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Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2010 National Olympiad First Round
2010 National Olympiad First Round
Part of
National Olympiad First Round
Subcontests
(36)
34
1
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Turkey NMO 2010 1st Round - P34 (Number Theory)
Which one divides
2
2
2010
+
2
2
2009
+
1
2^{2^{2010}}+2^{2^{2009}}+1
2
2
2010
+
2
2
2009
+
1
?
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<span class='latex-bold'>(A)</span>\ 19 \qquad<span class='latex-bold'>(B)</span>\ 17 \qquad<span class='latex-bold'>(C)</span>\ 13 \qquad<span class='latex-bold'>(D)</span>\ 11 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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None
30
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Turkey NMO 2010 1st Round - P30 (Number Theory)
If
N
=
⌊
2
5
⌋
+
⌊
2
2
5
⌋
+
…
⌊
2
2009
5
⌋
N=\lfloor \frac{2}{5} \rfloor + \lfloor \frac{2^2}{5} \rfloor +\dots \lfloor \frac{2^{2009}}{5} \rfloor
N
=
⌊
5
2
⌋
+
⌊
5
2
2
⌋
+
…
⌊
5
2
2009
⌋
, what is the remainder when
2
2010
2^{2010}
2
2010
is divided by
N
N
N
?
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5034
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5032
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5031
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5028
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5024
<span class='latex-bold'>(A)</span>\ 5034 \qquad<span class='latex-bold'>(B)</span>\ 5032 \qquad<span class='latex-bold'>(C)</span>\ 5031 \qquad<span class='latex-bold'>(D)</span>\ 5028 \qquad<span class='latex-bold'>(E)</span>\ 5024
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5024
26
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Turkey NMO 2010 1st Round - P26 (Number Theory)
For which value of
m
m
m
, there is no triple of integer
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
such that
3
x
2
+
4
y
2
−
5
z
2
=
m
3x^2+4y^2-5z^2=m
3
x
2
+
4
y
2
−
5
z
2
=
m
?
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<span class='latex-bold'>(A)</span>\ 16 \qquad<span class='latex-bold'>(B)</span>\ 14 \qquad<span class='latex-bold'>(C)</span>\ 12 \qquad<span class='latex-bold'>(D)</span>\ 10 \qquad<span class='latex-bold'>(E)</span>\ 8
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22
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Turkey NMO 2010 1st Round - P22 (Number Theory)
How many pairs of integers
(
x
,
y
)
(x,y)
(
x
,
y
)
are there such that
x
y
+
7
+
y
x
+
7
=
1
\frac{x}{y+7}+\frac{y}{x+7}=1
y
+
7
x
+
x
+
7
y
=
1
?
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<span class='latex-bold'>(A)</span>\ 18 \qquad<span class='latex-bold'>(B)</span>\ 17 \qquad<span class='latex-bold'>(C)</span>\ 15 \qquad<span class='latex-bold'>(D)</span>\ 14 \qquad<span class='latex-bold'>(E)</span>\ 11
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18
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Turkey NMO 2010 1st Round - P18 (Number Theory)
Which one does not divide the numbers of
500
500
500
-subset of a set with
1000
1000
1000
elements?
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<span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ 5 \qquad<span class='latex-bold'>(C)</span>\ 11 \qquad<span class='latex-bold'>(D)</span>\ 13 \qquad<span class='latex-bold'>(E)</span>\ 17
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Turkey NMO 2010 1st Round - P10 (Number Theory)
How many integers
n
n
n
with
0
≤
n
<
840
0\leq n < 840
0
≤
n
<
840
are there such that
840
840
840
divides
n
8
−
n
4
+
n
−
1
n^8-n^4+n-1
n
8
−
n
4
+
n
−
1
?
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<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>\ 6 \qquad<span class='latex-bold'>(E)</span>\ 8
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Turkey NMO 2010 1st Round - P06 (Number Theory)
How many ordered pairs of integers
(
x
,
y
)
(x,y)
(
x
,
y
)
are there such that
2011
y
2
=
2010
x
+
3
2011y^2=2010x+3
2011
y
2
=
2010
x
+
3
?
