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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
2000 Spain Mathematical Olympiad
2000 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(3)
2
2
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The probability that the persons meet [Spain 2000]
The figure shows a network of roads bounding
12
12
12
blocks. Person
P
P
P
goes from
A
A
A
to
B
,
B,
B
,
and person
Q
Q
Q
goes from
B
B
B
to
A
,
A,
A
,
each going by a shortest path (along roads). The persons start simultaneously and go at the same constant speed. At each point with two possible directions to take, both have the same probability. Find the probability that the persons meet.[asy] import graph; size(150); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; draw((0,3)--(4,3),linewidth(1.2pt)); draw((4,3)--(4,0),linewidth(1.2pt)); draw((4,0)--(0,0),linewidth(1.2pt)); draw((0,0)--(0,3),linewidth(1.2pt)); draw((1,3)--(1,0),linewidth(1.2pt)); draw((2,3)--(2,0),linewidth(1.2pt)); draw((3,3)--(3,0),linewidth(1.2pt)); draw((0,1)--(4,1),linewidth(1.2pt)); draw((4,2)--(0,2),linewidth(1.2pt)); dot((0,0),ds); label("
A
A
A
", (-0.3,-0.36),NE*lsf); dot((4,3),ds); label("
B
B
B
", (4.16,3.1),NE*lsf); clip((-4.3,-10.94)--(-4.3,6.3)--(16.18,6.3)--(16.18,-10.94)--cycle); [/asy]
Two points with distance at most 1 [Spain 2000]
Four points are given inside or on the boundary of a unit square. Prove that at least two of these points are on a mutual distance at most
1.
1.
1.
3
2
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There exists a point M depending on C1, C2 [Spain 2000]
Two circles
C
1
C_1
C
1
and
C
2
C_2
C
2
with the respective radii
r
1
r_1
r
1
and
r
2
r_2
r
2
intersect in
A
A
A
and
B
.
B.
B
.
A variable line
r
r
r
through
B
B
B
meets
C
1
C_1
C
1
and
C
2
C_2
C
2
again at
P
r
P_r
P
r
and
Q
r
Q_r
Q
r
respectively. Prove that there exists a point
M
,
M,
M
,
depending only on
C
1
C_1
C
1
and
C
2
,
C_2,
C
2
,
such that the perpendicular bisector of each segment
P
r
Q
r
P_rQ_r
P
r
Q
r
passes through
M
.
M.
M
.
There is no function f(f(n))=n+1 [Spain 2000] - Maybe old
Show that there is no function
f
:
N
→
N
f : \mathbb N \to \mathbb N
f
:
N
→
N
satisfying
f
(
f
(
n
)
)
=
n
+
1
f(f(n)) = n + 1
f
(
f
(
n
))
=
n
+
1
for each positive integer
n
.
n.
n
.
1
2
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Polynomials with common rootas [Spain 2000]
Consider the polynomials P(x) = x^4 + ax^3 + bx^2 + cx + 1 \text{and} Q(x) = x^4 + cx^3 + bx^2 + ax + 1. Find the conditions on the parameters
a
,
b
,
a, b,
a
,
b
,
c with
a
≠
c
a\neq c
a
=
c
for which
P
(
x
)
P(x)
P
(
x
)
and
Q
(
x
)
Q(x)
Q
(
x
)
have two common roots and, in such cases, solve the equations
P
(
x
)
=
0
P(x) = 0
P
(
x
)
=
0
and
Q
(
x
)
=
0.
Q(x) = 0.
Q
(
x
)
=
0.
Find the largest integer N [Spain 2000]
Find the largest integer
N
N
N
satisfying the following two conditions:(i)
[
N
3
]
\left[ \frac N3 \right]
[
3
N
]
consists of three equal digits;(ii)
[
N
3
]
=
1
+
2
+
3
+
⋯
+
n
\left[ \frac N3 \right] = 1 + 2 + 3 +\cdots + n
[
3
N
]
=
1
+
2
+
3
+
⋯
+
n
for some positive integer
n
.
n.
n
.