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Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2008 National Olympiad First Round
2008 National Olympiad First Round
Part of
National Olympiad First Round
Subcontests
(36)
33
1
Hide problems
Turkey NMO 2008 1st Round - P33 (Geometry)
Let
E
E
E
be a point inside the rhombus
A
B
C
D
ABCD
A
BC
D
such that
∣
A
E
∣
=
∣
E
B
∣
|AE|=|EB|
∣
A
E
∣
=
∣
EB
∣
,
m
(
E
A
B
^
)
=
1
2
∘
m(\widehat{EAB})=12^\circ
m
(
E
A
B
)
=
1
2
∘
, and
m
(
D
A
E
^
)
=
7
2
∘
m(\widehat{DAE})=72^\circ
m
(
D
A
E
)
=
7
2
∘
. What is
m
(
C
D
E
^
)
m(\widehat{CDE})
m
(
C
D
E
)
in degrees?
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64
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66
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>
(
C
)
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68
<
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a
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s
=
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x
−
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′
>
(
D
)
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70
<
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)
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72
<span class='latex-bold'>(A)</span>\ 64 \qquad<span class='latex-bold'>(B)</span>\ 66 \qquad<span class='latex-bold'>(C)</span>\ 68 \qquad<span class='latex-bold'>(D)</span>\ 70 \qquad<span class='latex-bold'>(E)</span>\ 72
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64
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(
B
)
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66
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(
C
)
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68
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x
−
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′
>
(
D
)
<
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70
<
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a
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=
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x
−
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>
(
E
)
<
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>
72
34
1
Hide problems
Turkey NMO 2008 1st Round - P34 (Number Theory)
We call a positive integer "special" if the number is divided by all of its digits (except the
0
0
0
s). At most how many consequtive special numbers are there?
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9
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10
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12
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(
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)
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13
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14
<span class='latex-bold'>(A)</span>\ 9 \qquad<span class='latex-bold'>(B)</span>\ 10 \qquad<span class='latex-bold'>(C)</span>\ 12 \qquad<span class='latex-bold'>(D)</span>\ 13 \qquad<span class='latex-bold'>(E)</span>\ 14
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)
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9
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(
B
)
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10
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12
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(
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)
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13
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>
(
E
)
<
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>
14
35
1
Hide problems
Turkey NMO 2008 1st Round - P35 (Algebra)
What is the least real value of the expression
x
2
−
6
x
+
13
+
x
2
−
14
x
+
58
\sqrt{x^2-6x+13} + \sqrt{x^2-14x+58}
x
2
−
6
x
+
13
+
x
2
−
14
x
+
58
where
x
x
x
is a real number?
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)
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39
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6
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43
6
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D
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2
2
+
13
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E
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None of the above
<span class='latex-bold'>(A)</span>\ \sqrt {39} \qquad<span class='latex-bold'>(B)</span>\ 6 \qquad<span class='latex-bold'>(C)</span>\ \frac {43}6 \qquad<span class='latex-bold'>(D)</span>\ 2\sqrt 2 + \sqrt {13} \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
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39
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(
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)
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6
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6
43
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(
D
)
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2
2
+
13
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−
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>
(
E
)
<
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>
None of the above
36
1
Hide problems
Turkey NMO 2008 1st Round - P36 (Combinatorics)
There is a white table with a pile of
2008
2008
2008
coins and there are two empty black tables. At each move, the uppermost coin on a table is transferred to an empty table or to the top of the pile on a non-empty table. What is the least number of moves required to reverse the pile at the beginning on the white table?
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6016
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)
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6017
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6022
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6023
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6024
<span class='latex-bold'>(A)</span>\ 6016 \qquad<span class='latex-bold'>(B)</span>\ 6017 \qquad<span class='latex-bold'>(C)</span>\ 6022 \qquad<span class='latex-bold'>(D)</span>\ 6023 \qquad<span class='latex-bold'>(E)</span>\ 6024
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6016
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>
(
B
)
<
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>
6017
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C
)
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6022
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(
D
)
<
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6023
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(
E
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6024
32
1
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Turkey NMO 2008 1st Round - P32 (Combinatorics)
At a party with
n
≥
4
n\geq 4
n
≥
4
people, if every
3
3
3
people have exactly
1
1
1
common friend, how many different values can
n
n
n
take?
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A
)
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1
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>
(
B
)
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2
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>
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C
)
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>
4
<
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−
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>
(
D
)
<
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>
Infinitely many
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None of the above
<span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 4 \qquad<span class='latex-bold'>(D)</span>\ \text{Infinitely many} \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
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(
B
)
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2
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(
C
)
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4
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(
D
)
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Infinitely many
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−
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d
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E
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None of the above
31
1
Hide problems
Turkey NMO 2008 1st Round - P31 (Algebra)
If the inequality
(
(
x
+
y
)
2
+
4
)
(
(
x
+
y
)
2
−
2
)
≥
A
⋅
(
x
−
y
)
2
((x+y)^2+4)((x+y)^2-2)\geq A\cdot (x-y)^2
((
x
+
y
)
2
+
4
)
((
x
+
y
)
2
−
2
)
≥
A
⋅
(
x
−
y
)
2
is hold for every real numbers
x
,
y
x,y
x
,
y
such that
x
y
=
1
xy=1
x
y
=
1
, what is the largest value of
A
A
A
?
