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Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2001 National Olympiad First Round
2001 National Olympiad First Round
Part of
National Olympiad First Round
Subcontests
(36)
36
1
Hide problems
P36 [Algebra] - Turkish NMO 1st Round - 2001
Let
a
a
a
and
b
b
b
be positive real numbers such that
a
b
(
a
−
b
)
=
1
ab(a-b)=1
ab
(
a
−
b
)
=
1
. Which of the followings can
a
2
+
b
2
a^2+b^2
a
2
+
b
2
take?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
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p
a
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>
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
>
11
<
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 preceding
<span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 2\sqrt 2 \qquad<span class='latex-bold'>(D)</span>\ \sqrt {11} \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
<
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
>
2
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
11
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the preceding
35
1
Hide problems
P35 [Number Theory] - Turkish NMO 1st Round - 2001
How many ordered pairs
(
p
,
n
)
(p,n)
(
p
,
n
)
are there such that
(
1
+
p
)
n
=
1
+
p
n
+
n
p
(1+p)^n = 1+pn + n^p
(
1
+
p
)
n
=
1
+
p
n
+
n
p
where
p
p
p
is a prime and
n
n
n
is a positive integer?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
5
<
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
>
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
0
<
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 preceding
<span class='latex-bold'>(A)</span>\ 5 \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 of the preceding}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
5
<
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
>
None of the preceding
34
1
Hide problems
P34 [Combinatorics] - Turkish NMO 1st Round - 2001
Let
f
f
f
be a real-valued function defined over ordered pairs of integers such that
f
(
x
+
3
m
−
2
n
,
y
−
4
m
+
5
n
)
=
f
(
x
,
y
)
f(x+3m-2n, y-4m+5n) = f(x,y)
f
(
x
+
3
m
−
2
n
,
y
−
4
m
+
5
n
)
=
f
(
x
,
y
)
for every integers
x
,
y
,
m
,
n
x,y,m,n
x
,
y
,
m
,
n
. At most how many elements does the range set of
f
f
f
have?
<
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p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
7
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
8
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
15
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
49
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
Infinitely many
<span class='latex-bold'>(A)</span>\ 7 \qquad<span class='latex-bold'>(B)</span>\ 8 \qquad<span class='latex-bold'>(C)</span>\ 15 \qquad<span class='latex-bold'>(D)</span>\ 49 \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
>
7
<
s
p
an
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
l
d
′
>
(
C
)
<
/
s
p
an
>
15
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
49
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
Infinitely many
33
1
Hide problems
P33 [Geometry] - Turkish NMO 1st Round - 2001
Let
A
B
C
ABC
A
BC
be a triangle such that
∣
A
C
∣
=
1
|AC|=1
∣
A
C
∣
=
1
and
∣
A
B
∣
=
2
|AB|=\sqrt 2
∣
A
B
∣
=
2
. Let
M
M
M
be a point such that
∣
M
A
∣
=
∣
A
B
∣
|MA|=|AB|
∣
M
A
∣
=
∣
A
B
∣
,
m
(
M
A
B
^
)
=
9
0
∘
m(\widehat{MAB}) = 90^\circ
m
(
M
A
B
)
=
9
0
∘
, and
C
C
C
and
M
M
M
are on the opposite sides of
A
B
AB
A
B
. Let
N
N
N
be a point such that
∣
N
A
∣
=
∣
A
X
∣
|NA|=|AX|
∣
N
A
∣
=
∣
A
X
∣
,
m
(
N
A
C
^
)
=
9
0
∘
m(\widehat{NAC}) = 90^\circ
m
(
N
A
C
)
=
9
0
∘
, and
B
B
B
and
N
N
N
are on the opposite sides of
A
C
AC
A
C
. If the line passing throung
A
A
A
and the circumcenter of triangle
M
A
N
MAN
M
A
N
meets
[
B
C
]
[BC]
[
BC
]
at
F
F
F
, what is
∣
B
F
∣
∣
F
C
∣
\dfrac {|BF|}{|FC|}
∣
FC
∣
∣
BF
∣
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
2
2
<
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
>
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
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
3
2
<span class='latex-bold'>(A)</span>\ 2\sqrt 2 \qquad<span class='latex-bold'>(B)</span>\ 2\sqrt 3 \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ 3 \qquad<span class='latex-bold'>(E)</span>\ 3\sqrt 2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
2
2
<
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
>
2
<
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
>
3
2
32
1
Hide problems
P32 [Algebra] - Turkish NMO 1st Round - 2001
What is the
33
33
33
-rd number after the decimal point of
(
10
+
3
)
2001
(\sqrt {10} + 3)^{2001}
(
10
+
3
)
2001
?
<
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
>
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
8
<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>\ 4 \qquad<span class='latex-bold'>(E)</span>\ 8
<
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
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
8
31
1
Hide problems
P31 [Number Theory] - Turkish NMO 1st Round - 2001
What is the largest integer
n
n
n
such that
2
n
+
65
2^n + 65
2
n
+
65
is equal to square of an integer?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
1024
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
268
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
10
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
None of the preceding
<span class='latex-bold'>(A)</span>\ 1024 \qquad<span class='latex-bold'>(B)</span>\ 268 \qquad<span class='latex-bold'>(C)</span>\ 10 \qquad<span class='latex-bold'>(D)</span>\ 4 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
1024
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
268
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
10
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the preceding
30
1
Hide problems
P30 [Combinatorics] - Turkish NMO 1st Round - 2001
There are
n
n
n
airports which form a regular
n
n
n
-gon. In the beginnig, there is exactly one plane at only
k
k
k
airports. Each of the planes flies to one of the nearest airport each day. For which of the following ordered pairs
(
n
,
k
)
(n,k)
(
n
,
k
)
, it is impossible to gather all planes at a airport on one day however the planes are arranged initially?
