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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1993 Spain Mathematical Olympiad
1993 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(6)
6
1
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probability of wining a specific game in a casino
A game in a casino uses the diagram shown. At the start a ball appears at
S
S
S
. Each time the player presses a button, the ball moves to one of the adjacent letters with equal probability. The game ends when one of the following two things happens: (i) The ball returns to
S
S
S
, the player loses. (ii) The ball reaches
G
G
G
, the player wins. Find the probability that the player wins and the expected duration of a game.
2
1
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sort of pascal's triange
In the arithmetic triangle below each number (apart from those in the first row) is the sum of the two numbers immediately above.
0
1
2
3
4
.
.
.
1991
1992
1993
0 \, 1\, 2\, 3 \,4\, ... \,1991 \,1992\, 1993
0
1
2
3
4
...
1991
1992
1993
1
3
5
7
.
.
.
.
.
.
3983
3985
\,\,1\, 3\, 5 \,7\, ......\,\,\,\,3983 \,3985
1
3
5
7
......
3983
3985
4
8
12
.
.
.
.
.
.
.
.
.
.
7968
\,\,\,4 \,8 \,12\, .......... \,\,\,7968
4
8
12
..........
7968
······································· Prove that the bottom number is a multiple of
1993
1993
1993
.
5
1
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4 points in a grid 4x4x such that 6 pairwise distances of them are distinct
Given a 4×4 grid of points, the points at two opposite corners are denoted
A
A
A
and
D
D
D
. We need to choose two other points
B
B
B
and
C
C
C
such that the six pairwise distances of these four points are all distinct. (a) How many such quadruples of points are there? (b) How many such quadruples of points are non-congruent? (c) If each point is assigned a pair of coordinates
(
x
i
,
y
i
)
(x_i,y_i)
(
x
i
,
y
i
)
, prove that the sum of the expressions
∣
x
i
−
x
j
∣
+
∣
y
i
−
y
j
∣
|x_i-x_j |+|y_i-y_j|
∣
x
i
−
x
j
∣
+
∣
y
i
−
y
j
∣
over all six pairs of points in a quadruple is constant.
1
1
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201 people from 5 countries, in each group of 6 at least 2 have same age
There is a reunion of
201
201
201
people from
5
5
5
different countries. It is known that in each group of
6
6
6
people, at least two have the same age. Show that there must be at least
5
5
5
people with the same country, age and sex.
4
1
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infinitely many multiples of prime p of the form 1111...1, besides 2 and 5
Prove that for each prime number distinct from
2
2
2
and
5
5
5
there exist infinitely many multiples of
p
p
p
of the form
1111...1
1111...1
1111...1
.
3
1
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R> = 2ρ, where R , ρ are circumradius , inradius resp.
Prove that in every triangle the diameter of the incircle is not greater than the radius of the circumcircle.