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s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
Infinitely many
<span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 1 \qquad<span class='latex-bold'>(D)</span>\ 0 \qquad<span class='latex-bold'>(E)</span>\ \text{Infinitely many}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
3
<
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
>
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
0
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
Infinitely many
2
1
Hide problems
Turkey NMO 2010 1st Round - P02 (Number Theory)
How many ordered pairs of positive integers
(
x
,
y
)
(x,y)
(
x
,
y
)
are there such that
y
2
−
x
2
=
2
y
+
7
x
+
4
y^2-x^2=2y+7x+4
y
2
−
x
2
=
2
y
+
7
x
+
4
?
<
s
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a
n
c
l
a
s
s
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′
l
a
t
e
x
−
b
o
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>
(
A
)
<
/
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a
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>
3
<
s
p
a
n
c
l
a
s
s
=
′
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a
t
e
x
−
b
o
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d
′
>
(
B
)
<
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>
2
<
s
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a
n
c
l
a
s
s
=
′
l
a
t
e
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−
b
o
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>
(
C
)
<
/
s
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a
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>
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
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a
n
>
0
<
s
p
a
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c
l
a
s
s
=
′
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a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
Infinitely many
<span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 1 \qquad<span class='latex-bold'>(D)</span>\ 0 \qquad<span class='latex-bold'>(E)</span>\ \text{Infinitely many}
<
s
p
an
c
l
a
ss
=
′
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a
t
e
x
−
b
o
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(
A
)
<
/
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>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
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an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
0
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
Infinitely many
35
1
Hide problems
Turkey NMO 2010 1st Round - P35 (Algebra)
Which one below is not less than
x
3
+
y
5
x^3+y^5
x
3
+
y
5
for all reals
x
,
y
x,y
x
,
y
such that
0
<
x
<
1
0<x<1
0
<
x
<
1
and
0
<
y
<
1
0<y<1
0
<
y
<
1
?
<
s
p
a
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c
l
a
s
s
=
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a
t
e
x
−
b
o
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′
>
(
A
)
<
/
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a
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>
x
2
y
<
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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
′
>
(
B
)
<
/
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a
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>
x
2
y
2
<
s
p
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c
l
a
s
s
=
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a
t
e
x
−
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o
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>
(
C
)
<
/
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a
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>
x
2
y
3
<
s
p
a
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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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>
x
3
y
<
s
p
a
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c
l
a
s
s
=
′
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a
t
e
x
−
b
o
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d
′
>
(
E
)
<
/
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>
x
y
4
<span class='latex-bold'>(A)</span>\ x^2y \qquad<span class='latex-bold'>(B)</span>\ x^2y^2 \qquad<span class='latex-bold'>(C)</span>\ x^2y^3 \qquad<span class='latex-bold'>(D)</span>\ x^3y \qquad<span class='latex-bold'>(E)</span>\ xy^4
<
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p
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c
l
a
ss
=
′
l
a
t
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x
−
b
o
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d
′
>
(
A
)
<
/
s
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an
>
x
2
y
<
s
p
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c
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a
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=
′
l
a
t
e
x
−
b
o
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′
>
(
B
)
<
/
s
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>
x
2
y
2
<
s
p
an
c
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a
ss
=
′
l
a
t
e
x
−
b
o
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>
(
C
)
<
/
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>
x
2
y
3
<
s
p
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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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an
>
x
3
y
<
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p
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c
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a
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=
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a
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−
b
o
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>
(
E
)
<
/
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>
x
y
4
31
1
Hide problems
Turkey NMO 2010 1st Round - P31 (Algebra)
For which pair
(
A
,
B
)
(A,B)
(
A
,
B
)
,
x
2
+
x
y
+
y
=
A
y
y
−
x
=
B
x^2+xy+y=A \\ \frac{y}{y-x}=B
x
2
+
x
y
+
y
=
A
y
−
x
y
=
B
has no real roots?
