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Problems
Contests
International Contests
Gulf Math Olympiad
2017 Gulf Math Olympiad
2017 Gulf Math Olympiad
Part of
Gulf Math Olympiad
Subcontests
(4)
4
1
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max power of 2 that divides \lceil(1+\sqrt{3})^{2n}\rceil for pos. integer n
1 - Prove that
55
<
(
1
+
3
)
4
<
56
55 < (1+\sqrt{3})^4 < 56
55
<
(
1
+
3
)
4
<
56
.2 - Find the largest power of
2
2
2
that divides
⌈
(
1
+
3
)
2
n
⌉
\lceil(1+\sqrt{3})^{2n}\rceil
⌈(
1
+
3
)
2
n
⌉
for the positive integer
n
n
n
3
1
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so much circles
Let
C
1
C_1
C
1
and
C
2
C_2
C
2
be two different circles , and let their radii be
r
1
r_1
r
1
and
r
2
r_2
r
2
, the two circles are passing through the two points
A
A
A
and
B
B
B
(i)Let
P
1
P_1
P
1
on
C
1
C_1
C
1
and
P
2
P_2
P
2
on
C
2
C_2
C
2
such that the line
P
1
P
2
P_1P_2
P
1
P
2
passes through
A
A
A
. Prove that
P
1
B
⋅
r
2
=
P
2
B
⋅
r
1
P_1B \cdot r_2 = P_2B \cdot r_1
P
1
B
⋅
r
2
=
P
2
B
⋅
r
1
(ii)Let
D
E
F
DEF
D
EF
be a triangle that it's inscribed in
C
1
C_1
C
1
, and let
D
′
E
′
F
′
D'E'F'
D
′
E
′
F
′
be a triangle that is inscribed in
C
2
C_2
C
2
. The lines
E
E
′
EE'
E
E
′
,
D
D
′
DD'
D
D
′
and
F
F
′
FF'
F
F
′
all pass through
A
A
A
. Prove that the triangles
D
E
F
DEF
D
EF
and
D
′
E
′
F
′
D'E'F'
D
′
E
′
F
′
are similar(iii)The circle
C
3
C_3
C
3
also passes through
A
A
A
and
B
B
B
. Let
l
l
l
be a line that passes through
A
A
A
and cuts circles
C
i
C_i
C
i
in
M
i
M_i
M
i
with
i
=
1
,
2
,
3
i = 1,2,3
i
=
1
,
2
,
3
. Prove that the value of
M
1
M
2
M
1
M
3
\frac{M_1M_2}{M_1M_3}
M
1
M
3
M
1
M
2
is constant regardless of the position of
l
l
l
Provided that
l
l
l
is different from
A
B
AB
A
B
2
1
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GMO 2017 #2
One country consists of islands
A
1
,
A
2
,
⋯
,
A
N
A_1,A_2,\cdots,A_N
A
1
,
A
2
,
⋯
,
A
N
,The ministry of transport decided to build some bridges such that anyone will can travel by car from any of the islands
A
1
,
A
2
,
⋯
,
A
N
A_1,A_2,\cdots,A_N
A
1
,
A
2
,
⋯
,
A
N
to any another island by one or more of these bridges. For technical reasons the only bridges that can be built is between
A
i
A_i
A
i
and
A
i
+
1
A_{i+1}
A
i
+
1
where
i
=
1
,
2
,
⋯
,
N
−
1
i = 1,2,\cdots,N-1
i
=
1
,
2
,
⋯
,
N
−
1
, and between
A
i
A_i
A
i
and
A
N
A_N
A
N
where
i
<
N
i<N
i
<
N
.We say that a plan to build some bridges is good if it is satisfies the above conditions , but when we remove any bridge it will not satisfy this conditions. We assume that there is
a
N
a_N
a
N
of good plans. Observe that
a
1
=
1
a_1 = 1
a
1
=
1
(The only good plan is to not build any bridge) , and
a
2
=
1
a_2 = 1
a
2
=
1
(We build one bridge).1-Prove that
a
3
=
3
a_3 = 3
a
3
=
3
2-Draw at least
5
5
5
different good plans in the case that
N
=
4
N=4
N
=
4
and the islands are the vertices of a square 3-Compute
a
4
a_4
a
4
4-Compute
a
6
a_6
a
6
5-Prove that there is a positive integer
i
i
i
such that
1438
1438
1438
divides
a
i
a_i
a
i
1
1
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GMO 2017 #1
1- Find a pair
(
m
,
n
)
(m,n)
(
m
,
n
)
of positive integers such that
K
=
∣
2
m
−
3
n
∣
K = |2^m-3^n|
K
=
∣
2
m
−
3
n
∣
in all of this cases :
a
)
K
=
5
a) K=5
a
)
K
=
5
b
)
K
=
11
b) K=11
b
)
K
=
11
c
)
K
=
19
c) K=19
c
)
K
=
19
2-Is there a pair
(
m
,
n
)
(m,n)
(
m
,
n
)
of positive integers such that :
∣
2
m
−
3
n
∣
=
2017
|2^m-3^n| = 2017
∣
2
m
−
3
n
∣
=
2017
3-Every prime number less than
41
41
41
can be represented in the form
∣
2
m
−
3
n
∣
|2^m-3^n|
∣
2
m
−
3
n
∣
by taking an Appropriate pair
(
m
,
n
)
(m,n)
(
m
,
n
)
of positive integers. Prove that the number
41
41
41
cannot be represented in the form
∣
2
m
−
3
n
∣
|2^m-3^n|
∣
2
m
−
3
n
∣
where
m
m
m
and
n
n
n
are positive integers4-Note that
2
5
+
3
2
=
41
2^5+3^2=41
2
5
+
3
2
=
41
. The number
53
53
53
is the least prime number that cannot be represented as a sum or an difference of a power of
2
2
2
and a power of
3
3
3
. Prove that the number
53
53
53
cannot be represented in any of the forms
2
m
−
3
n
2^m-3^n
2
m
−
3
n
,
3
n
−
2
m
3^n-2^m
3
n
−
2
m
,
2
m
−
3
n
2^m-3^n
2
m
−
3
n
where
m
m
m
and
n
n
n
are positive integers