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National and Regional Contests
Turkey Contests
National Olympiad First Round
2003 National Olympiad First Round
2003 National Olympiad First Round
Part of
National Olympiad First Round
Subcontests
(36)
36
1
Hide problems
P36 [Algebra] - Turkish NMO 1st Round - 2003
a
1
,
a
2
,
⋯
,
a
2003
a_1,a_2, \cdots , a_{2003}
a
1
,
a
2
,
⋯
,
a
2003
are integers such that
∣
a
1
∣
=
1
|a_1| = 1
∣
a
1
∣
=
1
and
∣
a
i
+
1
∣
=
∣
a
i
+
1
∣
|a_{i+1}|=|a_i+1|
∣
a
i
+
1
∣
=
∣
a
i
+
1∣
(
1
≤
i
≤
2002
)
(1\leq i \leq 2002)
(
1
≤
i
≤
2002
)
. What is the minimal value of
∣
a
1
+
a
2
+
⋯
+
a
2003
∣
|a_1+a_2+\cdots + a_{2003}|
∣
a
1
+
a
2
+
⋯
+
a
2003
∣
?
<
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a
s
s
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b
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>
(
A
)
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4
<
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a
s
s
=
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>
(
B
)
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34
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
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>
56
<
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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>
65
<
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c
l
a
s
s
=
′
l
a
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x
−
b
o
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d
′
>
(
E
)
<
/
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a
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>
None of the preceding
<span class='latex-bold'>(A)</span>\ 4 \qquad<span class='latex-bold'>(B)</span>\ 34 \qquad<span class='latex-bold'>(C)</span>\ 56 \qquad<span class='latex-bold'>(D)</span>\ 65 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
<
s
p
an
c
l
a
ss
=
′
l
a
t
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x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
4
<
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
>
34
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
56
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
65
<
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 [Combinatorics] - Turkish NMO 1st Round - 2003
n
+
m
−
1
n+m-1
n
+
m
−
1
unit squares are arranged in
L
L
L
-shape whose one side contains
n
n
n
squares and the other side contains
m
m
m
squares. Ayse and Betul plays a turn based game with following rules: Ayse plays first. At each move, the player captures desired number of adjacent squares in same side of
L
L
L
. The one who captures the last square loses the game. If four games are played for pairs
(
n
,
m
)
=
(
2003
,
2003
)
(n,m)=(2003,2003)
(
n
,
m
)
=
(
2003
,
2003
)
,
(
2002
,
2003
)
(2002,2003)
(
2002
,
2003
)
,
(
2003
,
3
)
(2003,3)
(
2003
,
3
)
,
(
2001
,
2003
)
(2001,2003)
(
2001
,
2003
)
; how many times can Ayse guarantee to win?
<
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s
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>
(
A
)
<
/
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0
<
s
p
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c
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a
s
s
=
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t
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x
−
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>
(
B
)
<
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>
1
<
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c
l
a
s
s
=
′
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a
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x
−
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>
(
C
)
<
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2
<
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c
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a
s
s
=
′
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a
t
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x
−
b
o
l
d
′
>
(
D
)
<
/
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>
3
<
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a
s
s
=
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−
b
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′
>
(
E
)
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4
<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>\ 4
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x
−
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d
′
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(
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
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an
>
4
34
1
Hide problems
P34 [Number Theory] - Turkish NMO 1st Round - 2003
If the sum of digits of only
m
m
m
and
m
+
n
m+n
m
+
n
from the numbers
m
m
m
,
m
+
1
m+1
m
+
1
,
⋯
\cdots
⋯
,
m
+
n
m+n
m
+
n
are divisible by
8
8
8
where
m
m
m
and
n
n
n
are positive integers, what is the largest possible value of
n
n
n
?