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12
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(
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14
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16
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D
)
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18
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20
<span class='latex-bold'>(A)</span>\ 12 \qquad<span class='latex-bold'>(B)</span>\ 14 \qquad<span class='latex-bold'>(C)</span>\ 16 \qquad<span class='latex-bold'>(D)</span>\ 18 \qquad<span class='latex-bold'>(E)</span>\ 20
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)
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12
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(
B
)
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14
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(
C
)
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16
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(
D
)
<
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18
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(
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)
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20
30
1
Hide problems
Turkey NMO 2008 1st Round - P30 (Number Theory)
In a sequence with the first term is positive integer, the next term is generated by adding the previous term and its largest digit. At most how many consequtive terms of this sequence are odd?
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3
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x
−
b
o
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d
′
>
(
C
)
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4
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(
D
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5
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6
<span class='latex-bold'>(A)</span>\ 2 \qquad<span class='latex-bold'>(B)</span>\ 3 \qquad<span class='latex-bold'>(C)</span>\ 4 \qquad<span class='latex-bold'>(D)</span>\ 5 \qquad<span class='latex-bold'>(E)</span>\ 6
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2
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(
B
)
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3
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4
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(
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5
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(
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6
29
1
Hide problems
Turkey NMO 2008 1st Round - P29 (Geometry)
[
A
B
]
[AB]
[
A
B
]
and
[
C
D
]
[CD]
[
C
D
]
are not parallel in the convex quadrilateral
A
B
C
D
ABCD
A
BC
D
. Let
E
E
E
and
F
F
F
be the midpoints of
[
A
D
]
[AD]
[
A
D
]
and
[
B
C
]
[BC]
[
BC
]
, respectively. If
∣
C
D
∣
=
12
|CD|=12
∣
C
D
∣
=
12
,
∣
A
B
∣
=
22
|AB|=22
∣
A
B
∣
=
22
, and
∣
E
F
∣
=
x
|EF|=x
∣
EF
∣
=
x
, what is the sum of integer values of
x
x
x
?
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(
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110
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(
B
)
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114
<
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(
C
)
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118
<
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D
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121
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None of the above
<span class='latex-bold'>(A)</span>\ 110 \qquad<span class='latex-bold'>(B)</span>\ 114 \qquad<span class='latex-bold'>(C)</span>\ 118 \qquad<span class='latex-bold'>(D)</span>\ 121 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
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A
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110
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(
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114
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118
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(
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121
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(
E
)
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None of the above
28
1
Hide problems
Turkey NMO 2008 1st Round - P28 (Combinatorics)
A unit square from one of the corners of a
8
×
8
8\times 8
8
×
8
chessboard is cut and thrown. At least how many triangles are necessary to divide the new board into triangles with equal areas?
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17
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(
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)
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19
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20
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(
D
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21
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None of the above
<span class='latex-bold'>(A)</span>\ 17 \qquad<span class='latex-bold'>(B)</span>\ 19 \qquad<span class='latex-bold'>(C)</span>\ 20 \qquad<span class='latex-bold'>(D)</span>\ 21 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
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17
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(
B
)
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19
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20
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(
D
)
<
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21
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−
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E
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None of the above
27
1
Hide problems
Turkey NMO 2008 1st Round - P27 (Algebra)
The angles
α
,
β
,
γ
\alpha, \beta, \gamma
α
,
β
,
γ
of a triangle are in arithmetic progression. If
sin
20
α
\sin 20\alpha
sin
20
α
,
sin
20
β
\sin 20\beta
sin
20
β
, and
sin
20
γ
\sin 20\gamma
sin
20
γ
are in arithmetic progression, how many different values can
α
\alpha
α
take?
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(
A
)
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1
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(
B
)
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2
<
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s
s
=
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t
e
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−
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>
(
C
)
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3
<
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a
s
s
=
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x
−
b
o
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>
(
D
)
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>
4
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>
(
E
)
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None of the above
<span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 4 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
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)
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1
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=
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(
B
)
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2
<
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−
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(
C
)
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3
<
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a
ss
=
′
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a
t
e
x
−
b
o
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d
′
>
(
D
)
<
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4
<
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p
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c
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a
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=
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−
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d
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>
(
E
)
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None of the above
26
1
Hide problems
Turkey NMO 2008 1st Round - P26 (Number Theory)
Let
A
=
2
2
+
3
⋅
2
+
1
3
!
⋅
4
!
+
3
2
+
3
⋅
3
+
1
4
!
⋅
5
!
+
4
2
+
3
⋅
4
+
1
5
!
⋅
6
!
+
⋯
+
1
0
2
+
3
⋅
10
+
1
11
!
⋅
12
!
A=\frac{2^2+3\cdot 2 + 1}{3! \cdot 4!} + \frac{3^2+3\cdot 3 + 1}{4! \cdot 5!} + \frac{4^2+3\cdot 4 + 1}{5! \cdot 6!} + \dots + \frac{10^2+3\cdot 10 + 1}{11! \cdot 12!}
A
=
3
!
⋅
4
!
2
2
+
3
⋅
2
+
1
+
4
!
⋅
5
!
