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Contests
International Contests
Gulf Math Olympiad
2013 Gulf Math Olympiad
2013 Gulf Math Olympiad
Part of
Gulf Math Olympiad
Subcontests
(4)
4
1
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Gulf Mathematical Olympiad 2013 - Problem 4
Let
m
,
n
m,n
m
,
n
be integers. It is known that there are integers
a
,
b
a,b
a
,
b
such that
a
m
+
b
n
=
1
am+bn=1
am
+
bn
=
1
if, and only if, the greatest common divisor of
m
,
n
m,n
m
,
n
is 1. You are not required to prove this.Now suppose that
p
,
q
p,q
p
,
q
are different odd primes. In each case determine if there are integers
a
,
b
a,b
a
,
b
such that
a
p
+
b
q
=
1
ap+bq=1
a
p
+
b
q
=
1
so that the given condition is satisfied:a.
p
p
p
divides
b
b
b
and
q
q
q
divides
a
a
a
; b.
p
p
p
divides
a
a
a
and
q
q
q
divides
b
b
b
; c.
p
p
p
does not divide
a
a
a
and
q
q
q
does not divide
b
b
b
.
3
1
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Gulf Mathematical Olympiad 2013 - Problem 3
There are
n
n
n
people standing on a circular track. We want to perform a number of moves so that we end up with a situation where the distance between every two neighbours is the same. The move that is allowed consists in selecting two people and asking one of them to walk a distance
d
d
d
on the circular track clockwise, and asking the other to walk the same distance on the track anticlockwise. The two people selected and the quantity
d
d
d
can vary from move to move.Prove that it is possible to reach the desired situation (where the distance between every two neighbours is the same) after at most
n
−
1
n-1
n
−
1
moves.
2
1
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Gulf Mathematical Olympiad 2013 - Problem 2
In triangle
A
B
C
ABC
A
BC
, the bisector of angle
B
B
B
meets the opposite side
A
C
AC
A
C
at
B
′
B'
B
′
. Similarly, the bisector of angle
C
C
C
meets the opposite side
A
B
AB
A
B
at
C
′
C'
C
′
. Prove that
A
=
6
0
∘
A=60^{\circ}
A
=
6
0
∘
if, and only if,
B
C
′
+
C
B
′
=
B
C
BC'+CB'=BC
B
C
′
+
C
B
′
=
BC
.
1
1
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Gulf Mathematical Olympiad 2013 - Problem 1
Let
a
1
,
a
2
,
…
,
a
2
n
a_1,a_2,\ldots,a_{2n}
a
1
,
a
2
,
…
,
a
2
n
be positive real numbers such that
a
j
a
n
+
j
=
1
a_ja_{n+j}=1
a
j
a
n
+
j
=
1
for the values
j
=
1
,
2
,
…
,
n
j=1,2,\ldots,n
j
=
1
,
2
,
…
,
n
. a. Prove that either the average of the numbers
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
is at least 1 or the average of the numbers
a
n
+
1
,
a
n
+
2
,
…
,
a
2
n
a_{n+1},a_{n+2},\ldots,a_{2n}
a
n
+
1
,
a
n
+
2
,
…
,
a
2
n
is at least 1.b. Assuming that
n
≥
2
n\ge2
n
≥
2
, prove that there exist two distinct numbers
j
,
k
j,k
j
,
k
in the set
{
1
,
2
,
…
,
2
n
}
\{1,2,\ldots,2n\}
{
1
,
2
,
…
,
2
n
}
such that
∣
a
j
−
a
k
∣
<
1
n
−
1
.
|a_j-a_k|<\frac{1}{n-1}.
∣
a
j
−
a
k
∣
<
n
−
1
1
.