<
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(
A
)
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12
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s
s
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>
(
B
)
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>
13
<
s
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c
l
a
s
s
=
′
l
a
t
e
x
−
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o
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′
>
(
C
)
<
/
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a
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>
14
<
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a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
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>
15
<
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a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
E
)
<
/
s
p
a
n
>
None of the preceding
<span class='latex-bold'>(A)</span>\ 12 \qquad<span class='latex-bold'>(B)</span>\ 13 \qquad<span class='latex-bold'>(C)</span>\ 14 \qquad<span class='latex-bold'>(D)</span>\ 15 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
<
s
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
−
b
o
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d
′
>
(
A
)
<
/
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an
>
12
<
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p
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c
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a
ss
=
′
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a
t
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x
−
b
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d
′
>
(
B
)
<
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>
13
<
s
p
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c
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a
ss
=
′
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a
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x
−
b
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d
′
>
(
C
)
<
/
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>
14
<
s
p
an
c
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a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
15
<
s
p
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c
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a
ss
=
′
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a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the preceding
33
1
Hide problems
P33 [Geometry] - Turkish NMO 1st Round - 2003
Let
G
G
G
be the intersection of medians of
△
A
B
C
\triangle ABC
△
A
BC
and
I
I
I
be the incenter of
△
A
B
C
\triangle ABC
△
A
BC
. If
∣
A
B
∣
=
c
|AB|=c
∣
A
B
∣
=
c
,
∣
A
C
∣
=
b
|AC|=b
∣
A
C
∣
=
b
and
G
I
⊥
B
C
GI \perp BC
G
I
⊥
BC
, what is
∣
B
C
∣
|BC|
∣
BC
∣
?
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−
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>
(
A
)
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b
+
c
2
<
s
p
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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′
>
(
B
)
<
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a
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>
b
+
c
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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>
(
C
)
<
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a
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>
b
2
+
c
2
2
<
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p
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c
l
a
s
s
=
′
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a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
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a
n
>
b
2
+
c
2
3
2
<
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p
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c
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a
s
s
=
′
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−
b
o
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′
>
(
E
)
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None of the preceding
<span class='latex-bold'>(A)</span>\ \dfrac{b+c}2 \qquad<span class='latex-bold'>(B)</span>\ \dfrac{b+c}{3} \qquad<span class='latex-bold'>(C)</span>\ \dfrac{\sqrt{b^2+c^2}}{2} \qquad<span class='latex-bold'>(D)</span>\ \dfrac{\sqrt{b^2+c^2}}{3\sqrt 2} \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
<
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c
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>
(
A
)
<
/
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>
2
b
+
c
<
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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>
3
b
+
c
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
C
)
<
/
s
p
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>
2
b
2
+
c
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
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x
−
b
o
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d
′
>
(
D
)
<
/
s
p
an
>
3
2
b
2
+
c
2
<
s
p
an
c
l
a
ss
=
′
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a
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x
−
b
o
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d
′
>
(
E
)
<
/
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>
None of the preceding
32
1
Hide problems
P32 [Algebra] - Turkish NMO 1st Round - 2003
The function
f
f
f
satisfies
f
(
x
)
+
3
f
(
1
−
x
)
=
x
2
f(x)+3f(1-x)=x^2
f
(
x
)
+
3
f
(
1
−
x
)
=
x
2
for every real
x
x
x
. If
S
=
{
x
∣
f
(
x
)
=
0
}
S=\{x \mid f(x)=0 \}
S
=
{
x
∣
f
(
x
)
=
0
}
, which one is true?
<
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(
A
)
<
/
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p
a
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>
<span class='latex-bold'>(A)</span>
<
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=
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x
−
b
o
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′
>
(
A
)
<
/
s
p
an
>
S
S
S
is an infinite set.