<
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a
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−
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>
(
A
)
<
/
s
p
a
n
>
(
1
/
2
,
2
)
<
s
p
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l
a
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s
=
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t
e
x
−
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>
(
B
)
<
/
s
p
a
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>
(
−
1
,
1
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
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o
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d
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>
(
C
)
<
/
s
p
a
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>
(
2
,
2
)
<
s
p
a
n
c
l
a
s
s
=
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a
t
e
x
−
b
o
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′
>
(
D
)
<
/
s
p
a
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>
(
1
,
1
/
2
)
<
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(
E
)
<
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>
(
2
,
2
/
3
)
<span class='latex-bold'>(A)</span>\ (1/2,2) \qquad<span class='latex-bold'>(B)</span>\ (-1,1) \qquad<span class='latex-bold'>(C)</span>\ (\sqrt 2, \sqrt 2) \qquad<span class='latex-bold'>(D)</span>\ (1,1/2) \qquad<span class='latex-bold'>(E)</span>\ (2,2/3)
<
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−
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(
A
)
<
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>
(
1/2
,
2
)
<
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p
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=
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a
t
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x
−
b
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>
(
B
)
<
/
s
p
an
>
(
−
1
,
1
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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′
>
(
C
)
<
/
s
p
an
>
(
2
,
2
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
D
)
<
/
s
p
an
>
(
1
,
1/2
)
<
s
p
an
c
l
a
ss
=
′
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a
t
e
x
−
b
o
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d
′
>
(
E
)
<
/
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an
>
(
2
,
2/3
)
27
1
Hide problems
Turkey NMO 2010 1st Round - P27 (Algebra)
Let
P
P
P
be a polynomial with each root is real and each coefficient is either
1
1
1
or
−
1
-1
−
1
. The degree of
P
P
P
can be at most ?
<
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(
A
)
<
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>
5
<
s
p
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c
l
a
s
s
=
′
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a
t
e
x
−
b
o
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d
′
>
(
B
)
<
/
s
p
a
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>
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
C
)
<
/
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a
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>
3
<
s
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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>
2
<
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
None
<span class='latex-bold'>(A)</span>\ 5 \qquad<span class='latex-bold'>(B)</span>\ 4 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 2 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
<
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−
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>
(
A
)
<
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>
5
<
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p
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c
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a
ss
=
′
l
a
t
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x
−
b
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d
′
>
(
B
)
<
/
s
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>
4
<
s
p
an
c
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a
ss
=
′
l
a
t
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x
−
b
o
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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
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None
23
1
Hide problems
Turkey NMO 2010 1st Round - P23 (Algebra)
For how many integers
1
≤
n
≤
2010
1\leq n \leq 2010
1
≤
n
≤
2010
,
2010
2010
2010
divides
1
2
−
2
2
+
3
2
−
4
2
+
⋯
+
(
2
n
−
1
)
2
−
(
2
n
)
2
1^2-2^2+3^2-4^2+\dots+(2n-1)^2-(2n)^2
1
2
−
2
2
+
3
2
−
4
2
+
⋯
+
(
2
n
−
1
)
2
−
(
2
n
)
2
?
<
s
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s
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(
A
)
<
/
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>
9
<
s
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c
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s
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−
b
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>
(
B
)
<
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>
8
<
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c
l
a
s
s
=
′
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a
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x
−
b
o
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′
>
(
C
)
<
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a
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>
7
<
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c
l
a
s
s
=
′
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a
t
e
x
−
b
o
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>
(
D
)
<
/
s
p
a
n
>
6
<
s
p
a
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c
l
a
s
s
=
′
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x
−
b
o
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′
>
(
E
)
<
/
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a
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>
5
<span class='latex-bold'>(A)</span>\ 9 \qquad<span class='latex-bold'>(B)</span>\ 8 \qquad<span class='latex-bold'>(C)</span>\ 7 \qquad<span class='latex-bold'>(D)</span>\ 6 \qquad<span class='latex-bold'>(E)</span>\ 5
<
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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
′
>
(
A
)
<
/
s
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>
9
<
s
p
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c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
8
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
C
)
<
/
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an
>
7
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
6
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
E
)
<
/
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an
>
5
19
1
Hide problems
Turkey NMO 2010 1st Round - P19 (Algebra)
What is the sum of distinct real roots of
x
5
−
2
x
2
−
9
x
−
6
x^5-2x^2-9x-6
x
5
−
2
x
2
−
9
x
−
6
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
0
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
−
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
6
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
−
17
<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>\ 6 \qquad<span class='latex-bold'>(E)</span>\ -17
<
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p
an
c
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a
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=
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a
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−
b
o
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d
′
>
(
A
)
<
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an
>
0
<
s
p
an
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−
17
15
1
Hide problems
Turkey NMO 2010 1st Round - P15 (Algebra)
If the real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
satisfies the equations
x
y
z
x
+
y
=
−
1
\frac{xyz}{x+y}=-1
x
+
y
x
yz
=
−
1
,
x
y
z
y
+
z
=
1
\frac{xyz}{y+z}=1
y
+
z
x
yz
=
1
, and
x
y
z
z
+
x
=
2
\frac{xyz}{z+x}=2
z
+
x
x
yz
=
2
, what can
x
y
z
xyz
x
yz
be?