3
2
+
3
⋅
3
+
1
+
5
!
⋅
6
!
4
2
+
3
⋅
4
+
1
+
⋯
+
11
!
⋅
12
!
1
0
2
+
3
⋅
10
+
1
. What is the remainder when
11
!
⋅
12
!
⋅
A
11!\cdot 12! \cdot A
11
!
⋅
12
!
⋅
A
is divided by
11
11
11
?
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)
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0
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(
B
)
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1
<
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c
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a
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s
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−
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(
C
)
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>
5
<
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a
s
s
=
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x
−
b
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(
D
)
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8
<
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10
<span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ 5 \qquad<span class='latex-bold'>(D)</span>\ 8 \qquad<span class='latex-bold'>(E)</span>\ 10
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(
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)
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=
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>
(
B
)
<
/
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1
<
s
p
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c
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a
ss
=
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x
−
b
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(
C
)
<
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5
<
s
p
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c
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a
ss
=
′
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a
t
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x
−
b
o
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d
′
>
(
D
)
<
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8
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10
25
1
Hide problems
Turkey NMO 2008 1st Round - P25 (Geometry)
Let
C
C
C
and
D
D
D
be points on the circle with center
O
O
O
and diameter
[
A
B
]
[AB]
[
A
B
]
where
C
C
C
and
D
D
D
are on different semicircles with diameter
[
A
B
]
[AB]
[
A
B
]
. Let
H
H
H
be the foot perpendicular from
B
B
B
to
[
C
D
]
[CD]
[
C
D
]
. If
∣
A
O
∣
=
13
|AO|=13
∣
A
O
∣
=
13
,
∣
A
C
∣
=
24
|AC|=24
∣
A
C
∣
=
24
, and
∣
H
D
∣
=
12
|HD|=12
∣
HD
∣
=
12
, what is
D
C
B
^
\widehat{DCB}
D
CB
in degrees?
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)
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30
<
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−
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(
B
)
<
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>
45
<
s
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c
l
a
s
s
=
′
l
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x
−
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>
(
C
)
<
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>
60
<
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c
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a
s
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=
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(
D
)
<
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>
75
<
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(
E
)
<
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80
<span class='latex-bold'>(A)</span>\ 30 \qquad<span class='latex-bold'>(B)</span>\ 45 \qquad<span class='latex-bold'>(C)</span>\ 60 \qquad<span class='latex-bold'>(D)</span>\ 75 \qquad<span class='latex-bold'>(E)</span>\ 80
<
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)
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30
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(
B
)
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45
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c
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a
ss
=
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a
t
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x
−
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d
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>
(
C
)
<
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60
<
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p
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c
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a
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=
′
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a
t
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x
−
b
o
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d
′
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(
D
)
<
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>
75
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p
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a
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=
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x
−
b
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>
(
E
)
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80
24
1
Hide problems
Turkey NMO 2008 1st Round - P24 (Combinatorics)
How many of the numbers
a
1
⋅
5
1
+
a
2
⋅
5
2
+
a
3
⋅
5
3
+
a
4
⋅
5
4
+
a
5
⋅
5
5
+
a
6
⋅
5
6
a_1\cdot 5^1+a_2\cdot 5^2+a_3\cdot 5^3+a_4\cdot 5^4+a_5\cdot 5^5+a_6\cdot 5^6
a
1
⋅
5
1
+
a
2
⋅
5
2
+
a
3
⋅
5
3
+
a
4
⋅
5
4
+
a
5
⋅
5
5
+
a
6
⋅
5
6
are negative if
a
1
,
a
2
,
a
3
,
a
4
,
a
5
,
a
6
∈
{
−
1
,
0
,
1
}
a_1,a_2,a_3,a_4,a_5,a_6 \in \{-1,0,1 \}
a
1
,
a
2
,
a
3
,
a
4
,
a
5
,
a
6
∈
{
−
1
,
0
,
1
}
?
<
s
p
a
n
c
l
a
s
s
=
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l
a
t
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x
−
b
o
l
d
′
>
(
A
)
<
/
s
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a
n
>
121
<
s
p
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n
c
l
a
s
s
=
′
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a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
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a
n
>
224
<
s
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n
c
l
a
s
s
=
′
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a
t
e
x
−
b
o
l
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′
>
(
C
)
<
/
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a
n
>
275
<
s
p
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c
l
a
s
s
=
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t
e
x
−
b
o
l
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>
(
D
)
<
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>
364
<
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>
(
E
)
<
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>
375
<span class='latex-bold'>(A)</span>\ 121 \qquad<span class='latex-bold'>(B)</span>\ 224 \qquad<span class='latex-bold'>(C)</span>\ 275 \qquad<span class='latex-bold'>(D)</span>\ 364 \qquad<span class='latex-bold'>(E)</span>\ 375
<
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(
A
)
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>
121
<
s
p
an
c
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a
ss
=
′
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a
t
e
x
−
b
o
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d
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>
(
B
)
<
/
s
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>
224
<
s
p
an
c
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a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
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an
>
275
<
s
p
an
c
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a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
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an
>
364
<
s
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an
c
l
a
ss
=
′
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a
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e
x
−
b
o
l
d
′
>
(
E
)
<
/
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an
>
375
23
1
Hide problems
Turkey NMO 2008 1st Round - P23 (Algebra)
If
a
2
+
b
2
+
c
2
+
d
2
−
a
b
−
b
c
−
c
d
−
d
+
2
5
=
0
a^2+b^2+c^2+d^2-ab-bc-cd-d+\frac 25 = 0
a
2
+
b
2
+
c
2
+
d
2
−
ab
−
b
c
−
c
d
−
d
+
5
2
=
0
where
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
are real numbers, what is
a
a
a
?