<
s
p
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c
l
a
s
s
=
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x
−
b
o
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>
(
B
)
<
/
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a
n
>
<span class='latex-bold'>(B)</span>
<
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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
′
>
(
B
)
<
/
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p
an
>
{
0
,
1
}
⊂
S
\{0,1\} \subset S
{
0
,
1
}
⊂
S
<
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s
=
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−
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′
>
(
C
)
<
/
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p
a
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>
<span class='latex-bold'>(C)</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
′
>
(
C
)
<
/
s
p
an
>
S
=
ϕ
S=\phi
S
=
ϕ
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
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x
−
b
o
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′
>
(
D
)
<
/
s
p
a
n
>
<span class='latex-bold'>(D)</span>
<
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p
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c
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a
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=
′
l
a
t
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x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
S
=
{
(
3
+
3
)
/
2
,
(
3
−
3
)
/
2
}
S = \{(3+\sqrt 3)/2, (3-\sqrt 3)/2\}
S
=
{(
3
+
3
)
/2
,
(
3
−
3
)
/2
}
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
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x
−
b
o
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d
′
>
(
E
)
<
/
s
p
a
n
>
<span class='latex-bold'>(E)</span>
<
s
p
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c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of above
31
1
Hide problems
P31 [Combinatorics] - Turkish NMO 1st Round - 2003
Positive integers are written into squares of a infinite chessboard such that a number
n
n
n
is written
n
n
n
times. If the absolute differences of numbers written into any two squares having a common side is not greater than
k
k
k
, what is the least possible value of
k
k
k
?
<
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a
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s
=
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>
(
A
)
<
/
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>
1
<
s
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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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′
>
(
B
)
<
/
s
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a
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>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
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o
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′
>
(
C
)
<
/
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a
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>
3
<
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p
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
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>
4
<
s
p
a
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c
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a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
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>
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>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 4 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
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1
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(
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2
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(
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3
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(
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4
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(
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None of the preceding
30
1
Hide problems
P30 [Number Theory] - Turkish NMO 1st Round - 2003
If the sum of digits in decimal representaion of positive integer
n
n
n
is
111
111
111
and the sum of digits in decimal representation of
7002
n
7002n
7002
n
is
990
990
990
, what is the sum of digits in decimal representation of
2003
n
2003n
2003
n
?
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309
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(
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330
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555
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(
E
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None of the preceding
<span class='latex-bold'>(A)</span>\ 309 \qquad<span class='latex-bold'>(B)</span>\ 330 \qquad<span class='latex-bold'>(C)</span>\ 550 \qquad<span class='latex-bold'>(D)</span>\ 555 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
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309
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(
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330
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(
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550
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(
D
)
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555
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(
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None of the preceding
29
1
Hide problems
P29 [Geometry] - Turkish NMO 1st Round - 2003
In right triangle
A
B
C
ABC
A
BC
, let
D
D
D
be the midpoint of hypotenuse
[
A
B
]
[AB]
[
A
B
]
, circumradius be
5
2
\dfrac 52
2
5
and
∣
B
C
∣
=
3
|BC|=3
∣
BC
∣
=
3
. What is the distance between circumcenter of
△
A
C
D
\triangle ACD
△
A
C
D
and incenter of
△
B
C
D
\triangle BCD
△
BC
D
?
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)
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29
2
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3
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2
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5
34
12
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2
2
<span class='latex-bold'>(A)</span>\ \dfrac {29}{2} \qquad<span class='latex-bold'>(B)</span>\ 3 \qquad<span class='latex-bold'>(C)</span>\ \dfrac 52 \qquad<span class='latex-bold'>(D)</span>\ \dfrac{5\sqrt{34}}{12} \qquad<span class='latex-bold'>(E)</span>\ 2\sqrt 2
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2
29
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3
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5
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12
5
34
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2
2
28
1
Hide problems
P28 [Algebra] - Turkish NMO 1st Round - 2003
Let
a
a
a
,
x
x
x
,
y
y
y
,
z
z
z
be real numbers such that
a
x
−
y
+
z
=
3
a
−
1
ax-y+z=3a-1
a
x
−
y
+
z
=
3
a
−
1
ve
x
−
a
y
+
z
=
a
2
−
1
x-ay+z=a^2-1
x
−
a
y
+
z
=
a
2
−
1
, which of the followings cannot be equal to
x
2
+
y
2
+
z
2
x^2+y^2+z^2
x
2
+
y
2
+
z
2
?
