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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1992 Spain Mathematical Olympiad
1992 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(6)
6
1
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(x+iy)^n+(x-iy)^n = 2x^n , z = x+iy, |z| = 1
For a positive integer
n
n
n
, let
S
(
n
)
S(n)
S
(
n
)
be the set of complex numbers
z
=
x
+
i
y
z = x+iy
z
=
x
+
i
y
(
x
,
y
∈
R
x,y \in R
x
,
y
∈
R
) with
∣
z
∣
=
1
|z| = 1
∣
z
∣
=
1
satisfying
(
x
+
i
y
)
n
+
(
x
−
i
y
)
n
=
2
x
n
(x+iy)^n+(x-iy)^n = 2x^n
(
x
+
i
y
)
n
+
(
x
−
i
y
)
n
=
2
x
n
. (a) Determine
S
(
n
)
S(n)
S
(
n
)
for
n
=
2
,
3
,
4
n = 2,3,4
n
=
2
,
3
,
4
. (b) Find an upper bound (depending on
n
n
n
) of the number of elements of
S
(
n
)
S(n)
S
(
n
)
for
n
>
5
n > 5
n
>
5
.
5
1
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Brocard point in Spanish Olympiad
Given a triangle
A
B
C
ABC
A
BC
, show how to construct the point
P
P
P
such that
∠
P
A
B
=
∠
P
B
C
=
∠
P
C
A
\angle PAB= \angle PBC= \angle PCA
∠
P
A
B
=
∠
PBC
=
∠
PC
A
. Express this angle in terms of
∠
A
,
∠
B
,
∠
C
\angle A,\angle B,\angle C
∠
A
,
∠
B
,
∠
C
using trigonometric functions.
3
1
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solving equations in integers, quadratics
Prove that if
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
are nonnegative integers satisfying
(
a
+
b
)
2
+
2
a
+
b
=
(
c
+
d
)
2
+
2
c
+
d
(a+b)^2+2a+b= (c+d)^2+2c+d
(
a
+
b
)
2
+
2
a
+
b
=
(
c
+
d
)
2
+
2
c
+
d
, then
a
=
c
a = c
a
=
c
and
b
=
d
b = d
b
=
d
. Show that the same is true if
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
satisfy
(
a
+
b
)
2
+
3
a
+
b
=
(
c
+
d
)
2
+
3
c
+
d
(a+b)^2+3a+b=(c+d)^2+3c+d
(
a
+
b
)
2
+
3
a
+
b
=
(
c
+
d
)
2
+
3
c
+
d
, but show that there exist
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
with
a
≠
c
a \ne c
a
=
c
and
b
≠
d
b \ne d
b
=
d
satisfying
(
a
+
b
)
2
+
4
a
+
b
=
(
c
+
d
)
2
+
4
c
+
d
(a+b)^2+4a+b = (c+d)^2+4c+d
(
a
+
b
)
2
+
4
a
+
b
=
(
c
+
d
)
2
+
4
c
+
d
.
2
1
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construct segment // to a line intersecting 2 circles with constant sum
Given two circles of radii
r
r
r
and
r
′
r'
r
′
exterior to each other, construct a line parallel to a given line and intersecting the two circles in chords with the sum of lengths
ℓ
\ell
ℓ
.
4
1
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infinitely many prime numbers in arithmetic progression 3,7,11,15, . .
Prove that the arithmetic progression
3
,
7
,
11
,
15
,
.
.
.
3,7,11,15,...
3
,
7
,
11
,
15
,
...
. contains infinitely many prime numbers.
1
1
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Olympiad of Spain 1992
Determine the smallest number N, multiple of 83, such that N^2 has 63 positive divisors.