<
s
p
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n
c
l
a
s
s
=
′
l
a
t
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x
−
b
o
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d
′
>
(
A
)
<
/
s
p
a
n
>
(
10
,
6
)
<
s
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a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
(
10
,
4
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
(
11
,
3
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
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>
(
11
,
5
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
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x
−
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o
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>
(
E
)
<
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a
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>
(
13
,
8
)
<span class='latex-bold'>(A)</span>\ (10,6) \qquad<span class='latex-bold'>(B)</span>\ (10,4) \qquad<span class='latex-bold'>(C)</span>\ (11,3) \qquad<span class='latex-bold'>(D)</span>\ (11,5) \qquad<span class='latex-bold'>(E)</span>\ (13,8)
<
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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
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(
A
)
<
/
s
p
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>
(
10
,
6
)
<
s
p
an
c
l
a
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=
′
l
a
t
e
x
−
b
o
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d
′
>
(
B
)
<
/
s
p
an
>
(
10
,
4
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
C
)
<
/
s
p
an
>
(
11
,
3
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
(
11
,
5
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
(
13
,
8
)
29
1
Hide problems
P29 [Geometry] - Turkish NMO 1st Round - 2001
Let
A
B
C
D
ABCD
A
BC
D
be a isosceles trapezoid such that
A
B
∣
∣
C
D
AB || CD
A
B
∣∣
C
D
and all of its sides are tangent to a circle.
[
A
D
]
[AD]
[
A
D
]
touches this circle at
N
N
N
.
N
C
NC
NC
and
N
B
NB
NB
meet the circle again at
K
K
K
and
L
L
L
, respectively. What is
∣
B
N
∣
∣
B
L
∣
+
∣
C
N
∣
∣
C
K
∣
\dfrac {|BN|}{|BL|} + \dfrac {|CN|}{|CK|}
∣
B
L
∣
∣
BN
∣
+
∣
C
K
∣
∣
CN
∣
?
<
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(
A
)
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4
<
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a
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(
B
)
<
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>
6
<
s
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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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>
8
<
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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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>
9
<
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c
l
a
s
s
=
′
l
a
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x
−
b
o
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>
(
E
)
<
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>
10
<span class='latex-bold'>(A)</span>\ 4 \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>\ 10
<
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p
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c
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a
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=
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l
a
t
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x
−
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o
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(
A
)
<
/
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>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
B
)
<
/
s
p
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>
6
<
s
p
an
c
l
a
ss
=
′
l
a
t
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x
−
b
o
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>
(
C
)
<
/
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>
8
<
s
p
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c
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a
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=
′
l
a
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x
−
b
o
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d
′
>
(
D
)
<
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>
9
<
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a
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x
−
b
o
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d
′
>
(
E
)
<
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>
10
28
1
Hide problems
P28 [Algebra] - Turkish NMO 1st Round - 2001
The towns
A
,
B
,
C
,
D
,
E
A,B,C,D,E
A
,
B
,
C
,
D
,
E
are located clockwise on a circular road such that the distance between
A
A
A
and
B
B
B
,
B
B
B
and
C
C
C
,
C
C
C
and
D
D
D
,
E
E
E
and
A
A
A
are
5
5
5
,
5
5
5
,
2
2
2
,
1
1
1
and
4
4
4
km respectively. A health center will be located on that road such that the maximum of the shortest distance to each town will be minimum. How many alternative locations are there for the health center?
<
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(
A
)
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>
0
<
s
p
a
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c
l
a
s
s
=
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l
a
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e
x
−
b
o
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>
(
B
)
<
/
s
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>
1
<
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p
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c
l
a
s
s
=
′
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−
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>
(
C
)
<
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>
2
<
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p
a
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c
l
a
s
s
=
′
l
a
t
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x
−
b
o
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>
(
D
)
<
/
s
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a
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>
3
<
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a
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c
l
a
s
s
=
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a
t
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x
−
b
o
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>
(
E
)
<
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a
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>
None of the preceding
<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 preceding}
<
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a
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x
−
b
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d
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>
(
A
)
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>
0
<
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p
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a
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=
′
l
a
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x
−
b
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d
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>
(
B
)
<
/
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>
1
<
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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
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>
(
C
)
<
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>
2
<
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p
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c
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a
ss
=
′
l
a
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x
−
b
o
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′
>
(
D
)
<
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>
3
<
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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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x
−
b
o
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d
′
>
(
E
)
<
/
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an
>
None of the preceding
27
1
Hide problems
P27 [Number Theory] - Turkish NMO 1st Round - 2001
If decimal representation of
2
n
2^n
2
n
starts with
7
7
7
, what is the first digit in decimal representation of
5
n
5^n
5
n
?