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B
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D
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4
3
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None of the preceding
<span class='latex-bold'>(A)</span>\ \sqrt 2 \qquad<span class='latex-bold'>(B)</span>\ \sqrt 3 \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ \sqrt[3]{4} \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
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2
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(
B
)
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3
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−
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(
C
)
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2
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=
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(
D
)
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3
4
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E
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None of the preceding
27
1
Hide problems
P27 [Combinatorics] - Turkish NMO 1st Round - 2003
A finite number of circles are placed into a
1
×
1
1 \times 1
1
×
1
square. Let
C
C
C
be the sum of the perimeters of the circles. For how many
C
C
C
s from
C
=
43
5
C=\dfrac {43}5
C
=
5
43
,
9
9
9
,
91
10
\dfrac{91}{10}
10
91
,
19
2
\dfrac{19}{2}
2
19
,
10
10
10
, we can definitely say there exists a line cutting four of the circles?
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(
E
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4
<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>\ 4
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A
)
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0
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x
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(
B
)
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/
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1
<
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(
C
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2
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D
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3
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(
E
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4
26
1
Hide problems
P26 [Number Theory] - Turkish NMO 1st Round - 2003
Each of the numbers
n
n
n
,
n
+
1
n+1
n
+
1
,
n
+
2
n+2
n
+
2
,
n
+
3
n+3
n
+
3
is divisible by its sum of digits in its decimal representation. How many different values can the tens column of
n
n
n
have, if the number in ones column of
n
n
n
is
8
8
8
?
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A
)
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1
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B
)
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2
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>
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C
)
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3
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−
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>
(
D
)
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4
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(
E
)
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5
<span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 4 \qquad<span class='latex-bold'>(E)</span>\ 5
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x
−
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A
)
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1
<
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p
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a
ss
=
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t
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x
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>
(
B
)
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2
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ss
=
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t
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x
−
b
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(
C
)
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3
<
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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
′
>
(
D
)
<
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4
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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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>
(
E
)
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5
25
1
Hide problems
P25 [Geometry] - Turkish NMO 1st Round - 2003
Let
A
B
C
ABC
A
BC
be an acute triangle and
O
O
O
be its circumcenter. Let
D
D
D
be the midpoint of
[
A
B
]
[AB]
[
A
B
]
. The circumcircle of
△
A
D
O
\triangle ADO
△
A
D
O
meets
[
A
C
]
[AC]
[
A
C
]
at
A
A
A
and
E
E
E
. If
∣
A
E
∣
=
7
|AE|=7
∣
A
E
∣
=
7
,
∣
D
E
∣
=
8
|DE|=8
∣
D
E
∣
=
8
, and
m
(
A
O
D
^
)
=
4
5
∘
m(\widehat{AOD}) = 45^\circ
m
(
A
O
D
)
=
4
5
∘
, what is the area of
△
A
B
C
\triangle ABC
△
A
BC
?
