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Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2007 National Olympiad First Round
2007 National Olympiad First Round
Part of
National Olympiad First Round
Subcontests
(36)
36
1
Hide problems
Turkey NMO 2007 1st Round - P36 (Combinatorics)
Five points, no three of which are collinear, are given. What is the least possible value of the numbers of convex polygons whose some corners are from these five points?
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<span class='latex-bold'>(A)</span>\ 10 \qquad<span class='latex-bold'>(B)</span>\ 11 \qquad<span class='latex-bold'>(C)</span>\ 12 \qquad<span class='latex-bold'>(D)</span>\ 15 \qquad<span class='latex-bold'>(E)</span>\ 16
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35
1
Hide problems
Turkey NMO 2007 1st Round - P35 (Algebra)
What is the third digit after the decimal point of the decimal representation of
2
+
5
3
+
2
−
5
3
\sqrt[3]{2+\sqrt 5} + \sqrt[3]{2-\sqrt 5}
3
2
+
5
+
3
2
−
5
?
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8
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<span class='latex-bold'>(A)</span>\ 8 \qquad<span class='latex-bold'>(B)</span>\ 5 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 1 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
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None of the above
34
1
Hide problems
Turkey NMO 2007 1st Round - P34 (Number Theory)
For how many primes
p
p
p
less than
15
15
15
, there exists integer triples
(
m
,
n
,
k
)
(m,n,k)
(
m
,
n
,
k
)
such that
m
+
n
+
k
≡
0
(
m
o
d
p
)
m
n
+
m
k
+
n
k
≡
1
(
m
o
d
p
)
m
n
k
≡
2
(
m
o
d
p
)
.
\begin{array}{rcl} m+n+k &\equiv& 0 \pmod p \\ mn+mk+nk &\equiv& 1 \pmod p \\ mnk &\equiv& 2 \pmod p. \end{array}
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+
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+
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mnk
≡
≡
≡
0
(
mod
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1
(
mod
p
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2
(
mod
p
)
.
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<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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33
1
Hide problems
Turkey NMO 2007 1st Round - P33 (Geometry)
The tangent lines from the point
A
A
A
to the circle
C
C
C
touches the circle at
M
M
M
and
N
N
N
. Let
P
P
P
a point on
[
A
N
]
[AN]
[
A
N
]
. Let
M
P
MP
MP
meet
C
C
C
at
Q
Q
Q
. Let
M
N
MN
MN
meet the line through
P
P
P
and parallel to
M
A
MA
M
A
at
R
R
R
. If
∣
M
A
∣
=
2
|MA|=2
∣
M
A
∣
=
2
,
∣
M
N
∣
=
3
|MN|=\sqrt 3
∣
MN
∣
=
3
, and
Q
R
∥
A
N
QR \parallel AN
QR
∥
A
N
, what is
∣
P
N
∣
|PN|
∣
PN
∣
?
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<span class='latex-bold'>(A)</span>\ \dfrac 32 \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ \dfrac {\sqrt 3} 2 \qquad<span class='latex-bold'>(D)</span>\ \sqrt 2 \qquad<span class='latex-bold'>(E)</span>\ \sqrt 3
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32
1
Hide problems
Turkey NMO 2007 1st Round - P32 (Combinatorics)
We are writing either
0
0
0
or
1
1
1
to unit squares of an
8
×
8
8\times 8
8
×
8
chessboard. If the sum of numbers is even vertically, horizontally, or diagonally, what is the greatest possible value of the sum of the all numbers on the board?
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<span class='latex-bold'>(A)</span>\ 32 \qquad<span class='latex-bold'>(B)</span>\ 48 \qquad<span class='latex-bold'>(C)</span>\ 52 \qquad<span class='latex-bold'>(D)</span>\ 56 \qquad<span class='latex-bold'>(E)</span>\ 64
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31
1
Hide problems
Turkey NMO 2007 1st Round - P31 (Algebra)
A square-shaped field is divided into
n
n
n
rectangular farms whose sides are parallel to the sides of the field. What is the greatest value of
n
n
n
, if the sum of the perimeters of the farms is equal to
100
100
100
times of the perimeter of the field?