<
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−
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(
A
)
<
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>
1
<
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a
s
s
=
′
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a
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x
−
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o
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′
>
(
B
)
<
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a
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>
3
<
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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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>
5
<
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c
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a
s
s
=
′
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a
t
e
x
−
b
o
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>
(
D
)
<
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>
7
<
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−
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(
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)
<
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9
<span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 3 \qquad<span class='latex-bold'>(C)</span>\ 5 \qquad<span class='latex-bold'>(D)</span>\ 7 \qquad<span class='latex-bold'>(E)</span>\ 9
<
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a
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=
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a
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x
−
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o
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d
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(
A
)
<
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>
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
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x
−
b
o
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d
′
>
(
B
)
<
/
s
p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
C
)
<
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5
<
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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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x
−
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d
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>
(
D
)
<
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7
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−
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>
(
E
)
<
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>
9
26
1
Hide problems
P26 [Combinatorics] - Turkish NMO 1st Round - 2001
Berk tries to guess the two-digit number that Ayca picks. After each guess, Ayca gives a hint indicating the number of digits which match the number picked. If Berk can guarantee to guess Ayca's number in
n
n
n
guesses, what is the smallest possible value of
n
n
n
?
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(
A
)
<
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>
9
<
s
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a
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l
a
s
s
=
′
l
a
t
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x
−
b
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d
′
>
(
B
)
<
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a
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>
10
<
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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
′
>
(
C
)
<
/
s
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a
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>
11
<
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a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
D
)
<
/
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a
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>
15
<
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a
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=
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a
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−
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>
(
E
)
<
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>
20
<span class='latex-bold'>(A)</span>\ 9 \qquad<span class='latex-bold'>(B)</span>\ 10 \qquad<span class='latex-bold'>(C)</span>\ 11 \qquad<span class='latex-bold'>(D)</span>\ 15 \qquad<span class='latex-bold'>(E)</span>\ 20
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x
−
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(
A
)
<
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>
9
<
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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
o
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d
′
>
(
B
)
<
/
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>
10
<
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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
′
>
(
C
)
<
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>
11
<
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c
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a
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=
′
l
a
t
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x
−
b
o
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d
′
>
(
D
)
<
/
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p
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>
15
<
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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
′
>
(
E
)
<
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>
20
25
1
Hide problems
P25 [Geometry] - Turkish NMO 1st Round - 2001
The circumradius of acute triangle
A
B
C
ABC
A
BC
is twice of the distance of its circumcenter to
A
B
AB
A
B
. If
∣
A
C
∣
=
2
|AC|=2
∣
A
C
∣
=
2
and
∣
B
C
∣
=
3
|BC|=3
∣
BC
∣
=
3
, what is the altitude passing through
C
C
C
?
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(
A
)
<
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>
14
<
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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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d
′
>
(
B
)
<
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a
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>
3
7
21
<
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p
a
n
c
l
a
s
s
=
′
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a
t
e
x
−
b
o
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d
′
>
(
C
)
<
/
s
p
a
n
>
4
7
21
<
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
)
<
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a
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>
1
2
21
<
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a
s
s
=
′
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x
−
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>
(
E
)
<
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>
2
3
14
<span class='latex-bold'>(A)</span>\ \sqrt {14} \qquad<span class='latex-bold'>(B)</span>\ \dfrac{3}{7}\sqrt{21} \qquad<span class='latex-bold'>(C)</span>\ \dfrac{4}{7}\sqrt{21} \qquad<span class='latex-bold'>(D)</span>\ \dfrac{1}{2}\sqrt{21} \qquad<span class='latex-bold'>(E)</span>\ \dfrac{2}{3}\sqrt{14}
<
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<
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14
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=
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(
B
)
<
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>
7
3
21
<
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=
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a
t
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x
−
b
o
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d
′
>
(
C
)
<
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>
7
4
21
<
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p
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c
l
a
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=
′
l
a
t
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x
−
b
o
l
d
′
>
(
D
)
<
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>
2
1
21
<
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c
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a
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=
′
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x
−
b
o
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>
(
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)
<
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3
2
14
24
1
Hide problems
P24 [Algebra] - Turkish NMO 1st Round - 2001
How many real roots of the equation
x
2
−
18
[
x
]
+
77
=
0
x^2 - 18[x]+77=0
x
2
−
18
[
x
]
+
77
=
0
are not integer, where
[
x
]
[x]
[
x
]
denotes the greatest integer not exceeding the real number
x
x
x
?
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a
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>
(
A
)
<
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a
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>
0
<
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c
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a
s
s
=
′
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a
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x
−
b
o
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d
′
>
(
B
)
<
/
s
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a
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>
1
<
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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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>
2
<
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
D
)
<
/
s
p
a
n
>
3
<
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a
s
s
=
′
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x
−
b
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′
>
(
E
)
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a
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>
None of the preceding
<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 preceding}
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a
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x
−
b
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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
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>
1
<
s
p
an
c
l
a
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=
′
l
a
t
e
x
−
b
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d
′
>
(
C
)
<
/
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p
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>
2
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p
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c
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a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
D
)
<
/
s
p
an
>
3
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c
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a
ss
=
′
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a
t
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x
−
b
o
l
d
′
>
(
E
)
<
/
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p
an
>
None of the preceding
23
1
Hide problems
P23 [Number Theory] - Turkish NMO 1st Round - 2001
Which of the followings is false for the sequence
9
,
99
,
999
,
…
9,99,999,\dots
9
,
99
,
999
,
…
?