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8
15
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5
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3
5
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15
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None
<span class='latex-bold'>(A)</span>\ -\frac{8}{\sqrt {15}} \qquad<span class='latex-bold'>(B)</span>\ \frac{8}{\sqrt 5} \qquad<span class='latex-bold'>(C)</span>\ -8\sqrt{\frac{3}{5}} \qquad<span class='latex-bold'>(D)</span>\ \frac{7}{\sqrt{15}} \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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8
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3
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7
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None
14
1
Hide problems
Turkey NMO 2010 1st Round - P14 (Number Theory)
A grasshopper jumps either
364
364
364
or
715
715
715
units on the real number line. If it starts from the point
0
0
0
, what is the smallest distance that the grasshoper can be away from the point
2010
2010
2010
?
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34
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164
<span class='latex-bold'>(A)</span>\ 5 \qquad<span class='latex-bold'>(B)</span>\ 8 \qquad<span class='latex-bold'>(C)</span>\ 18 \qquad<span class='latex-bold'>(D)</span>\ 34 \qquad<span class='latex-bold'>(E)</span>\ 164
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164
11
1
Hide problems
Turkey NMO 2010 1st Round - P11 (Algebra)
At most how many points with integer coordinates are there over a circle with center of
(
20
,
10
)
(\sqrt{20}, \sqrt{10})
(
20
,
10
)
in the
x
y
xy
x
y
-plane?
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A
)
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8
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)
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4
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1
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None
<span class='latex-bold'>(A)</span>\ 8 \qquad<span class='latex-bold'>(B)</span>\ 4 \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ 1 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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None
7
1
Hide problems
Turkey NMO 2010 1st Round - P07 (Algebra)
A frog is at the center of a circular shaped island with radius
r
r
r
. The frog jumps
1
/
2
1/2
1/2
meters at first. After the first jump, it turns right or left at exactly
9
0
∘
90^\circ
9
0
∘
, and it always jumps one half of its previous jump. After a finite number of jumps, what is the least
r
r
r
that yields the frog can never fall into the water?
<
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3
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)
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5
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6
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2
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4
<span class='latex-bold'>(A)</span>\ \frac{\sqrt 5}{3} \qquad<span class='latex-bold'>(B)</span>\ \frac{\sqrt {13}}{5} \qquad<span class='latex-bold'>(C)</span>\ \frac{\sqrt {19}}{6} \qquad<span class='latex-bold'>(D)</span>\ \frac{1}{\sqrt 2} \qquad<span class='latex-bold'>(E)</span>\ \frac34
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13
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19
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3
3
1
Hide problems
Turkey NMO 2010 1st Round - P03 (Algebra)
How many real pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
are there such that
x
2
+
2
y
=
2
x
y
x
3
+
x
2
y
=
y
2
x^2+2y = 2xy \\ x^3+x^2y = y^2
x
2
+
2
y
=
2
x
y
x
3
+
x
2
y
=
y
2
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)
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3
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None
<span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 1 \qquad<span class='latex-bold'>(D)</span>\ 0 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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(
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36
1
Hide problems
Turkey NMO 2010 1st Round - P36 (Combinatorics)
Two players are playing a turn based game on a
n
×
n
n \times n
n
×
n
chessboard. At the beginning, only the bottom left corner of the chessboard contains a piece. At each turn, the player moves the piece to either the square just above, or the square just right, or the diagonal square just right-top. If a player cannot make a move, he loses the game. The game is played once on each
6
×
7
6\times 7
6
×
7
,
6
×
8
6 \times 8
6
×
8
,
7
×
7
7 \times 7
7
×
7
,
7
×
8
7 \times 8
7
×
8
, and
8
×
8
8 \times 8
8
×
8
chessboard. In how many of them, can the first player guarantee to win?
