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Problems
Contests
International Contests
Gulf Math Olympiad
2014 Gulf Math Olympiad
2014 Gulf Math Olympiad
Part of
Gulf Math Olympiad
Subcontests
(4)
2
1
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Ahmad and Salem play a game, sum and product of 2 numbers
Ahmad and Salem play the following game. Ahmad writes two integers (not necessarily different) on a board. Salem writes their sum and product. Ahmad does the same thing: he writes the sum and product of the two numbers which Salem has just written. They continue in this manner, not stopping unless the two players write the same two numbers one after the other (for then they are stuck!). The order of the two numbers which each player writes is not important. Thus if Ahmad starts by writing
3
3
3
and
−
2
-2
−
2
, the first five moves (or steps) are as shown: (a) Step 1 (Ahmad)
3
3
3
and
−
2
-2
−
2
(b) Step 2 (Salem)
1
1
1
and
−
6
-6
−
6
(c) Step 3 (Ahmad)
−
5
-5
−
5
and
−
6
-6
−
6
(d) Step 4 (Salem)
−
11
-11
−
11
and
30
30
30
(e) Step 5 (Ahmad)
19
19
19
and
−
330
-330
−
330
(i) Describe all pairs of numbers that Ahmad could write, and ensure that Salem must write the same numbers, and so the game stops at step 2. (ii) What pair of integers should Ahmad write so that the game finishes at step 4? (iii) Describe all pairs of integers which Ahmad could write at step 1, so that the game will finish after finitely many steps. (iv) Ahmad and Salem decide to change the game. The first player writes three numbers on the board,
u
,
v
u, v
u
,
v
and
w
w
w
. The second player then writes the three numbers
u
+
v
+
w
,
u
v
+
v
w
+
w
u
u + v + w,uv + vw + wu
u
+
v
+
w
,
uv
+
v
w
+
w
u
and
u
v
w
uvw
uv
w
, and they proceed as before, taking turns, and using this new rule describing how to work out the next three numbers. If Ahmad goes first, determine all collections of three numbers which he can write down, ensuring that Salem has to write the same three numbers at the next step.
3
1
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2 geometry problems, equal segments, right triangle, semicirle, angle wanted
(i)
A
B
C
ABC
A
BC
is a triangle with a right angle at
A
A
A
, and
P
P
P
is a point on the hypotenuse
B
C
BC
BC
. The line
A
P
AP
A
P
produced beyond
P
P
P
meets the line through
B
B
B
which is perpendicular to
B
C
BC
BC
at
U
U
U
. Prove that
B
U
=
B
A
BU = BA
B
U
=
B
A
if, and only if,
C
P
=
C
A
CP = CA
CP
=
C
A
. (ii)
A
A
A
is a point on the semicircle
C
B
CB
CB
, and points
X
X
X
and
Y
Y
Y
are on the line segment
B
C
BC
BC
. The line
A
X
AX
A
X
, produced beyond
X
X
X
, meets the line through
B
B
B
which is perpendicular to
B
C
BC
BC
at
U
U
U
. Also the line
A
Y
AY
A
Y
, produced beyond
Y
Y
Y
, meets the line through
C
C
C
which is perpendicular to
B
C
BC
BC
at
V
V
V
. Given that
B
Y
=
B
A
BY = BA
B
Y
=
B
A
and
C
X
=
C
A
CX = CA
CX
=
C
A
, determine the angle
∠
V
A
U
\angle VAU
∠
V
A
U
.
4
1
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112 numbers adding each 2 with a common edge in chessboard of 1-64
The numbers from
1
1
1
to
64
64
64
must be written on the small squares of a chessboard, with a different number in each small square. Consider the
112
112
112
numbers you can make by adding the numbers in two small squares which have a common edge. Is it possible to write the numbers in the squares so that these
112
112
112
sums are all different?
1
1
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GMO 2014 #1
A sequence
a
0
,
a
1
,
a
2
,
⋯
a_0,a_1,a_2,\cdots
a
0
,
a
1
,
a
2
,
⋯
satisfies the conditions
a
0
=
0
a_0 = 0
a
0
=
0
,
a
n
−
1
2
−
a
n
−
1
=
a
n
2
+
a
n
a_{n-1}^2 - a_{n-1} = a_n^2 + a_n
a
n
−
1
2
−
a
n
−
1
=
a
n
2
+
a
n
1) determine the two possible values of
a
1
a_1
a
1
. then determine all possible values of
a
2
a_2
a
2
. 2)for each
n
n
n
, prove that
a
n
+
1
=
a
n
+
1
a_{n+1}=a_n+1
a
n
+
1
=
a
n
+
1
or
a
n
+
1
=
−
a
n
a_{n+1} = -a_n
a
n
+
1
=
−
a
n
3)Describe the possible values of
a
1435
a_{1435}
a
1435
4)Prove that the values that you got in (3) are correct