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=
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a
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−
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′
>
(
A
)
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a
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>
<span class='latex-bold'>(A)</span>
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a
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a
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x
−
b
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d
′
>
(
A
)
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p
an
>
The primes which do not divide any term of the sequence are finite.
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=
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a
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x
−
b
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′
>
(
B
)
<
/
s
p
a
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>
<span class='latex-bold'>(B)</span>
<
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p
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c
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a
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=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
Infinitely many primes divide infinitely many terms of the sequence.
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=
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x
−
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>
(
C
)
<
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a
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>
<span class='latex-bold'>(C)</span>
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a
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=
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a
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e
x
−
b
o
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d
′
>
(
C
)
<
/
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p
an
>
For every positive integer
n
n
n
, there is a term which is divisible by at least
n
n
n
distinct prime numbers.
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a
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x
−
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>
(
D
)
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p
a
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>
<span class='latex-bold'>(D)</span>
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=
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x
−
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>
(
D
)
<
/
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p
an
>
There is an inteter
n
n
n
such that every prime number greater than
n
n
n
divides infinitely many terms of the sequence.
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a
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=
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x
−
b
o
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d
′
>
(
E
)
<
/
s
p
a
n
>
<span class='latex-bold'>(E)</span>
<
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p
an
c
l
a
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=
′
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a
t
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x
−
b
o
l
d
′
>
(
E
)
<
/
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p
an
>
None of above
22
1
Hide problems
P22 [Combinatorics] - Turkish NMO 1st Round - 2001
A ladder is formed by removing some consecutive unit squares of a
10
×
10
10\times 10
10
×
10
chessboard such that for each
k
−
k-
k
−
th row (
k
∈
{
1
,
2
,
…
,
10
}
k\in \{1,2,\dots, 10\}
k
∈
{
1
,
2
,
…
,
10
}
), the leftmost
k
−
1
k-1
k
−
1
unit squares are removed. How many rectangles formed by composition of unit squares does the ladder have?
<
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a
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x
−
b
o
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d
′
>
(
A
)
<
/
s
p
a
n
>
625
<
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a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
715
<
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a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
1024
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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
)
<
/
s
p
a
n
>
1512
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a
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s
=
′
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a
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e
x
−
b
o
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′
>
(
E
)
<
/
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p
a
n
>
None of the preceding
<span class='latex-bold'>(A)</span>\ 625 \qquad<span class='latex-bold'>(B)</span>\ 715 \qquad<span class='latex-bold'>(C)</span>\ 1024 \qquad<span class='latex-bold'>(D)</span>\ 1512 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
<
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a
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x
−
b
o
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d
′
>
(
A
)
<
/
s
p
an
>
625
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
715
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
1024
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
1512
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c
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a
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=
′
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x
−
b
o
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d
′
>
(
E
)
<
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>
None of the preceding
21
1
Hide problems
P21 [Geometry] - Turkish NMO 1st Round - 2001
Let
b
b
b
be the length of the largest diagonal and
c
c
c
be the length of the smallest diagonal of a regular nonagon with side length
a
a
a
. Which one of the followings is true?
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−
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>
(
A
)
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>
b
=
a
+
c
2
<
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a
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s
=
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x
−
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′
>
(
B
)
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>
b
=
a
c
<
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a
s
s
=
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a
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x
−
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>
(
C
)
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p
a
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>
b
2
=
a
2
+
c
2
2
<
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c
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a
s
s
=
′
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a
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x
−
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′
>
(
D
)
<
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a
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>
c
=
a
+
b
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a
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−
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>
(
E
)
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>
c
2
=
a
2
+
b
2
<span class='latex-bold'>(A)</span>\ b=\dfrac{a+c}2 \qquad<span class='latex-bold'>(B)</span>\ b=\sqrt {ac} \qquad<span class='latex-bold'>(C)</span>\ b^2=\dfrac{a^2+c^2}2 \\ <span class='latex-bold'>(D)</span>\ c=a+b \qquad<span class='latex-bold'>(E)</span>\ c^2=a^2+b^2
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−
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>
(
A
)
<
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p
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>
b
=
2
a
+
c
<
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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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x
−
b
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>
(
B
)
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/
s
p
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>
b
=
a
c
<
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p
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c
l
a
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=
′
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a
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x
−
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′
>
(
C
)
<
/
s
p
an
>
b
2
=
2
a
2
+
c
2
<
s
p
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c
l
a
ss
=
′
l
a
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x
−
b
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d
′
>
(
D
)
<
/
s
p
an
>
c
=
a
+
b
<
s
p
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c
l
a
ss
=
′
l
a
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x
−
b
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d
′
>
(
E
)
<
/
s
p
an
>
c
2
=
a
2
+
b
2
20
1
Hide problems
P20 [Algebra] - Turkish NMO 1st Round - 2001
If the sum of any
10
10
10
of
21
21
21
real numbers is less than the sum of remaining
11
11
11
of them, at least how many of these
21
21
21
numbers are positive?