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(
D
)
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<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}
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A
)
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(
B
)
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(
C
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(
D
)
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32
1
Hide problems
Turkey NMO 2010 1st Round - P32 (Combinatorics)
At least two of any three students of a school with
1001
1001
1001
students are friends. How many of the numbers
334
,
412
,
450
,
499
334,412,450,499
334
,
412
,
450
,
499
can be the number of friends of the one with the highest number of friends?
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<span class='latex-bold'>(A)</span>\ 4 \qquad<span class='latex-bold'>(B)</span>\ 3 \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ 1 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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28
1
Hide problems
Turkey NMO 2010 1st Round - P28 (Combinatorics)
Only
A
A
A
and
B
B
B
have
n
n
n
friends in a village of
2010
2010
2010
people. The other
2008
2008
2008
people have all different numbers of friends. How many possible values of
n
n
n
are there?
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<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}
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24
1
Hide problems
Turkey NMO 2010 1st Round - P24 (Combinatorics)
How many
7
7
7
-digit positive integers are there such that the number remains same when its digits are reversed and is multiple of
11
11
11
?
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900
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854
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818
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726
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<span class='latex-bold'>(A)</span>\ 900 \qquad<span class='latex-bold'>(B)</span>\ 854 \qquad<span class='latex-bold'>(C)</span>\ 818 \qquad<span class='latex-bold'>(D)</span>\ 726 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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854
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818
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726
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20
1
Hide problems
Turkey NMO 2010 1st Round - P20 (Combinatorics)
Starting from
0
0
0
, at each step we take
1
1
1
more or
2
2
2
times of the previous number. Which one below can be get in a less number of steps?
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2011
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2008
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<span class='latex-bold'>(A)</span>\ 2011 \qquad<span class='latex-bold'>(B)</span>\ 2010 \qquad<span class='latex-bold'>(C)</span>\ 2009 \qquad<span class='latex-bold'>(D)</span>\ 2008 \qquad<span class='latex-bold'>(E)</span>\ 2007
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2007
16
1
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Turkey NMO 2010 1st Round - P16 (Combinatorics)
11
11
11
different books are on a
3
3
3
-shelf bookcase. In how many different ways can the books be arranged such that at most one shelf is empty?
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75
⋅
11
!
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⋅
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!
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⋅
12
!
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13
!
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13
!
<span class='latex-bold'>(A)</span>\ 75\cdot 11! \qquad<span class='latex-bold'>(B)</span>\ 62\cdot 11! \qquad<span class='latex-bold'>(C)</span>\ 68\cdot 12! \qquad<span class='latex-bold'>(D)</span>\ 12\cdot 13! \qquad<span class='latex-bold'>(E)</span>\ 6 \cdot 13!
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13
!
12
1
Hide problems
Turkey NMO 2010 1st Round - P12 (Combinatorics)
How many integer quadruples
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
are there such that
7
7
7
divides
a
b
−
c
d
ab-cd
ab
−
c
d
where
0
≤
a
,
b
,
c
,
d
<
7
0\leq a,b,c,d < 7
0
≤
a
,
b
,
c
,
d
<
7
?
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412
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252
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<span class='latex-bold'>(A)</span>\ 412 \qquad<span class='latex-bold'>(B)</span>\ 385 \qquad<span class='latex-bold'>(C)</span>\ 294 \qquad<span class='latex-bold'>(D)</span>\ 252 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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412
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8
1
Hide problems
Turkey NMO 2010 1st Round - P08 (Combinatorics)
What is the sum of the digits of the first
2010
2010
2010
positive integers?
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30516
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28068
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<span class='latex-bold'>(A)</span>\ 30516 \qquad<span class='latex-bold'>(B)</span>\ 28068 \qquad<span class='latex-bold'>(C)</span>\ 25020 \qquad<span class='latex-bold'>(D)</span>\ 20100 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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30516
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28068
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25020
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4
1
Hide problems
Turkey NMO 2010 1st Round - P04 (Combinatorics)
How many positive integers less than
2010
2010
2010
are there such that the sum of factorials of its digits is equal to itself?