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200000
<span class='latex-bold'>(A)</span>\ 10000 \qquad<span class='latex-bold'>(B)</span>\ 20000 \qquad<span class='latex-bold'>(C)</span>\ 50000 \qquad<span class='latex-bold'>(D)</span>\ 100000 \qquad<span class='latex-bold'>(E)</span>\ 200000
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30
1
Hide problems
Turkey NMO 2007 1st Round - P30 (Number Theory)
Let
(
a
n
)
n
=
1
∞
(a_n)_{n=1}^{\infty}
(
a
n
)
n
=
1
∞
be an integer sequence such that
a
n
+
48
≡
a
n
(
m
o
d
35
)
a_{n+48} \equiv a_n \pmod {35}
a
n
+
48
≡
a
n
(
mod
35
)
for every
n
≥
1
n \geq 1
n
≥
1
. Let
i
i
i
and
j
j
j
be the least numbers satisfying the conditions
a
n
+
i
≡
a
n
(
m
o
d
5
)
a_{n+i} \equiv a_n \pmod {5}
a
n
+
i
≡
a
n
(
mod
5
)
and
a
n
+
j
≡
a
n
(
m
o
d
7
)
a_{n+j} \equiv a_n \pmod {7}
a
n
+
j
≡
a
n
(
mod
7
)
for every
n
≥
1
n\geq 1
n
≥
1
. Which one below cannot be an
(
i
,
j
)
(i,j)
(
i
,
j
)
pair?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
(
16
,
4
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
(
3
,
16
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
(
8
,
6
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
(
1
,
48
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
(
16
,
18
)
<span class='latex-bold'>(A)</span>\ (16,4) \qquad<span class='latex-bold'>(B)</span>\ (3,16) \qquad<span class='latex-bold'>(C)</span>\ (8,6) \qquad<span class='latex-bold'>(D)</span>\ (1,48) \qquad<span class='latex-bold'>(E)</span>\ (16,18)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
(
16
,
4
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
(
3
,
16
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
(
8
,
6
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
(
1
,
48
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
(
16
,
18
)
29
1
Hide problems
Turkey NMO 2007 1st Round - P29 (Geometry)
Let
M
M
M
and
N
N
N
be points on the sides
B
C
BC
BC
and
C
D
CD
C
D
, respectively, of a square
A
B
C
D
ABCD
A
BC
D
. If
∣
B
M
∣
=
21
|BM|=21
∣
BM
∣
=
21
,
∣
D
N
∣
=
4
|DN|=4
∣
D
N
∣
=
4
, and
∣
N
C
∣
=
24
|NC|=24
∣
NC
∣
=
24
, what is
m
(
M
A
N
^
)
m(\widehat{MAN})
m
(
M
A
N
)
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
1
5
∘
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
3
0
∘
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
3
7
∘
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
4
5
∘
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
6
0
∘
<span class='latex-bold'>(A)</span>\ 15^\circ \qquad<span class='latex-bold'>(B)</span>\ 30^\circ \qquad<span class='latex-bold'>(C)</span>\ 37^\circ \qquad<span class='latex-bold'>(D)</span>\ 45^\circ \qquad<span class='latex-bold'>(E)</span>\ 60^\circ
<
s
p
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c
l
a
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=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
1
5
∘
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
3
0
∘
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
3
7
∘
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
4
5
∘
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
6
0
∘
28
1
Hide problems
Turkey NMO 2007 1st Round - P28 (Combinatorics)
n
n
n
integers are arranged along a circle in such a way that each number is equal to the absolute value of the difference of the two numbers following that number in clockwise direction. If the sum of all numbers is
278
278
278
, how many different values can
n
n
n
take?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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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
>
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
139
<
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>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 4 \qquad<span class='latex-bold'>(D)</span>\ 139 \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
>
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
>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
139
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the above
27
1
Hide problems
Turkey NMO 2007 1st Round - P27 (Algebra)
What is the sum of real roots of the equation
(
x
+
1
)
(
x
+
1
4
)
(
x
+
1
2
)
(
x
+
3
4
)
=
45
32
?
\left ( x + 1\right )\left ( x + \dfrac 14\right )\left ( x + \dfrac 12\right )\left ( x + \dfrac 34\right )= \dfrac {45}{32}?