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(
A
)
<
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a
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>
18
<
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c
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a
s
s
=
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a
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x
−
b
o
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d
′
>
(
B
)
<
/
s
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a
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>
19
<
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c
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a
s
s
=
′
l
a
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x
−
b
o
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d
′
>
(
C
)
<
/
s
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a
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>
20
<
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
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d
′
>
(
D
)
<
/
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a
n
>
21
<
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s
=
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a
t
e
x
−
b
o
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′
>
(
E
)
<
/
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p
a
n
>
None of the preceding
<span class='latex-bold'>(A)</span>\ 18 \qquad<span class='latex-bold'>(B)</span>\ 19 \qquad<span class='latex-bold'>(C)</span>\ 20 \qquad<span class='latex-bold'>(D)</span>\ 21 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
<
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p
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c
l
a
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=
′
l
a
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x
−
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o
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>
(
A
)
<
/
s
p
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>
18
<
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c
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a
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=
′
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a
t
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x
−
b
o
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′
>
(
B
)
<
/
s
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>
19
<
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p
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c
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a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
C
)
<
/
s
p
an
>
20
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
21
<
s
p
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c
l
a
ss
=
′
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a
t
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x
−
b
o
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d
′
>
(
E
)
<
/
s
p
an
>
None of the preceding
19
1
Hide problems
P19 [Number Theory] - Turkish NMO 1st Round - 2001
If the integers
m
,
n
,
k
m,n,k
m
,
n
,
k
hold the equation
221
m
+
247
n
+
323
k
=
2001
221m+247n+323k=2001
221
m
+
247
n
+
323
k
=
2001
, what is the smallest possible value of
k
k
k
greater than
100
100
100
?
<
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−
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>
(
A
)
<
/
s
p
a
n
>
124
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
111
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
107
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
101
<
s
p
a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
None of the preceding
<span class='latex-bold'>(A)</span>\ 124 \qquad<span class='latex-bold'>(B)</span>\ 111 \qquad<span class='latex-bold'>(C)</span>\ 107 \qquad<span class='latex-bold'>(D)</span>\ 101 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
124
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
111
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
107
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
101
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the preceding
18
1
Hide problems
P18 [Combinatorics] - Turkish NMO 1st Round - 2001
A convex polygon has at least one side with length
1
1
1
. If all diagonals of the polygon have integer lengths, at most how many sides does the polygon have?
<
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s
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=
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−
b
o
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d
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>
(
A
)
<
/
s
p
a
n
>
3
<
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
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a
n
>
7
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
10
<
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 preceding
<span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ 5 \qquad<span class='latex-bold'>(C)</span>\ 7 \qquad<span class='latex-bold'>(D)</span>\ 10 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
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>
(
A
)
<
/
s
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an
>
3
<
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p
an
c
l
a
ss
=
′
l
a
t
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−
b
o
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d
′
>
(
B
)
<
/
s
p
an
>
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
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>
7
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
10
<
s
p
an
c
l
a
ss
=
′
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a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the preceding
17
1
Hide problems
P17 [Geometry] - Turkish NMO 1st Round - 2001
Let
A
B
C
ABC
A
BC
be a triangle such that midpoints of three altitudes are collinear. If the largest side of triangle is
10
10
10
, what is the largest possible area of the triangle?
<
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c
l
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s
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−
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(
A
)
<
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>
20
<
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c
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a
s
s
=
′
l
a
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e
x
−
b
o
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d
′
>
(
B
)
<
/
s
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>
25
<
s
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
C
)
<
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>
30
<
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n
c
l
a
s
s
=
′
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t
e
x
−
b
o
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>
(
D
)
<
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>
40
<
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=
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−
b
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>
(
E
)
<
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>
50
<span class='latex-bold'>(A)</span>\ 20 \qquad<span class='latex-bold'>(B)</span>\ 25 \qquad<span class='latex-bold'>(C)</span>\ 30 \qquad<span class='latex-bold'>(D)</span>\ 40 \qquad<span class='latex-bold'>(E)</span>\ 50
<
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p
an
c
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a
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=
′
l
a
t
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x
−
b
o
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d
′
>
(
A
)
<
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>
20
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
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>
25
<
s
p
an
c
l
a
ss
=
′
l
a
t
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x
−
b
o
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d
′
>
(
C
)
<
/
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>
30
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
40
<
s
p
an
c
l
a
ss
=
′
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a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
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an
>
50
16
1
Hide problems
P16 [Algebra] - Turkish NMO 1st Round - 2001
The polynomial
P
(
x
)
=
x
3
+
a
x
+
1
P(x)=x^3+ax+1
P
(
x
)
=
x
3
+
a
x
+
1
has exactly one solution on the interval
[
−
2
,
0
)
[-2,0)
[
−
2
,
0
)
and has exactly one solution on the interval
(
0
,
1
]
(0,1]
(
0
,
1
]
where
a
a
a
is a real number. Which of the followings cannot be equal to
P
(
2
)
P(2)
P
(
2
)
?