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<span class='latex-bold'>(A)</span>\ 5 \qquad<span class='latex-bold'>(B)</span>\ 4 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 2 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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33
1
Hide problems
Turkey NMO 2010 1st Round - P33 (Geometry)
Let
D
D
D
be the midpoint of
[
A
C
]
[AC]
[
A
C
]
of
△
A
B
C
\triangle ABC
△
A
BC
with
m
(
A
B
C
^
)
=
9
0
∘
m(\widehat{ABC})=90^\circ
m
(
A
BC
)
=
9
0
∘
and
∣
A
C
∣
=
10
|AC|=10
∣
A
C
∣
=
10
. Let
E
E
E
be the point of intersections of bisectors of
[
A
D
]
[AD]
[
A
D
]
and
[
B
D
]
[BD]
[
B
D
]
. Let
F
F
F
be the point of intersections of bisectors of
[
B
D
]
[BD]
[
B
D
]
and
[
C
D
]
[CD]
[
C
D
]
. If
∣
E
F
∣
=
13
|EF|=13
∣
EF
∣
=
13
, then
∣
A
B
∣
|AB|
∣
A
B
∣
can be
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2
13
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2
13
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13
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13
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None
<span class='latex-bold'>(A)</span>\ 20\sqrt{\frac 2{13}} \qquad<span class='latex-bold'>(B)</span>\ 15\sqrt{\frac 2{13}} \qquad<span class='latex-bold'>(C)</span>\ 10\sqrt{\frac 2{13}} \qquad<span class='latex-bold'>(D)</span>\ 5\sqrt{\frac 2{13}} \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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2
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2
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29
1
Hide problems
Turkey NMO 2010 1st Round - P29 (Geometry)
Let
I
I
I
be the incenter of
△
A
B
C
\triangle ABC
△
A
BC
, and
O
O
O
be the excenter corresponding to
B
B
B
. If
∣
B
I
∣
=
12
|BI|=12
∣
B
I
∣
=
12
,
∣
I
O
∣
=
18
|IO|=18
∣
I
O
∣
=
18
, and
∣
B
C
∣
=
15
|BC|=15
∣
BC
∣
=
15
, then what is
∣
A
B
∣
|AB|
∣
A
B
∣
?
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<span class='latex-bold'>(A)</span>\ 16 \qquad<span class='latex-bold'>(B)</span>\ 18 \qquad<span class='latex-bold'>(C)</span>\ 20 \qquad<span class='latex-bold'>(D)</span>\ 22 \qquad<span class='latex-bold'>(E)</span>\ 24
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24
25
1
Hide problems
Turkey NMO 2010 1st Round - P25 (Geometry)
Let
P
P
P
and
Q
Q
Q
be points on the plane
A
B
C
ABC
A
BC
such that
m
(
B
A
C
^
)
=
9
0
∘
m(\widehat{BAC})=90^\circ
m
(
B
A
C
)
=
9
0
∘
,
∣
A
B
∣
=
1
|AB|=1
∣
A
B
∣
=
1
,
∣
A
C
∣
=
2
|AC|=\sqrt 2
∣
A
C
∣
=
2
,
∣
P
B
∣
=
1
=
∣
Q
B
∣
|PB|=1=|QB|
∣
PB
∣
=
1
=
∣
QB
∣
,
∣
P
C
∣
=
2
=
∣
Q
C
∣
|PC|=2=|QC|
∣
PC
∣
=
2
=
∣
QC
∣
, and
∣
P
A
∣
>
∣
Q
A
∣
|PA|>|QA|
∣
P
A
∣
>
∣
Q
A
∣
. What is
∣
P
A
∣
/
∣
Q
A
∣
|PA|/|QA|
∣
P
A
∣/∣
Q
A
∣
?