<
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c
l
a
s
s
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−
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(
A
)
<
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>
2
3
<
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p
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c
l
a
s
s
=
′
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a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
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>
2
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
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a
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>
3
2
<
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p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
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s
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a
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>
1
5
<
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−
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>
(
E
)
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>
None of the above
<span class='latex-bold'>(A)</span>\ \frac 23 \qquad<span class='latex-bold'>(B)</span>\ \frac {\sqrt 2} 3 \qquad<span class='latex-bold'>(C)</span>\ \frac {\sqrt 3} 2 \qquad<span class='latex-bold'>(D)</span>\ \frac 15 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
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(
A
)
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3
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
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>
(
B
)
<
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3
2
<
s
p
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c
l
a
ss
=
′
l
a
t
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x
−
b
o
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d
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>
(
C
)
<
/
s
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an
>
2
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
5
1
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c
l
a
ss
=
′
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a
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e
x
−
b
o
l
d
′
>
(
E
)
<
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>
None of the above
22
1
Hide problems
Turkey NMO 2008 1st Round - P22 (Number Theory)
How many pairs of postive integers
(
a
,
b
)
(a,b)
(
a
,
b
)
with
a
≥
b
a\geq b
a
≥
b
are there such that
a
2
+
b
2
a^2+b^2
a
2
+
b
2
divides both
a
3
+
b
a^3+b
a
3
+
b
and
a
+
b
3
a+b^3
a
+
b
3
?
<
s
p
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c
l
a
s
s
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a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
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a
n
>
0
<
s
p
a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
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a
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>
1
<
s
p
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c
l
a
s
s
=
′
l
a
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−
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o
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>
(
C
)
<
/
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>
2
<
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c
l
a
s
s
=
′
l
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−
b
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>
(
D
)
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>
3
<
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c
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a
s
s
=
′
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a
t
e
x
−
b
o
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d
′
>
(
E
)
<
/
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p
a
n
>
Infinitely many
<span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ 3 \qquad<span class='latex-bold'>(E)</span>\ \text{Infinitely many}
<
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p
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c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
0
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
1
<
s
p
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c
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a
ss
=
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l
a
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−
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o
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d
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>
(
C
)
<
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>
2
<
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a
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=
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o
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>
(
D
)
<
/
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>
3
<
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c
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a
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=
′
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t
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x
−
b
o
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d
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>
(
E
)
<
/
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>
Infinitely many
21
1
Hide problems
Turkey NMO 2008 1st Round - P21 (Geometry)
Let
A
B
C
ABC
A
BC
be a right triangle with
m
(
A
^
)
=
9
0
∘
m(\widehat{A})=90^\circ
m
(
A
)
=
9
0
∘
. Let
A
P
Q
R
APQR
A
PQR
be a square with area
9
9
9
such that
P
∈
[
A
C
]
P\in [AC]
P
∈
[
A
C
]
,
Q
∈
[
B
C
]
Q\in [BC]
Q
∈
[
BC
]
,
R
∈
[
A
B
]
R\in [AB]
R
∈
[
A
B
]
. Let
K
L
M
N
KLMN
K
L
MN
be a square with area
8
8
8
such that
N
,
K
∈
[
B
C
]
N,K\in [BC]
N
,
K
∈
[
BC
]
,
M
∈
[
A
B
]
M\in [AB]
M
∈
[
A
B
]
, and
L
∈
[
A
C
]
L\in [AC]
L
∈
[
A
C
]
. What is
∣
A
B
∣
+
∣
A
C
∣
|AB|+|AC|
∣
A
B
∣
+
∣
A
C
∣
?
<
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
A
)
<
/
s
p
a
n
>
8
<
s
p
a
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c
l
a
s
s
=
′
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a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
10
<
s
p
a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
12
<
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p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
D
)
<
/
s
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a
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>
14
<
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c
l
a
s
s
=
′
l
a
t
e
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−
b
o
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d
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>
(
E
)
<
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16
<span class='latex-bold'>(A)</span>\ 8 \qquad<span class='latex-bold'>(B)</span>\ 10 \qquad<span class='latex-bold'>(C)</span>\ 12 \qquad<span class='latex-bold'>(D)</span>\ 14 \qquad<span class='latex-bold'>(E)</span>\ 16
<
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c
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a
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′
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a
t
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x
−
b
o
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d
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>
(
A
)
<
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s
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>
8
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
10
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
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>
12
<
s
p
an
c
l
a
ss
=
′
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a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
14
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
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an
>
16
20
1
Hide problems
Turkey NMO 2008 1st Round - P20 (Combinatorics)
Each of the integers
a
1
,
a
2
,
a
3
,
…
,
a
2008
a_1,a_2,a_3,\dots,a_{2008}
a
1
,
a
2
,
a
3
,
…
,
a
2008
is at least
1
1
1
and at most
5
5
5
. If
a
n
<
a
n
+
1
a_n < a_{n+1}
a
n
<
a
n
+
1
, the pair
(
a
n
,
a
n
+
1
)
(a_n, a_{n+1})
(
a
n
,
a
n
+
1
)
will be called as an increasing pair. If
a
n
>
a
n
+
1
a_n > a_{n+1}
a
n
>
a
n
+
1
, the pair
(
a
n
,
a
n
+
1
)
(a_n, a_{n+1})
(
a
n
,
a
n
+
1
)
will be called as an decreasing pair. If the sequence contains
103
103
103
increasing pairs, at least how many decreasing pairs are there?