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(
A
)
<
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>
17
<
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c
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=
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x
−
b
o
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>
(
B
)
<
/
s
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>
30
3
<
s
p
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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′
>
(
C
)
<
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>
26
−
1
<
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n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
D
)
<
/
s
p
a
n
>
30
<
s
p
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c
l
a
s
s
=
′
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a
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x
−
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d
′
>
(
E
)
<
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>
10
3
<span class='latex-bold'>(A)</span>\ \sqrt{17} \qquad<span class='latex-bold'>(B)</span>\ \sqrt[3]{30} \qquad<span class='latex-bold'>(C)</span>\ \sqrt{26}-1 \qquad<span class='latex-bold'>(D)</span>\ \sqrt {30} \qquad<span class='latex-bold'>(E)</span>\ \sqrt [3]{10}
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−
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A
)
<
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>
17
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=
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(
B
)
<
/
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>
3
30
<
s
p
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c
l
a
ss
=
′
l
a
t
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x
−
b
o
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d
′
>
(
C
)
<
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>
26
−
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
D
)
<
/
s
p
an
>
30
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
3
10
15
1
Hide problems
P15 [Number Theory] - Turkish NMO 1st Round - 2001
How many different solutions does the congruence
x
3
+
3
x
2
+
x
+
3
≡
0
(
m
o
d
25
)
x^3+3x^2+x+3 \equiv 0 \pmod{25}
x
3
+
3
x
2
+
x
+
3
≡
0
(
mod
25
)
have?
<
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−
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(
A
)
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>
0
<
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a
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s
=
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−
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>
(
B
)
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2
<
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=
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C
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>
4
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a
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−
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>
(
D
)
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>
5
<
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=
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x
−
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>
(
E
)
<
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>
6
<span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 2 \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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c
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x
−
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o
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d
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>
(
A
)
<
/
s
p
an
>
0
<
s
p
an
c
l
a
ss
=
′
l
a
t
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x
−
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o
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d
′
>
(
B
)
<
/
s
p
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>
2
<
s
p
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c
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a
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=
′
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a
t
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−
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>
(
C
)
<
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>
4
<
s
p
an
c
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a
ss
=
′
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a
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x
−
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(
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)
<
/
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>
5
<
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=
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−
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o
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d
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>
(
E
)
<
/
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>
6
14
1
Hide problems
P14 [Combinatorics] - Turkish NMO 1st Round - 2001
Let
x
1
,
x
2
,
…
,
x
n
x_1, x_2, \dots, x_n
x
1
,
x
2
,
…
,
x
n
be a positive integer sequence such that each term is less than or equal to
2001
2001
2001
and for every
i
≥
3
i\geq 3
i
≥
3
,
x
i
=
∣
x
i
−
1
−
x
i
−
2
∣
x_i = |x_{i-1}-x_{i-2}|
x
i
=
∣
x
i
−
1
−
x
i
−
2
∣
. What is the largest possible value of
n
n
n
?
<
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s
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a
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−
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(
A
)
<
/
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a
n
>
1000
<
s
p
a
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c
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a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
B
)
<
/
s
p
a
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>
2001
<
s
p
a
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c
l
a
s
s
=
′
l
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t
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x
−
b
o
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d
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>
(
C
)
<
/
s
p
a
n
>
3002
<
s
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c
l
a
s
s
=
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l
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t
e
x
−
b
o
l
d
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>
(
D
)
<
/
s
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a
n
>
4003
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=
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x
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None of the preceding
<span class='latex-bold'>(A)</span>\ 1000 \qquad<span class='latex-bold'>(B)</span>\ 2001 \qquad<span class='latex-bold'>(C)</span>\ 3002 \qquad<span class='latex-bold'>(D)</span>\ 4003 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
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>
1000
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2001
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
3002
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
4003
<
s
p
an
c
l
a
ss
=
′
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a
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x
−
b
o
l
d
′
>
(
E
)
<
/
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an
>
None of the preceding
13
1
Hide problems
P13 [Geometry] - Turkish NMO 1st Round - 2001
Let
A
B
C
ABC
A
BC
be a triangle such that
∣
B
C
∣
=
7
|BC|=7
∣
BC
∣
=
7
and
∣
A
B
∣
=
9
|AB|=9
∣
A
B
∣
=
9
. If
m
(
A
B
C
^
)
=
2
m
(
B
C
A
^
)
m(\widehat{ABC}) = 2m(\widehat{BCA})
m
(
A
BC
)
=
2
m
(
BC
A
)
, then what is the area of the triangle?
<
s
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c
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a
s
s
=
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l
a
t
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x
−
b
o
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d
′
>
(
A
)
<
/
s
p
a
n
>
14
5
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
30
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
10
6
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
20
2
<
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a
s
s
=
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−
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(
E
)
<
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12
3
<span class='latex-bold'>(A)</span>\ 14\sqrt 5 \qquad<span class='latex-bold'>(B)</span>\ 30 \qquad<span class='latex-bold'>(C)</span>\ 10\sqrt 6 \qquad<span class='latex-bold'>(D)</span>\ 20 \sqrt 2 \qquad<span class='latex-bold'>(E)</span>\ 12 \sqrt 3
<
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a
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=
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a
t
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x
−
b
o
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d
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(
A
)
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/
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>
14
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
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an
>
30
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
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an
>
10
6
<
s
p
an
c
l
a
ss
=
′
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a
t
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x
−
b
o
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d
′
>
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D
)
<
/
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an
>
20
2
<
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p
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c
l
a
ss
=
′
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a
t
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x
−
b
o
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d
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>
(
E
)
<
/
s
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an
>
12
3
12
1
Hide problems
P12 [Algebra] - Turkish NMO 1st Round - 2001
A circle with center
O
O
O
and radius
15
15
15
is given. Let
P
P
P
be a point such that
∣
O
P
∣
=
9
|OP|=9
∣
OP
∣
=
9
. How many of the chords of the circle pass through
P
P
P
and have integer length?