(
x
+
1
)
(
x
+
4
1
)
(
x
+
2
1
)
(
x
+
4
3
)
=
32
45
?
<
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
>
−
3
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
−
5
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
−
7
12
<span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ -1 \qquad<span class='latex-bold'>(C)</span>\ -\dfrac {3}{2} \qquad<span class='latex-bold'>(D)</span>\ -\dfrac {5}{4} \qquad<span class='latex-bold'>(E)</span>\ -\dfrac {7}{12}
<
s
p
an
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
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
−
2
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
−
4
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
−
12
7
26
1
Hide problems
Turkey NMO 2007 1st Round - P26 (Number Theory)
Let
c
c
c
be the least common multiple of positive integers
a
a
a
and
b
b
b
, and
d
d
d
be the greatest common divisor of
a
a
a
and
b
b
b
. How many pairs of positive integers
(
a
,
b
)
(a,b)
(
a
,
b
)
are there such that
1
a
+
1
b
+
1
c
+
1
d
=
1
?
\dfrac {1}{a} + \dfrac {1}{b} + \dfrac {1}{c} + \dfrac {1}{d} = 1?
a
1
+
b
1
+
c
1
+
d
1
=
1
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
6
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
5
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
2
<span class='latex-bold'>(A)</span>\ 6 \qquad<span class='latex-bold'>(B)</span>\ 5 \qquad<span class='latex-bold'>(C)</span>\ 4 \qquad<span class='latex-bold'>(D)</span>\ 3 \qquad<span class='latex-bold'>(E)</span>\ 2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
6
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
2
25
1
Hide problems
Turkey NMO 2007 1st Round - P25 (Geometry)
Let
A
,
B
,
C
A, B, C
A
,
B
,
C
be points on a unit circle such that
∣
A
B
∣
=
∣
B
C
∣
|AB|=|BC|
∣
A
B
∣
=
∣
BC
∣
and
m
(
A
B
C
^
)
=
7
2
∘
m(\widehat{ABC})=72^\circ
m
(
A
BC
)
=
7
2
∘
. Let
D
D
D
be a point such that
△
B
C
D
\triangle BCD
△
BC
D
is equilateral. If
A
D
AD
A
D
meets the circle at
D
D
D
, what is
∣
D
E
∣
|DE|
∣
D
E
∣
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
1
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
3
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
2
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
3
−
1
<
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>\ \dfrac 12 \qquad<span class='latex-bold'>(B)</span>\ \dfrac {\sqrt 3}2 \qquad<span class='latex-bold'>(C)</span>\ \dfrac {\sqrt 2}2 \qquad<span class='latex-bold'>(D)</span>\ \sqrt 3 -1 \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
>
2
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
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2
2
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(
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−
1
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None of the above
24
1
Hide problems
Turkey NMO 2007 1st Round - P24 (Combinatorics)
The integers from
1
1
1
to
n
n
n
are arranged along a circle such that each number is a multiple of difference of its adjacents. For which
n
n
n
below such an arrangement is possible?
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(
A
)
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5
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s
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(
B
)
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6
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7
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9
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13
<span class='latex-bold'>(A)</span>\ 5 \qquad<span class='latex-bold'>(B)</span>\ 6 \qquad<span class='latex-bold'>(C)</span>\ 7 \qquad<span class='latex-bold'>(D)</span>\ 9 \qquad<span class='latex-bold'>(E)</span>\ 13
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5
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B
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6
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7
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(
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9
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13
23
1
Hide problems
Turkey NMO 2007 1st Round - P23 (Algebra)
A unit equilateral triangle is given. Divide each side into three equal parts. Remove the equilateral triangles whose bases are middle one-third segments. Now we have a new polygon. Remove the equilateral triangles whose bases are middle one-third segments of the sides of the polygon. After repeating these steps for infinite times, what is the area of the new shape?