<
s
p
a
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c
l
a
s
s
=
′
l
a
t
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x
−
b
o
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>
(
A
)
<
/
s
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a
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>
21
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
24
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
36
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
102
<
s
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
E
)
<
/
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a
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>
None of the above
<span class='latex-bold'>(A)</span>\ 21 \qquad<span class='latex-bold'>(B)</span>\ 24 \qquad<span class='latex-bold'>(C)</span>\ 36 \qquad<span class='latex-bold'>(D)</span>\ 102 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
<
s
p
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c
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a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
21
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
24
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
36
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
102
<
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p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the above
18
1
Hide problems
Turkey NMO 2008 1st Round - P18 (Number Theory)
How many positive integers
n
n
n
are there such that
n
+
n
+
n
+
n
\sqrt{n+\sqrt{n+\sqrt{n+\sqrt{n}}}}
n
+
n
+
n
+
n
is an integer?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
Infinitely many
<
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p
a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
E
)
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a
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>
None of the above
<span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ \text{Infinitely many} \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
<
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p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
Infinitely many
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the above
19
1
Hide problems
Turkey NMO 2008 1st Round - P19 (Algebra)
Let
f
:
(
0
,
∞
)
→
(
0
,
∞
)
f:(0,\infty) \rightarrow (0,\infty)
f
:
(
0
,
∞
)
→
(
0
,
∞
)
be a function such that
10
⋅
x
+
y
x
y
=
f
(
x
)
⋅
f
(
y
)
−
f
(
x
y
)
−
90
10\cdot \frac{x+y}{xy}=f(x)\cdot f(y)-f(xy)-90
10
⋅
x
y
x
+
y
=
f
(
x
)
⋅
f
(
y
)
−
f
(
x
y
)
−
90
for every
x
,
y
∈
(
0
,
∞
)
x,y \in (0,\infty)
x
,
y
∈
(
0
,
∞
)
. What is
f
(
1
11
)
f(\frac 1{11})
f
(
11
1
)
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
11
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
21
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
31
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
There is more than one solution
<span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 11 \qquad<span class='latex-bold'>(C)</span>\ 21 \qquad<span class='latex-bold'>(D)</span>\ 31 \qquad<span class='latex-bold'>(E)</span>\ \text{There is more than one solution}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
11
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
21
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
31
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
There is more than one solution
17
1
Hide problems
Turkey NMO 2008 1st Round - P17 (Geometry)
Let the vertices
A
A
A
and
C
C
C
of a right triangle
A
B
C
ABC
A
BC
be on the arc with center
B
B
B
and radius
20
20
20
. A semicircle with diameter
[
A
B
]
[AB]
[
A
B
]
is drawn to the inner region of the arc. The tangent from
C
C
C
to the semicircle touches the semicircle at
D
D
D
other than
B
B
B
. Let
C
D
CD
C
D
intersect the arc at
F
F
F
. What is
∣
F
D
∣
|FD|
∣
F
D
∣
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
5
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
5
<span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ \frac 52 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 4 \qquad<span class='latex-bold'>(E)</span>\ 5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
5
16
1
Hide problems
Turkey NMO 2008 1st Round - P16 (Combinatorics)
A class of
50
50
50
students took an exam with
4
4
4
questions. At least
1
1
1
of any
40
40
40
students gave exactly
3
3
3
, at least
2
2
2
of any
40
40
40
gave exactly
2
2
2
, and at least
3
3
3
of any
40
40
40
gave exactly
1
1
1
correct answers. At least
4
4
4
of any
40
40
40
students gave exactly
4
4
4
wrong answers. What is the least number of students who gave an odd number of correct answers?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
18
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
24
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
26
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
28
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
None of the above
<span class='latex-bold'>(A)</span>\ 18 \qquad<span class='latex-bold'>(B)</span>\ 24 \qquad<span class='latex-bold'>(C)</span>\ 26 \qquad<span class='latex-bold'>(D)</span>\ 28 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
18
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
24
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
26
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
28
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the above
15
1
Hide problems
Turkey NMO 2008 1st Round - P15 (Algebra)
Let the sequence
(
a
n
)
(a_n)
(
a
n
)
be defined as
a
1
=
1
3
a_1=\frac 13
a
1
=
3
1
and
a
n
+
1
=
a
n
1
+
13
a
n
2
a_{n+1}=\frac{a_n}{\sqrt{1+13a_n^2}}
a
n
+
1
=
1
+
13
a
n
2
a
n
for every
n
≥
1
n\geq 1
n
≥
1
. If
a
k
a_k
a
k
is the largest term of the sequence satisfying
a
k
<
1
50
a_k < \frac 1{50}
a
k
<
50
1
, what is
k
k
k
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
194
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
193
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
192
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
191