<
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
>
11
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
12
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
13
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
14
<
s
p
a
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c
l
a
s
s
=
′
l
a
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e
x
−
b
o
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d
′
>
(
E
)
<
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a
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>
29
<span class='latex-bold'>(A)</span>\ 11 \qquad<span class='latex-bold'>(B)</span>\ 12 \qquad<span class='latex-bold'>(C)</span>\ 13 \qquad<span class='latex-bold'>(D)</span>\ 14 \qquad<span class='latex-bold'>(E)</span>\ 29
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
11
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
12
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
13
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
14
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
29
11
1
Hide problems
P11 [Number Theory] - Turkish NMO 1st Round - 2001
For how many integers
n
n
n
, does the equation system
2
x
+
3
y
=
7
5
x
+
n
y
=
n
2
\begin{array}{rcl} 2x+3y &=& 7\\ 5x + ny &=& n^2 \end{array}
2
x
+
3
y
5
x
+
n
y
=
=
7
n
2
have a solution over integers?
<
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
>
3
<
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
>
8
<
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 preceding
<span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 3 \qquad<span class='latex-bold'>(C)</span>\ 4 \qquad<span class='latex-bold'>(D)</span>\ 8 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
<
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
>
3
<
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
>
8
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the preceding
10
1
Hide problems
P10 [Combinatorics] - Turkish NMO 1st Round - 2001
At each step, we are changing the places of exactly two numbers from the sequence
1
1
1
,
2
2
2
,
3
3
3
,
4
4
4
,
5
5
5
,
6
6
6
,
7
7
7
. How many different arrangements can be formed after two steps?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
88
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
100
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
120
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
176
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
441
<span class='latex-bold'>(A)</span>\ 88 \qquad<span class='latex-bold'>(B)</span>\ 100 \qquad<span class='latex-bold'>(C)</span>\ 120 \qquad<span class='latex-bold'>(D)</span>\ 176 \qquad<span class='latex-bold'>(E)</span>\ 441
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
88
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
100
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
120
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
176
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
441
9
1
Hide problems
P09 [Geometry] - Turkish NMO 1st Round - 2001
What is the largest possible area of an isosceles trapezoid in which the largest side is
13
13
13
and the perimeter is
28
28
28
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
13
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
24
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
27
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
28
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
30
<span class='latex-bold'>(A)</span>\ 13 \qquad<span class='latex-bold'>(B)</span>\ 24 \qquad<span class='latex-bold'>(C)</span>\ 27 \qquad<span class='latex-bold'>(D)</span>\ 28 \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
>
13
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
24
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
27
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
28
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
30
8
1
Hide problems
P08 [Algebra] - Turkish NMO 1st Round - 2001
Which of the followings gives the product of the real roots of the equation
x
4
+
3
x
3
+
5
x
2
+
21
x
−
14
=
0
x^4+3x^3+5x^2 + 21x -14=0
x
4
+
3
x
3
+
5
x
2
+
21
x
−
14
=
0
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
−
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
7
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
−
14
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
21
<
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 preceding
<span class='latex-bold'>(A)</span>\ -2 \qquad<span class='latex-bold'>(B)</span>\ 7 \qquad<span class='latex-bold'>(C)</span>\ -14 \qquad<span class='latex-bold'>(D)</span>\ 21 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
−
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
7
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
−
14
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
21
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the preceding
7
1
Hide problems
P07 [Number Theory] - Turkish NMO 1st Round - 2001
How many ordered triples of positive integers
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
are there such that
(
2
a
+
b
)
(
2
b
+
a
)
=
2
c
(2a+b)(2b+a)=2^c
(
2
a
+
b
)
(
2
b
+
a
)
=
2
c
?
<
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
>
3
<
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 preceding
<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 preceding}
<
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
<
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
>
None of the preceding
6
1
Hide problems
P06 [Combinatorics] - Turkish NMO 1st Round - 2001
How many
5
−
5-
5
−
digit positive numbers which contain only odd numbers are there such that there is at least one pair of consecutive digits whose sum is
10
10
10
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
3125
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
2500
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
1845
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
1190
<
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 preceding
<span class='latex-bold'>(A)</span>\ 3125 \qquad<span class='latex-bold'>(B)</span>\ 2500 \qquad<span class='latex-bold'>(C)</span>\ 1845 \qquad<span class='latex-bold'>(D)</span>\ 1190 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
3125
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2500
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
1845
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
1190
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the preceding
5
1
Hide problems
P05 [Geometry] - Turkish NMO 1st Round - 2001
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid such that
A
B
∥
C
D
AB \parallel CD
A
B
∥
C
D
,
∣
A
B
∣
<
∣
C
D
∣
|AB|<|CD|
∣
A
B
∣
<
∣
C
D
∣
, and
Area
(
A
B
C
)
=
30
\text{Area}(ABC)=30
Area
(
A
BC
)
=
30
. Let the line through
B
B
B
parallel to
A
D
AD
A
D
meet
[
A
C
]
[AC]
[
A
C
]
at
E
E
E
. If
∣
A
E
∣
:
∣
E
C
∣
=
3
:
2
|AE|:|EC|=3:2
∣
A
E
∣
:
∣
EC
∣
=
3
:
2
, then what is the area of trapezoid
A
B
C
D
ABCD
A
BC
D
?