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1
2
3
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3
8
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10
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4
3
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None of the above
<span class='latex-bold'>(A)</span>\ \dfrac {1}{2\sqrt 3} \qquad<span class='latex-bold'>(B)</span>\ \dfrac {\sqrt 3}{8} \qquad<span class='latex-bold'>(C)</span>\ \dfrac {\sqrt 3}{10} \qquad<span class='latex-bold'>(D)</span>\ \dfrac {1}{4\sqrt 3} \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
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3
1
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3
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3
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3
1
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None of the above
22
1
Hide problems
Turkey NMO 2007 1st Round - P22 (Number Theory)
Let
n
n
n
and
m
m
m
be integers such that
n
≤
2007
≤
m
n\leq 2007 \leq m
n
≤
2007
≤
m
and
n
n
≡
−
1
≡
m
m
(
m
o
d
5
)
n^n \equiv -1 \equiv m^m \pmod 5
n
n
≡
−
1
≡
m
m
(
mod
5
)
. What is the least possible value of
m
−
n
m-n
m
−
n
?
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4
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5
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6
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7
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8
<span class='latex-bold'>(A)</span>\ 4 \qquad<span class='latex-bold'>(B)</span>\ 5 \qquad<span class='latex-bold'>(C)</span>\ 6 \qquad<span class='latex-bold'>(D)</span>\ 7 \qquad<span class='latex-bold'>(E)</span>\ 8
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4
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5
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6
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7
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8
21
1
Hide problems
Turkey NMO 2007 1st Round - P21 (Geometry)
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral such that
m
(
A
^
)
=
m
(
D
^
)
=
9
0
∘
m(\widehat{A}) = m(\widehat{D}) = 90^\circ
m
(
A
)
=
m
(
D
)
=
9
0
∘
. Let
M
M
M
be the midpoint of
[
D
C
]
[DC]
[
D
C
]
. If
A
C
⊥
B
M
AC\perp BM
A
C
⊥
BM
,
∣
D
C
∣
=
12
|DC|=12
∣
D
C
∣
=
12
, and
∣
A
B
∣
=
9
|AB|=9
∣
A
B
∣
=
9
, then what is
∣
A
D
∣
|AD|
∣
A
D
∣
?
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6
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9
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12
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None of the above
<span class='latex-bold'>(A)</span>\ 4 \qquad<span class='latex-bold'>(B)</span>\ 6 \qquad<span class='latex-bold'>(C)</span>\ 9 \qquad<span class='latex-bold'>(D)</span>\ 12 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
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4
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=
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(
B
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6
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=
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(
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)
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9
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(
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12
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None of the above
20
1
Hide problems
Turkey NMO 2007 1st Round - P20 (Combinatorics)
The
9
9
9
consequtive sections of a paper strip are colored either red or white. If no two consequtive sections are white, in how many ways can this coloring be made?
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34
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89
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128
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360
<span class='latex-bold'>(A)</span>\ 34 \qquad<span class='latex-bold'>(B)</span>\ 89 \qquad<span class='latex-bold'>(C)</span>\ 128 \qquad<span class='latex-bold'>(D)</span>\ 144 \qquad<span class='latex-bold'>(E)</span>\ 360
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34
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)
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89
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128
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144
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360
19
1
Hide problems
Turkey NMO 2007 1st Round - P19 (Algebra)
If
x
1
=
5
,
x
2
=
401
x_1=5, x_2=401
x
1
=
5
,
x
2
=
401
, and
x
n
=
x
n
−
2
−
1
x
n
−
1
x_n=x_{n-2}-\frac 1{x_{n-1}}
x
n
=
x
n
−
2
−
x
n
−
1
1
for every
3
≤
n
≤
m
3\leq n \leq m
3
≤
n
≤
m
, what is the largest value of
m
m
m
?