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6
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2
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<span class='latex-bold'>(A)</span>\ \sqrt 2 +\sqrt 3 \qquad<span class='latex-bold'>(B)</span>\ 5-\sqrt 6 \qquad<span class='latex-bold'>(C)</span>\ \sqrt 6 -\sqrt 2 \qquad<span class='latex-bold'>(D)</span>\ \sqrt 6 + 1 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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21
1
Hide problems
Turkey NMO 2010 1st Round - P21 (Geometry)
A right circular cone and a right cylinder with same height
20
20
20
does not have same circular base but the circles are coplanar and their centers are same. If the cone and the cylinder are at the same side of the plane and their base radii are
20
20
20
and
10
10
10
, respectively, what is the ratio of the volume of the part of the cone inside the cylinder over the volume of the part of the cone outside the cylinder?
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3
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3
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<span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ \frac{5}{3} \qquad<span class='latex-bold'>(D)</span>\ \frac{4}{3} \qquad<span class='latex-bold'>(E)</span>\ 1
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4
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1
17
1
Hide problems
Turkey NMO 2010 1st Round - P17 (Geometry)
Let
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
be points in the space such that
∣
A
B
∣
=
∣
A
C
∣
=
3
|AB|=|AC|=3
∣
A
B
∣
=
∣
A
C
∣
=
3
,
∣
D
B
∣
=
∣
D
C
∣
=
5
|DB|=|DC|=5
∣
D
B
∣
=
∣
D
C
∣
=
5
,
∣
A
D
∣
=
6
|AD|=6
∣
A
D
∣
=
6
, and
∣
B
C
∣
=
2
|BC|=2
∣
BC
∣
=
2
. Let
P
P
P
be the nearest point of
B
C
BC
BC
to the point
D
D
D
, and
Q
Q
Q
be the nearest point of the plane
A
B
C
ABC
A
BC
to the point
D
D
D
. What is
∣
P
Q
∣
|PQ|
∣
PQ
∣
?
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c
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s
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2
11
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<span class='latex-bold'>(A)</span>\ \frac{1}{\sqrt 2} \qquad<span class='latex-bold'>(B)</span>\ \frac{3\sqrt 7}{2} \qquad<span class='latex-bold'>(C)</span>\ \frac{57}{2\sqrt{11}} \qquad<span class='latex-bold'>(D)</span>\ \frac{9}{2\sqrt 2} \qquad<span class='latex-bold'>(E)</span>\ 2\sqrt 2
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57
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2
13
1
Hide problems
Turkey NMO 2010 1st Round - P13 (Geometry)
Let
D
D
D
and
E
E
E
be points on respectively
[
A
B
]
[AB]
[
A
B
]
and
[
A
C
]
[AC]
[
A
C
]
of
△
A
B
C
\triangle ABC
△
A
BC
where
∣
A
B
∣
=
∣
A
C
∣
|AB|=|AC|
∣
A
B
∣
=
∣
A
C
∣
,
m
(
B
A
C
^
)
=
4
0
∘
m(\widehat{BAC})=40^\circ
m
(
B
A
C
)
=
4
0
∘
. Let
F
F
F
be a point on
B
C
BC
BC
such that
C
C
C
is between
B
B
B
and
F
F
F
. If
∣
B
E
∣
=
∣
C
F
∣
|BE|=|CF|
∣
BE
∣
=
∣
CF
∣
,
∣
A
D
∣
=
∣
A
E
∣
|AD|=|AE|
∣
A
D
∣
=
∣
A
E
∣
, and
m
(
B
E
C
^
)
=
6
0
∘
m(\widehat{BEC})=60^\circ
m
(
BEC
)
=
6
0
∘
, then what is
m
(
D
F
B
^
)
m(\widehat{DFB})
m
(
D
FB
)
?
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o
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A
)
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4
5
∘
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p
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a
s
s
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e
x
−
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o
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>
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B
)
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4
0
∘
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p
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l
a
s
s
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x
−
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o
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C
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3
5
∘
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a
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s
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D
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>
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0
∘
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∘
<span class='latex-bold'>(A)</span>\ 45^\circ \qquad<span class='latex-bold'>(B)</span>\ 40^\circ \qquad<span class='latex-bold'>(C)</span>\ 35^\circ \qquad<span class='latex-bold'>(D)</span>\ 30^\circ \qquad<span class='latex-bold'>(E)</span>\ 25^\circ
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5
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0
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5
∘
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s
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0
∘
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s
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E
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5
∘
9
1
Hide problems
Turkey NMO 2010 1st Round - P09 (Geometry)
Let
E
E
E
be a point outside of square
A
B
C
D
ABCD
A
BC
D
. If the distance of
E
E
E
to
A
C
AC
A
C
is
6
6
6
, to
B
D
BD
B
D
is
17
17
17
, and to the nearest vertex of the square is
10
10
10
, what is the area of the square?