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
None of the above
<span class='latex-bold'>(A)</span>\ 194 \qquad<span class='latex-bold'>(B)</span>\ 193 \qquad<span class='latex-bold'>(C)</span>\ 192 \qquad<span class='latex-bold'>(D)</span>\ 191 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
194
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
193
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
192
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
191
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the above
14
1
Hide problems
Turkey NMO 2008 1st Round - P14 (Number Theory)
What is the last three digits of
4
9
303
⋅
399
3
202
⋅
3
9
606
49^{303}\cdot 3993^{202}\cdot 39^{606}
4
9
303
⋅
399
3
202
⋅
3
9
606
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
001
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
081
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
561
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
721
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
961
<span class='latex-bold'>(A)</span>\ 001 \qquad<span class='latex-bold'>(B)</span>\ 081 \qquad<span class='latex-bold'>(C)</span>\ 561 \qquad<span class='latex-bold'>(D)</span>\ 721 \qquad<span class='latex-bold'>(E)</span>\ 961
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
001
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
081
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
561
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
721
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
961
13
1
Hide problems
Turkey NMO 2008 1st Round - P13 (Geometry)
Let
A
B
C
ABC
A
BC
be a triangle such that angle
C
C
C
is obtuse. Let
D
∈
[
A
B
]
D\in [AB]
D
∈
[
A
B
]
and
[
D
C
]
⊥
[
B
C
]
[DC]\perp [BC]
[
D
C
]
⊥
[
BC
]
. If
m
(
A
B
C
^
)
=
α
m(\widehat{ABC})=\alpha
m
(
A
BC
)
=
α
,
m
(
B
C
A
^
)
=
3
α
m(\widehat{BCA})=3\alpha
m
(
BC
A
)
=
3
α
, and
∣
A
C
∣
−
∣
A
D
∣
=
10
|AC|-|AD|=10
∣
A
C
∣
−
∣
A
D
∣
=
10
, what is
∣
B
D
∣
|BD|
∣
B
D
∣
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
10
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
14
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
18
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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>
(
D
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20
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22
<span class='latex-bold'>(A)</span>\ 10 \qquad<span class='latex-bold'>(B)</span>\ 14 \qquad<span class='latex-bold'>(C)</span>\ 18 \qquad<span class='latex-bold'>(D)</span>\ 20 \qquad<span class='latex-bold'>(E)</span>\ 22
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10
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(
B
)
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14
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(
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18
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20
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22
12
1
Hide problems
Turkey NMO 2008 1st Round - P12 (Combinatorics)
In how many ways a cube can be painted using seven different colors in such a way that no two faces are in same color?
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(
A
)
<
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154
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(
B
)
<
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>
203
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210
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D
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240
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None of the above
<span class='latex-bold'>(A)</span>\ 154 \qquad<span class='latex-bold'>(B)</span>\ 203 \qquad<span class='latex-bold'>(C)</span>\ 210 \qquad<span class='latex-bold'>(D)</span>\ 240 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
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154
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B
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203
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210
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240
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None of the above
11
1
Hide problems
Turkey NMO 2008 1st Round - P11 (Algebra)
Sequence
(
a
n
)
(a_n)
(
a
n
)
is defined as
a
n
+
1
−
2
a
n
+
a
n
−
1
=
7
a_{n+1}-2a_n+a_{n-1}=7
a
n
+
1
−
2
a
n
+
a
n
−
1
=
7
for every
n
≥
2
n\geq 2
n
≥
2
, where
a
1
=
1
,
a
2
=
5
a_1 = 1, a_2=5
a
1
=
1
,
a
2
=
5
. What is
a
17
a_{17}
a
17
?
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A
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>
895
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−
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>
(
B
)
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>
900
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−
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(
C
)
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905
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(
D
)
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>
910
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None of the above
<span class='latex-bold'>(A)</span>\ 895 \qquad<span class='latex-bold'>(B)</span>\ 900 \qquad<span class='latex-bold'>(C)</span>\ 905 \qquad<span class='latex-bold'>(D)</span>\ 910 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
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895
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(
B
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900
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905
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(
D
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910
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None of the above
10
1
Hide problems
Turkey NMO 2008 1st Round - P10 (Number Theory)
How many pairs of positive integers
(
x
,
y
)
(x,y)
(
x
,
y
)
are there such that
x
y
−
71
x
+
30
=
0
\sqrt{xy}-71\sqrt x + 30 = 0
x
y
−
71
x
+
30
=
0
?