<
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(
A
)
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>
406
<
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s
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x
−
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o
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(
B
)
<
/
s
p
a
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>
2005
<
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a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
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>
2006
<
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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
′
>
(
D
)
<
/
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p
a
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>
2007
<
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None of the above
<span class='latex-bold'>(A)</span>\ 406 \qquad<span class='latex-bold'>(B)</span>\ 2005 \qquad<span class='latex-bold'>(C)</span>\ 2006 \qquad<span class='latex-bold'>(D)</span>\ 2007 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
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A
)
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406
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d
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(
B
)
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/
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>
2005
<
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p
an
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a
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=
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a
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x
−
b
o
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d
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(
C
)
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/
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>
2006
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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
′
>
(
D
)
<
/
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>
2007
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a
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=
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None of the above
18
1
Hide problems
Turkey NMO 2007 1st Round - P18 (Number Theory)
How many integers
n
n
n
are there such that
n
3
+
8
n^3+8
n
3
+
8
has at most
3
3
3
positive divisors?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
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a
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>
2
<
s
p
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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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>
1
<
s
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c
l
a
s
s
=
′
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t
e
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−
b
o
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>
(
E
)
<
/
s
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a
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>
None of the above
<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 of the above}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
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(
A
)
<
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s
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>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
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an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the above
17
1
Hide problems
Turkey NMO 2007 1st Round - P17 (Geometry)
Let
K
K
K
be the point of intersection of
A
B
AB
A
B
and the line touching the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
at
C
C
C
where
m
(
A
^
)
>
m
(
B
^
)
m(\widehat {A}) > m(\widehat {B})
m
(
A
)
>
m
(
B
)
. Let
L
L
L
be a point on
[
B
C
]
[BC]
[
BC
]
such that
m
(
A
L
B
^
)
=
m
(
C
A
K
^
)
m(\widehat{ALB})=m(\widehat{CAK})
m
(
A
L
B
)
=
m
(
C
A
K
)
,
5
∣
L
C
∣
=
4
∣
B
L
∣
5|LC|=4|BL|
5∣
L
C
∣
=
4∣
B
L
∣
, and
∣
K
C
∣
=
12
|KC|=12
∣
K
C
∣
=
12
. What is
∣
A
K
∣
|AK|
∣
A
K
∣
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
A
)
<
/
s
p
a
n
>
4
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
6
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
8
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
9
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
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a
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>
None of the above
<span class='latex-bold'>(A)</span>\ 4\sqrt 2 \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>\ \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
>
4
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
6
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
8
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
9
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the above
16
1
Hide problems
Turkey NMO 2007 1st Round - P16 (Combinatorics)
The sum of the largest number and the smallest number of a triple of positive integers
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
is called to be the power of the triple. What is the sum of powers of all triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
where
x
,
y
,
z
≤
9
x,y,z \leq 9
x
,
y
,
z
≤
9
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
9000
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
8460
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
7290
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
6150
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
6000
<span class='latex-bold'>(A)</span>\ 9000 \qquad<span class='latex-bold'>(B)</span>\ 8460 \qquad<span class='latex-bold'>(C)</span>\ 7290 \qquad<span class='latex-bold'>(D)</span>\ 6150 \qquad<span class='latex-bold'>(E)</span>\ 6000
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
9000
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
8460
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
7290
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
6150
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
6000
15
1
Hide problems
Turkey NMO 2007 1st Round - P15 (Algebra)
What is the minimum value of
a
b
+
c
d
ab+cd
ab
+
c
d
, if
a
b
+
c
d
=
e
f
+
g
h
ab+cd = ef+gh
ab
+
c
d
=
e
f
+
g
h
where
a
,
b
,
c
,
d
,
e
,
f
,
g
,
h
a,b,c,d,e,f,g,h
a
,
b
,
c
,
d
,
e
,
f
,
g
,
h
are distinct positive integers?