<
s
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c
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a
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s
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a
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−
b
o
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′
>
(
A
)
<
/
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p
a
n
>
45
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
60
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
72
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
80
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
90
<span class='latex-bold'>(A)</span>\ 45 \qquad<span class='latex-bold'>(B)</span>\ 60 \qquad<span class='latex-bold'>(C)</span>\ 72 \qquad<span class='latex-bold'>(D)</span>\ 80 \qquad<span class='latex-bold'>(E)</span>\ 90
<
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p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
45
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
60
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
72
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
80
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
90
4
1
Hide problems
P04 [Algebra] - Turkish NMO 1st Round - 2001
How many real solution does the equation
x
2000
2001
+
2
3
x
2
−
2
5
x
+
3
\dfrac{x^{2000}}{2001} + 2\sqrt 3 x^2 - 2\sqrt 5 x + \sqrt 3
2001
x
2000
+
2
3
x
2
−
2
5
x
+
3
have?
<
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
>
11
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
12
<
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 preceding
<span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ 11 \qquad<span class='latex-bold'>(D)</span>\ 12 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
<
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
>
11
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
12
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the preceding
3
1
Hide problems
P03 [Number Theory] - Turkish NMO 1st Round - 2001
How many primes
p
p
p
are there such that
2
p
4
−
7
p
2
+
1
2p^4-7p^2+1
2
p
4
−
7
p
2
+
1
is equal to square of an integer?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
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
>
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
Infinitely many
<
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 preceding
<span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ 4 \qquad<span class='latex-bold'>(D)</span>\ \text{Infinitely many} \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
<
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
>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
Infinitely many
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
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p
an
>
None of the preceding
2
1
Hide problems
P02 [Combinatorics] - Turkish NMO 1st Round - 2001
Each of the football teams Istanbulspor, Yesildirek, Vefa, Karagumruk, and Adalet, played exactly one match against the other four teams. Istanbulspor defeated all teams except Yesildirek; Yesildirek defeated Istanbulspor but lost to all the other teams. Vefa defeated all except Istanbulspor. The winner of the game Karagumruk-Adalet is Karagumruk. In how many ways one can order these five teams such that each team except the last, defeated the next team?
<
s
p
a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
5
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
7
<
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
p
a
n
>
None of the preceding
<span class='latex-bold'>(A)</span>\ 5 \qquad<span class='latex-bold'>(B)</span>\ 7 \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 preceding}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
7
<
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 preceding
1
1
Hide problems
P01 [Geometry] - Turkish NMO 1st Round - 2001
Let
A
,
B
,
C
A,B,C
A
,
B
,
C
be points on
[
O
X
[OX
[
OX
and
D
,
E
,
F
D,E,F
D
,
E
,
F
be points on
[
O
Y
[OY
[
O
Y
such that
∣
O
A
∣
=
∣
A
B
∣
=
∣
B
C
∣
|OA|=|AB|=|BC|
∣
O
A
∣
=
∣
A
B
∣
=
∣
BC
∣
and
∣
O
D
∣
=
∣
D
E
∣
=
∣
E
F
∣
|OD|=|DE|=|EF|
∣
O
D
∣
=
∣
D
E
∣
=
∣
EF
∣
. If
∣
O
A
∣
>
∣
O
D
∣
|OA|>|OD|
∣
O
A
∣
>
∣
O
D
∣
, which one below is true?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
<span class='latex-bold'>(A)</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
For every
X
O
Y
^
\widehat{XOY}
XO
Y
,
Area
(
A
E
C
)
>
Area
(
D
B
F
)
\text{ Area}(AEC)>\text{Area}(DBF)
Area
(
A
EC
)
>
Area
(
D
BF
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
<span class='latex-bold'>(B)</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
For every
X
O
Y
^
\widehat{XOY}
XO
Y
,
Area
(
A
E
C
)
=
Area
(
D
B
F
)
\text{ Area}(AEC)=\text{Area}(DBF)
Area
(
A
EC
)
=
Area
(
D
BF
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
<span class='latex-bold'>(C)</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
For every
X
O
Y
^
\widehat{XOY}
XO
Y
,
Area
(
A
E
C
)
<
Area
(
D
B
F
)
\text{ Area}(AEC)<\text{Area}(DBF)
Area
(
A
EC
)
<
Area
(
D
BF
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
<span class='latex-bold'>(D)</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
If
m
(
X
O
Y
^
)
<
4
5
∘
m(\widehat{XOY})<45^\circ
m
(
XO
Y
)
<
4
5
∘
then
Area
(
A
E
C
)
<
Area
(
D
B
F
)
\text{Area}(AEC)<\text{Area}(DBF)
Area
(
A
EC
)
<
Area
(
D
BF
)
, and if
4
5
∘
<
m
(
X
O
Y
^
)
<
9
0
∘
45^\circ < m(\widehat{XOY})<90^\circ
4
5
∘
<
m
(
XO
Y
)
<
9
0
∘
then
Area
(
A
E
C
)
>
Area
(
D
B
F
)
\text{Area}(AEC)>\text{Area}(DBF)
Area
(
A
EC
)
>
Area
(
D
BF
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
<span class='latex-bold'>(E)</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of above