=
x
2
+
y
2
. p4. At the exit of a stadium I met my friends Carlos, Pedro and José, who are still to witness the classic match. I asked them what the result was, and their responses were:
∙
\bullet
∙
Carlos: U won and UC scored the first goal.
∙
\bullet
∙
Pedro: U won and scored the first goal.
∙
\bullet
∙
José: Pedro never tells the truth and the U lost. Also, I knew that each of them was telling either two truths or two lies. Could I deduce from this the final score of the match? How? p5. A natural
n
n
n
is the product of three odd primes. The sum of the primes is
1993
1993
1993
, the sum of its squares is
1363347
1363347
1363347
, and the sum of the divisors of
n
n
n
, including
1
1
1
and
n
n
n
, is
280411488
280411488
280411488
. Determine
n
n
n
. p6. To calculate the area of a quadrilateral, where
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
are the lengths of its sides (in that order), we have Humbertito's Formula:
A
=
a
+
c
2
⋅
b
+
d
2
.
A =\frac{a + c}{2}\cdot \frac{b + d}{2}.
A
=
2
a
+
c
⋅
2
b
+
d
.
https://cdn.artofproblemsolving.com/attachments/6/9/01a5812d194de4e31e4ce53643d99377925201.png Prove that this formula is correct only in the case of rectangles. p7. For each natural
k
k
k
, let
d
(
k
)
d (k)
d
(
k
)
be the number of positive divisors of
k
k
k
, including
1
1
1
and
k
k
k
. Find the maximum value of
d
(
k
)
d (k)
d
(
k
)
, when
k
k
k
is an integer, with
1
≤
k
≤
1993
1 \le k \le1993
1
≤
k
≤
1993
.