<
s
p
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n
c
l
a
s
s
=
′
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a
t
e
x
−
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o
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>
(
A
)
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/
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n
>
34
<
s
p
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n
c
l
a
s
s
=
′
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t
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−
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o
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d
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>
(
B
)
<
/
s
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a
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>
33
<
s
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n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
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a
n
>
32
<
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
>
31
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
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a
n
>
30
<span class='latex-bold'>(A)</span>\ 34 \qquad<span class='latex-bold'>(B)</span>\ 33 \qquad<span class='latex-bold'>(C)</span>\ 32 \qquad<span class='latex-bold'>(D)</span>\ 31 \qquad<span class='latex-bold'>(E)</span>\ 30
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
34
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
33
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
32
<
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
>
30
14
1
Hide problems
Turkey NMO 2007 1st Round - P14 (Number Theory)
What is the largest integer
n
n
n
that satisfies
(
10
0
2
−
9
9
2
)
(
9
9
2
−
9
8
2
)
…
(
3
2
−
2
2
)
(
2
2
−
1
2
)
(100^2-99^2)(99^2-98^2)\dots(3^2-2^2)(2^2-1^2)
(
10
0
2
−
9
9
2
)
(
9
9
2
−
9
8
2
)
…
(
3
2
−
2
2
)
(
2
2
−
1
2
)
is divisible by
3
n
3^n
3
n
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
49
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
53
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
97
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
103
<
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>\ 49 \qquad<span class='latex-bold'>(B)</span>\ 53 \qquad<span class='latex-bold'>(C)</span>\ 97 \qquad<span class='latex-bold'>(D)</span>\ 103 \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
>
49
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
53
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
97
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
103
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the above
13
1
Hide problems
Turkey NMO 2007 1st Round - P13 (Geometry)
Let
A
B
C
D
ABCD
A
BC
D
be an circumscribed quadrilateral such that
m
(
A
^
)
=
m
(
B
^
)
=
12
0
∘
m(\widehat{A})=m(\widehat{B})=120^\circ
m
(
A
)
=
m
(
B
)
=
12
0
∘
,
m
(
C
^
)
=
3
0
∘
m(\widehat{C})=30^\circ
m
(
C
)
=
3
0
∘
, and
∣
B
C
∣
=
2
|BC|=2
∣
BC
∣
=
2
. What is
∣
A
D
∣
|AD|
∣
A
D
∣
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
3
−
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
−
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
6
−
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
2
−
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
3
−
3
<span class='latex-bold'>(A)</span>\ \sqrt 3 - 1 \qquad<span class='latex-bold'>(B)</span>\ \sqrt 2 - 3 \qquad<span class='latex-bold'>(C)</span>\ \sqrt 6 - \sqrt 2 \qquad<span class='latex-bold'>(D)</span>\ 2 - \sqrt 2 \qquad<span class='latex-bold'>(E)</span>\ 3 - \sqrt 3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
3
−
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2
−
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
6
−
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
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12
1
Hide problems
Turkey NMO 2007 1st Round - P12 (Combinatorics)
In how many ways can
10
10
10
distinct books be placed onto
3
3
3
-shelf bookcase in such a way that no shelf is empty?
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36
⋅
10
!
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50
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!
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⋅
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!
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⋅
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!
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None of the above
<span class='latex-bold'>(A)</span>\ 36\cdot 10! \qquad<span class='latex-bold'>(B)</span>\ 50 \cdot 10! \qquad<span class='latex-bold'>(C)</span>\ 55 \cdot 10! \qquad<span class='latex-bold'>(D)</span>\ 81 \cdot 10! \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
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None of the above
11
1
Hide problems
Turkey NMO 2007 1st Round - P11 (Algebra)
If
8
/
19
8/19
8/19
of the product of largest two elements of a positive integer set is not greater than the sum of other elements, what is the minimum possible value of the largest number in the set?
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20
<span class='latex-bold'>(A)</span>\ 8 \qquad<span class='latex-bold'>(B)</span>\ 12 \qquad<span class='latex-bold'>(C)</span>\ 13 \qquad<span class='latex-bold'>(D)</span>\ 19 \qquad<span class='latex-bold'>(E)</span>\ 20
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10
1
Hide problems
Turkey NMO 2007 1st Round - P10 (Number Theory)
How many positive integers
n
<
1
0
6
n<10^6
n
<
1
0
6
are there such that
n
n
n
is equal to twice of square of an integer and is equal to three times of cube of an integer?
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None of the above
<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{None of the above}
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None of the above
9
1
Hide problems
Turkey NMO 2007 1st Round - P09 (Geometry)
Let
∣
A
B
∣
=
3
|AB|=3
∣
A
B
∣
=
3
and the length of the altitude from
C
C
C
be
2
2
2
in
△
A
B
C
\triangle ABC
△
A
BC
. What is the maximum value of the product of the lengths of the other two altitudes?
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144
25
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None of the above
<span class='latex-bold'>(A)</span>\ \frac{144}{25} \qquad<span class='latex-bold'>(B)</span>\ 5 \qquad<span class='latex-bold'>(C)</span>\ 3\sqrt 2 \qquad<span class='latex-bold'>(D)</span>\ 6 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
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None of the above
8
1
Hide problems
Turkey NMO 2007 1st Round - P08 (Combinatorics)
Starting from the number
123456789
123456789
123456789
, at each step, we are swaping two adjacent numbers which are different from zero, and then decreasing the two numbers by
1
1
1
. What is the sum of digits of the least number that can be get after finite steps?