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A
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8
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18
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72
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2130
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Infinitely many
<span class='latex-bold'>(A)</span>\ 8 \qquad<span class='latex-bold'>(B)</span>\ 18 \qquad<span class='latex-bold'>(C)</span>\ 72 \qquad<span class='latex-bold'>(D)</span>\ 2130 \qquad<span class='latex-bold'>(E)</span>\ \text{Infinitely many}
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8
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18
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72
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2130
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Infinitely many
9
1
Hide problems
Turkey NMO 2008 1st Round - P09 (Geometry)
Let
E
E
E
be a point outside the square
A
B
C
D
ABCD
A
BC
D
such that
m
(
B
E
C
^
)
=
9
0
∘
m(\widehat{BEC})=90^{\circ}
m
(
BEC
)
=
9
0
∘
,
F
∈
[
C
E
]
F\in [CE]
F
∈
[
CE
]
,
[
A
F
]
⊥
[
C
E
]
[AF]\perp [CE]
[
A
F
]
⊥
[
CE
]
,
∣
A
B
∣
=
25
|AB|=25
∣
A
B
∣
=
25
, and
∣
B
E
∣
=
7
|BE|=7
∣
BE
∣
=
7
. What is
∣
A
F
∣
|AF|
∣
A
F
∣
?
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29
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(
B
)
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30
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31
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32
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33
<span class='latex-bold'>(A)</span>\ 29 \qquad<span class='latex-bold'>(B)</span>\ 30 \qquad<span class='latex-bold'>(C)</span>\ 31 \qquad<span class='latex-bold'>(D)</span>\ 32 \qquad<span class='latex-bold'>(E)</span>\ 33
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(
B
)
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30
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(
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31
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32
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33
8
1
Hide problems
Turkey NMO 2008 1st Round - P08 (Combinatorics)
Numbers
0
,
1
,
2
,
…
,
9
0,1,2,\dots,9
0
,
1
,
2
,
…
,
9
are placed left to right into the squares of the first row of
10
×
10
10 \times 10
10
×
10
chessboard. Similarly,
10
,
11
,
…
,
19
10,11,\dots,19
10
,
11
,
…
,
19
are placed into the second row, and so on. We are changing signs of exactly five numbers into the squares of each row and each column. What is the minimum value of the sum of the numbers on the chessboard?
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−
10
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(
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)
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>
−
2
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(
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)
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10
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None of the above
<span class='latex-bold'>(A)</span>\ -10 \qquad<span class='latex-bold'>(B)</span>\ -2 \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ 10 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
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−
10
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(
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)
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2
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E
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None of the above
7
1
Hide problems
Turkey NMO 2008 1st Round - P07 (Algebra)
If
a
=
9
3
−
3
3
+
1
a=\sqrt[3]{9}-\sqrt[3]{3}+1
a
=
3
9
−
3
3
+
1
, what is
(
4
−
a
a
)
6
(\frac {4-a}a)^6
(
a
4
−
a
)
6
?
<
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c
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s
s
=
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a
t
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−
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o
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>
(
A
)
<
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a
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>
3
<
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c
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a
s
s
=
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a
t
e
x
−
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o
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′
>
(
B
)
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>
6
<
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c
l
a
s
s
=
′
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a
t
e
x
−
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>
(
C
)
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>
8
<
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c
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a
s
s
=
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x
−
b
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>
(
D
)
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>
9
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)
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12
<span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ 6 \qquad<span class='latex-bold'>(C)</span>\ 8 \qquad<span class='latex-bold'>(D)</span>\ 9 \qquad<span class='latex-bold'>(E)</span>\ 12
<
s
p
an
c
l
a
ss
=
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a
t
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d
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>
(
A
)
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3
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(
B
)
<
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6
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(
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8
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9
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12
6
1
Hide problems
Turkey NMO 2008 1st Round - P06 (Number Theory)
A positive integer
n
n
n
is called a good number if every integer multiple of
n
n
n
is divisible by
n
n
n
however its digits are rearranged. How many good numbers are there?
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(
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3
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(
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4
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6
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12
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Infinitely many
<span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ 4 \qquad<span class='latex-bold'>(C)</span>\ 6 \qquad<span class='latex-bold'>(D)</span>\ 12 \qquad<span class='latex-bold'>(E)</span>\ \text{Infinitely many}
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3
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(
B
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4
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(
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6
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12
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Infinitely many
5
1
Hide problems
Turkey NMO 2008 1st Round - P05 (Geometry)
A triangle with sides
a
,
b
,
c
a,b,c
a
,
b
,
c
is called a good triangle if
a
2
,
b
2
,
c
2
a^2,b^2,c^2
a
2
,
b
2
,
c
2
can form a triangle. How many of below triangles are good?(i)
4
0
∘
,
6
0
∘
,
8
0
∘
40^{\circ}, 60^{\circ}, 80^{\circ}
4
0
∘
,
6
0
∘
,
8
0
∘
(ii)
1
0
∘
,
1
0
∘
,
16
0
∘
10^{\circ}, 10^{\circ}, 160^{\circ}
1
0
∘
,
1
0
∘
,
16
0
∘
(iii)
11
0
∘
,
3
5
∘
,
3
5
∘
110^{\circ}, 35^{\circ}, 35^{\circ}
11
0
∘
,
3
5
∘
,
3
5
∘
(iv)
5
0
∘
,
3
0
∘
,
10
0
∘
50^{\circ}, 30^{\circ}, 100^{\circ}
5
0
∘
,
3
0
∘
,
10
0
∘
(v)
9
0
∘
,
4
0
∘
,
5
0
∘
90^{\circ}, 40^{\circ}, 50^{\circ}
9
0
∘
,
4
0
∘
,
5
0
∘
(vi)
8
0
∘
,
2
0
∘
,
8
0
∘
80^{\circ}, 20^{\circ}, 80^{\circ}
8
0
∘
,
2
0
∘
,
8
0
∘
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1
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(
B
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2
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4
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5
<span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 4 \qquad<span class='latex-bold'>(E)</span>\ 5
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(
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)
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1
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(
B
)
<
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2
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(
C
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3
<
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(
D
)
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4
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5
4
1
Hide problems
Turkey NMO 2008 1st Round - P04 (Combinatorics)
How many different sentences with two words can be written using all letters of the word
YARI
S
¸
MA
\text{YARI\c{S}MA}
YARI
S
¸
MA
? (The Turkish word
YARI
S
¸
MA
\text{YARI\c{S}MA}
YARI
S
¸
MA
means
CONTEST
\text{CONTEST}
CONTEST
. It will produce same result.)