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<span class='latex-bold'>(A)</span>\ 200 \qquad<span class='latex-bold'>(B)</span>\ 196 \qquad<span class='latex-bold'>(C)</span>\ 169 \qquad<span class='latex-bold'>(D)</span>\ 162 \qquad<span class='latex-bold'>(E)</span>\ 144
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E
)
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144
5
1
Hide problems
Turkey NMO 2010 1st Round - P05 (Geometry)
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral such that
∣
A
B
∣
=
10
|AB|=10
∣
A
B
∣
=
10
,
∣
C
D
∣
=
3
6
|CD|=3\sqrt 6
∣
C
D
∣
=
3
6
,
m
(
A
B
D
^
)
=
6
0
∘
m(\widehat{ABD})=60^\circ
m
(
A
B
D
)
=
6
0
∘
,
m
(
B
D
C
^
)
=
4
5
∘
m(\widehat{BDC})=45^\circ
m
(
B
D
C
)
=
4
5
∘
, and
∣
B
D
∣
=
13
+
3
3
|BD|=13+3\sqrt 3
∣
B
D
∣
=
13
+
3
3
. What is
∣
A
C
∣
|AC|
∣
A
C
∣
?
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)
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18
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)
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<span class='latex-bold'>(A)</span>\ 20 \qquad<span class='latex-bold'>(B)</span>\ 18 \qquad<span class='latex-bold'>(C)</span>\ 16 \qquad<span class='latex-bold'>(D)</span>\ 14 \qquad<span class='latex-bold'>(E)</span>\ 12
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)
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)
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)
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−
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>
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E
)
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12
1
1
Hide problems
Turkey NMO 2010 1st Round - P01 (Geometry)
Let
D
D
D
be a point inside of equilateral
△
A
B
C
\triangle ABC
△
A
BC
, and
E
E
E
be a point outside of equilateral
△
A
B
C
\triangle ABC
△
A
BC
such that
m
(
B
A
D
^
)
=
m
(
A
B
D
^
)
=
m
(
C
A
E
^
)
=
m
(
A
C
E
^
)
=
5
∘
m(\widehat{BAD})=m(\widehat{ABD})=m(\widehat{CAE})=m(\widehat{ACE})=5^\circ
m
(
B
A
D
)
=
m
(
A
B
D
)
=
m
(
C
A
E
)
=
m
(
A
CE
)
=
5
∘
. What is
m
(
E
D
C
^
)
m(\widehat{EDC})
m
(
E
D
C
)
?
<
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s
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−
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o
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A
)
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4
5
∘
<
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s
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x
−
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o
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>
(
B
)
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4
0
∘
<
s
p
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c
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a
s
s
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−
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o
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>
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C
)
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3
5
∘
<
s
p
a
n
c
l
a
s
s
=
′
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a
t
e
x
−
b
o
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>
(
D
)
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a
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>
3
0
∘
<
s
p
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c
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a
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s
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E
)
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5
∘
<span class='latex-bold'>(A)</span>\ 45^\circ \qquad<span class='latex-bold'>(B)</span>\ 40^\circ \qquad<span class='latex-bold'>(C)</span>\ 35^\circ \qquad<span class='latex-bold'>(D)</span>\ 30^\circ \qquad<span class='latex-bold'>(E)</span>\ 25^\circ
<
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p
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−
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o
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A
)
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4
5
∘
<
s
p
an
c
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a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
B
)
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an
>
4
0
∘
<
s
p
an
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
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>
(
C
)
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>
3
5
∘
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
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>
3
0
∘
<
s
p
an
c
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a
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a
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x
−
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o
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d
′
>
(
E
)
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5
∘