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9
<span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 5 \qquad<span class='latex-bold'>(E)</span>\ 9
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7
1
Hide problems
Turkey NMO 2007 1st Round - P07 (Algebra)
What is the sum of real numbers satisfying the equation
⌊
6
x
+
5
8
⌋
=
15
x
−
7
5
\left \lfloor \frac{6x+5}{8} \right \rfloor = \frac{15x-7}{5}
⌊
8
6
x
+
5
⌋
=
5
15
x
−
7
?
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90
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15
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15
<span class='latex-bold'>(A)</span>\ 2 \qquad<span class='latex-bold'>(B)</span>\ \frac{81}{90} \qquad<span class='latex-bold'>(C)</span>\ \frac{7}{15} \qquad<span class='latex-bold'>(D)</span>\ \frac{4}{5} \qquad<span class='latex-bold'>(E)</span>\ \frac{19}{15}
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81
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7
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19
6
1
Hide problems
Turkey NMO 2007 1st Round - P06 (Number Theory)
How many positive integers
n
n
n
are there such that
n
!
(
2
n
+
1
)
n!(2n+1)
n
!
(
2
n
+
1
)
and
221
221
221
are relatively prime?
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)
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)
<
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a
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None of the above
<span class='latex-bold'>(A)</span>\ 10 \qquad<span class='latex-bold'>(B)</span>\ 11 \qquad<span class='latex-bold'>(C)</span>\ 12 \qquad<span class='latex-bold'>(D)</span>\ 13 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
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10
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B
)
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11
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(
C
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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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None of the above
5
1
Hide problems
Turkey NMO 2007 1st Round - P05 (Geometry)
Let
C
C
C
and
D
D
D
be points on the semicircle with center
O
O
O
and diameter
A
B
AB
A
B
such that
A
B
C
D
ABCD
A
BC
D
is a convex quadrilateral. Let
Q
Q
Q
be the intersection of the diagonals
[
A
C
]
[AC]
[
A
C
]
and
[
B
D
]
[BD]
[
B
D
]
, and
P
P
P
be the intersection of the lines tangent to the semicircle at
C
C
C
and
D
D
D
. If
m
(
A
Q
B
^
)
=
2
m
(
C
O
D
^
)
m(\widehat{AQB})=2m(\widehat{COD})
m
(
A
QB
)
=
2
m
(
CO
D
)
and
∣
A
B
∣
=
2
|AB|=2
∣
A
B
∣
=
2
, then what is
∣
P
O
∣
|PO|
∣
PO
∣
?
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2
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(
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)
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3
<
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s
s
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(
C
)
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1
+
3
2
<
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s
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(
D
)
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1
+
3
2
2
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(
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2
3
3
<span class='latex-bold'>(A)</span>\ \sqrt 2 \qquad<span class='latex-bold'>(B)</span>\ \sqrt 3 \qquad<span class='latex-bold'>(C)</span>\ \frac{1+\sqrt 3} 2 \qquad<span class='latex-bold'>(D)</span>\ \frac{1+\sqrt 3}{2\sqrt 2} \qquad<span class='latex-bold'>(E)</span>\ \frac{2\sqrt 3} 3
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2
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(
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)
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3
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(
C
)
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2
1
+
3
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(
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2
2
1
+
3
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(
E
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3
2
3
4
1
Hide problems
Turkey NMO 2007 1st Round - P04 (Combinatorics)
How many ways are there to partition
7
7
7
students into the groups of
2
2
2
or
3
3
3
?