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A
)
<
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>
2520
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>
(
B
)
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>
5040
<
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−
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>
(
C
)
<
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a
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>
15120
<
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x
−
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o
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>
(
D
)
<
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a
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>
20160
<
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>
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E
)
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>
None of the above
<span class='latex-bold'>(A)</span>\ 2520 \qquad<span class='latex-bold'>(B)</span>\ 5040 \qquad<span class='latex-bold'>(C)</span>\ 15120 \qquad<span class='latex-bold'>(D)</span>\ 20160 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
<
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(
A
)
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2520
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(
B
)
<
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>
5040
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>
(
C
)
<
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15120
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a
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=
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x
−
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d
′
>
(
D
)
<
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>
20160
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−
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>
(
E
)
<
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>
None of the above
3
1
Hide problems
Turkey NMO 2008 1st Round - P03 (Algebra)
Let
P
(
x
)
=
1
−
x
+
x
2
−
x
3
+
⋯
+
x
18
−
x
19
P(x) = 1-x+x^2-x^3+\dots+x^{18}-x^{19}
P
(
x
)
=
1
−
x
+
x
2
−
x
3
+
⋯
+
x
18
−
x
19
and
Q
(
x
)
=
P
(
x
−
1
)
Q(x)=P(x-1)
Q
(
x
)
=
P
(
x
−
1
)
. What is the coefficient of
x
2
x^2
x
2
in polynomial
Q
Q
Q
?
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(
A
)
<
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>
840
<
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−
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′
>
(
B
)
<
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>
816
<
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−
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>
(
C
)
<
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>
969
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(
D
)
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1020
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1140
<span class='latex-bold'>(A)</span>\ 840 \qquad<span class='latex-bold'>(B)</span>\ 816 \qquad<span class='latex-bold'>(C)</span>\ 969 \qquad<span class='latex-bold'>(D)</span>\ 1020 \qquad<span class='latex-bold'>(E)</span>\ 1140
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)
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840
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(
B
)
<
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>
816
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(
C
)
<
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969
<
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(
D
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<
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1020
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1140
2
1
Hide problems
Turkey NMO 2008 1st Round - P02 (Number Theory)
For which value of
A
A
A
, does the equation
3
m
2
n
=
n
3
+
A
3m^2n = n^3 + A
3
m
2
n
=
n
3
+
A
have a solution in natural numbers?
<
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>
(
A
)
<
/
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a
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>
301
<
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s
s
=
′
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a
t
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x
−
b
o
l
d
′
>
(
B
)
<
/
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a
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>
403
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s
=
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a
t
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−
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o
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d
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>
(
C
)
<
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a
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>
415
<
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c
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a
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s
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e
x
−
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o
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>
(
D
)
<
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>
427
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481
<span class='latex-bold'>(A)</span>\ 301 \qquad<span class='latex-bold'>(B)</span>\ 403 \qquad<span class='latex-bold'>(C)</span>\ 415 \qquad<span class='latex-bold'>(D)</span>\ 427 \qquad<span class='latex-bold'>(E)</span>\ 481
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(
A
)
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301
<
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a
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−
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o
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>
(
B
)
<
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>
403
<
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(
C
)
<
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415
<
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a
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x
−
b
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(
D
)
<
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>
427
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(
E
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481
1
1
Hide problems
Turkey NMO 2008 1st Round - P01 (Geometry)
Let
A
D
AD
A
D
be a median of
△
A
B
C
\triangle ABC
△
A
BC
such that
m
(
A
D
B
^
)
=
4
5
∘
m(\widehat{ADB})=45^{\circ}
m
(
A
D
B
)
=
4
5
∘
and
m
(
A
C
B
^
)
=
3
0
∘
m(\widehat{ACB})=30^{\circ}
m
(
A
CB
)
=
3
0
∘
. What is the measure of
A
B
C
^
\widehat{ABC}
A
BC
in degrees?
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)
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75
<
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a
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−
b
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>
(
B
)
<
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>
90
<
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=
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−
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o
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>
(
C
)
<
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105
<
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o
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(
D
)
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120
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135
<span class='latex-bold'>(A)</span>\ 75 \qquad<span class='latex-bold'>(B)</span>\ 90 \qquad<span class='latex-bold'>(C)</span>\ 105 \qquad<span class='latex-bold'>(D)</span>\ 120 \qquad<span class='latex-bold'>(E)</span>\ 135
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75
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90
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105
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120
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135