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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1989 Spain Mathematical Olympiad
1989 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(6)
6
1
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among 7 real exist two a,b so that \sqrt3|a-b| < = |1+ab|
Prove that among any seven real numbers there exist two,
a
a
a
and
b
b
b
, such that
3
∣
a
−
b
∣
≤
∣
1
+
a
b
∣
\sqrt3|a-b|\le |1+ab|
3
∣
a
−
b
∣
≤
∣1
+
ab
∣
. Give an example of six real numbers not having this property.
5
1
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number of ways of writing 14 as product of 2 elements of set: a+13bi, a,b in Z
Consider the set
D
D
D
of all complex numbers of the form
a
+
b
−
13
a+b\sqrt{-13}
a
+
b
−
13
with
a
,
b
∈
Z
a,b \in Z
a
,
b
∈
Z
. The number
14
=
14
+
0
−
13
14 = 14+0\sqrt{-13}
14
=
14
+
0
−
13
can be written as a product of two elements of
D
D
D
:
14
=
2
⋅
7
14 = 2 \cdot 7
14
=
2
⋅
7
. Find all possible ways to express
14
14
14
as a product of two elements of
D
D
D
.
4
1
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1989^n = a^2 + b^2, in at least two ways for every n
Show that the number
1989
1989
1989
as well as each of its powers
198
9
n
1989^n
198
9
n
(
n
∈
N
n \in N
n
∈
N
), can be expressed as a sum of two positive squares in at least two ways.
2
1
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points defined by equal ratios on sides of triangle, area of triangle
Points
A
′
,
B
′
,
C
′
A' ,B' ,C'
A
′
,
B
′
,
C
′
on the respective sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
of triangle
A
B
C
ABC
A
BC
satisfy
A
C
′
A
B
=
B
A
′
B
C
=
C
B
′
C
A
=
k
\frac{AC' }{AB} = \frac{BA' }{BC} = \frac{CB' }{CA} = k
A
B
A
C
′
=
BC
B
A
′
=
C
A
C
B
′
=
k
. The lines
A
A
′
,
B
B
′
,
C
C
′
AA' ,BB' ,CC'
A
A
′
,
B
B
′
,
C
C
′
form a triangle
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
(possibly degenerate). Given
k
k
k
and the area
S
S
S
of
△
A
B
C
\triangle ABC
△
A
BC
, compute the area of
△
A
1
B
1
C
1
\triangle A_1B_1C_1
△
A
1
B
1
C
1
.
1
1
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probability of passing an exam of 1 question (1 of n), taking the exam n times
An exam at a university consists of one question randomly selected from the
n
n
n
possible questions. A student knows only one question, but he can take the exam
n
n
n
times. Express as a function of
n
n
n
the probability
p
n
p_n
p
n
that the student will pass the exam. Does
p
n
p_n
p
n
increase or decrease as
n
n
n
increases? Compute
l
i
m
n
→
∞
p
n
lim_{n\to \infty}p_n
l
i
m
n
→
∞
p
n
. What is the largest lower bound of the probabilities
p
n
p_n
p
n
?
3
1
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spain 1989
Prove
1
10
2
<
1
2
3
4
5
6
.
.
.
99
100
<
1
10
\frac{1}{10\sqrt2}<\frac{1}{2}\frac{3}{4}\frac{5}{6}...\frac{99}{100}<\frac{1}{10}
10
2
1
<
2
1
4
3
6
5
...
100
99
<
10
1