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(
A
)
<
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>
70
<
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s
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′
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−
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>
(
B
)
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>
105
<
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s
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(
C
)
<
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210
<
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l
a
s
s
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′
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(
D
)
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>
280
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(
E
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630
<span class='latex-bold'>(A)</span>\ 70 \qquad<span class='latex-bold'>(B)</span>\ 105 \qquad<span class='latex-bold'>(C)</span>\ 210 \qquad<span class='latex-bold'>(D)</span>\ 280 \qquad<span class='latex-bold'>(E)</span>\ 630
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70
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(
B
)
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105
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(
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)
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210
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280
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(
E
)
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>
630
3
1
Hide problems
Turkey NMO 2007 1st Round - P03 (Algebra)
Let
a
<
b
<
c
<
d
a<b<c<d
a
<
b
<
c
<
d
be integers. If one of the roots of the equation
(
x
−
a
)
(
x
−
b
)
(
x
−
c
)
(
x
−
d
)
−
9
(x-a)(x-b)(x-c)(x-d)-9
(
x
−
a
)
(
x
−
b
)
(
x
−
c
)
(
x
−
d
)
−
9
is
x
=
7
x=7
x
=
7
, what is
a
+
b
+
c
+
d
a+b+c+d
a
+
b
+
c
+
d
?
<
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s
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a
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(
A
)
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>
14
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s
s
=
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−
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>
(
B
)
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>
21
<
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s
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−
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(
C
)
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28
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s
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(
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42
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63
<span class='latex-bold'>(A)</span>\ 14 \qquad<span class='latex-bold'>(B)</span>\ 21 \qquad<span class='latex-bold'>(C)</span>\ 28 \qquad<span class='latex-bold'>(D)</span>\ 42 \qquad<span class='latex-bold'>(E)</span>\ 63
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14
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>
(
B
)
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21
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>
(
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28
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a
ss
=
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x
−
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>
(
D
)
<
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42
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E
)
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63
2
1
Hide problems
Turkey NMO 2007 1st Round - P02 (Number Theory)
What is the last three digits of base-4 representation of
10
⋅
3
195
⋅
4
9
49
10\cdot 3^{195}\cdot 49^{49}
10
⋅
3
195
⋅
4
9
49
?
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(
A
)
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>
112
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>
(
B
)
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130
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)
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132
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)
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212
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232
<span class='latex-bold'>(A)</span>\ 112 \qquad<span class='latex-bold'>(B)</span>\ 130 \qquad<span class='latex-bold'>(C)</span>\ 132 \qquad<span class='latex-bold'>(D)</span>\ 212 \qquad<span class='latex-bold'>(E)</span>\ 232
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112
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)
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130
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132
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212
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232
1
1
Hide problems
Turkey NMO 2007 1st Round - P01 (Geometry)
Let
A
B
C
ABC
A
BC
be a triangle with
m
(
A
^
)
=
9
0
∘
m(\widehat{A})=90^\circ
m
(
A
)
=
9
0
∘
,
∣
A
B
∣
=
4
|AB|=4
∣
A
B
∣
=
4
, and
∣
A
C
∣
=
3
|AC|=3
∣
A
C
∣
=
3
. Let
D
D
D
be the foot of perpendicular from
A
A
A
to
[
B
C
]
[BC]
[
BC
]
. If
P
P
P
a point on
[
B
D
]
[BD]
[
B
D
]
such that
5
∣
A
P
∣
=
13
∣
P
D
∣
5|AP|=13|PD|
5∣
A
P
∣
=
13∣
P
D
∣
, what is
∣
C
P
∣
|CP|
∣
CP
∣
?
<
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(
A
)
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>
9
+
4
3
5
<
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l
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−
b
o
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d
′
>
(
B
)
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>
56
15
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s
=
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>
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)
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14
5
<
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s
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−
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o
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>
(
D
)
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>
37
13
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E
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5
5
+
3
5
<span class='latex-bold'>(A)</span>\ \frac{9 + 4\sqrt 3}{5} \qquad<span class='latex-bold'>(B)</span>\ \frac{56}{15} \qquad<span class='latex-bold'>(C)</span>\ \frac{14}{5} \qquad<span class='latex-bold'>(D)</span>\ \frac{37}{13} \qquad<span class='latex-bold'>(E)</span>\ \frac{5\sqrt 5 + 3}{5}
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A
)
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5
9
+
4
3
<
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p
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c
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a
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a
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x
−
b
o
l
d
′
>
(
B
)
<
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>
15
56
<
s
p
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c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
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5
14
<
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c
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x
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o
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d
′
>
(
D
)
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13
37
<
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l
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x
−
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o
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>
(
E
)
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5
5
5
+
3