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A
)
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56
3
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B
)
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56
2
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C
)
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50
2
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(
D
)
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84
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(
E
)
<
/
s
p
a
n
>
None of the preceding
<span class='latex-bold'>(A)</span>\ 56\sqrt 3 \qquad<span class='latex-bold'>(B)</span>\ 56 \sqrt 2 \qquad<span class='latex-bold'>(C)</span>\ 50 \sqrt 2 \qquad<span class='latex-bold'>(D)</span>\ 84 \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
>
56
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
56
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
50
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
84
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the preceding
24
1
Hide problems
P24 [Algebra] - Turkish NMO 1st Round - 2003
If
3
a
=
1
+
2
3a=1+\sqrt 2
3
a
=
1
+
2
, what is the largest integer not exceeding
9
a
4
−
6
a
3
+
8
a
2
−
6
a
+
9
9a^4-6a^3+8a^2-6a+9
9
a
4
−
6
a
3
+
8
a
2
−
6
a
+
9
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
8
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
9
<
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
>
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>\ 8 \qquad<span class='latex-bold'>(B)</span>\ 9 \qquad<span class='latex-bold'>(C)</span>\ 10 \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
>
8
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
9
<
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
>
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
23
1
Hide problems
P23 [Combinatorics] - Turkish NMO 1st Round - 2003
Ayse knows the weights of nine balls with different colors are
1
,
2
,
⋯
,
9
1,2,\cdots, 9
1
,
2
,
⋯
,
9
grams, but she doesn't know the weight of a specific ball. But Baris knows the weight of each ball. Baris wants to prove his knowledge to Ayse. There is a double pan balance which shows the heavier pan and the difference of the two pans. At least how many weighs are required for proof of Ali's knowledge?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
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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
>
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
>
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
<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
<
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
>
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
>
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
6
22
1
Hide problems
P22 [Number Theory] - Turkish NMO 1st Round - 2003
For which of the following integers
n
n
n
, there is at least one integer
x
x
x
such that
x
2
≡
−
1
(
m
o
d
n
)
x^2 \equiv -1 \pmod{n}
x
2
≡
−
1
(
mod
n
)
?
<
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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
>
97
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
98
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
99
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
100
<
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>\ 97 \qquad<span class='latex-bold'>(B)</span>\ 98 \qquad<span class='latex-bold'>(C)</span>\ 99 \qquad<span class='latex-bold'>(D)</span>\ 100 \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
>
97
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
98
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
99
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
100
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the preceding
21
1
Hide problems
P21 [Geometry] - Turkish NMO 1st Round - 2003
The circle
C
1
C_1
C
1
and
C
2
C_2
C
2
are externally tangent to each other at
T
T
T
. A line passing through
T
T
T
meets
C
1
C_1
C
1
at
A
A
A
and meets
C
2
C_2
C
2
at
B
B
B
. The line which is tangent to
C
1
C_1
C
1
at
A
A
A
meets
C
2
C_2
C
2
at
D
D
D
and
E
E
E
. If
D
∈
[
A
E
]
D \in [AE]
D
∈
[
A
E
]
,
∣
T
A
∣
=
a
|TA|=a
∣
T
A
∣
=
a
,
∣
T
B
∣
=
b
|TB|=b
∣
TB
∣
=
b
, what is
∣
B
E
∣
|BE|
∣
BE
∣
?
<
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=
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−
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>
(
A
)
<
/
s
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a
n
>
a
(
a
+
b
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
a
2
+
b
2
+
a
b
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
a
2
+
b
2
−
a
b
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
a
2
+
b
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
(
a
+
b
)
b
<span class='latex-bold'>(A)</span>\ \sqrt{a(a+b)} \qquad<span class='latex-bold'>(B)</span>\ \sqrt{a^2+b^2+ab} \qquad<span class='latex-bold'>(C)</span>\ \sqrt{a^2+b^2-ab} \qquad<span class='latex-bold'>(D)</span>\ \sqrt{a^2+b^2} \qquad<span class='latex-bold'>(E)</span>\ \sqrt{(a+b)b}
<
s
p
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c
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a
ss
=
′
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a
t
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x
−
b
o
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>
(
A
)
<
/
s
p
an
>
a
(
a
+
b
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
a
2
+
b
2
+
ab
<
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p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
a
2
+
b
2
−
ab
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
a
2
+
b
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
(
a
+
b
)
b
20
1
Hide problems
P20 [Algebra] - Turkish NMO 1st Round - 2003
How many real numbers
x
x
x
are there such that
x
+
1
−
4
x
−
3
+
x
+
6
−
6
x
−
3
=
1
\sqrt{ x + 1 - 4\sqrt{x-3}} + \sqrt{ x + 6 - 6\sqrt{x-3}} = 1
x
+
1
−
4
x
−
3
+
x
+
6
−
6
x
−
3
=
1
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
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
>
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
6
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
7
<
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>\ 4 \qquad<span class='latex-bold'>(C)</span>\ 6 \qquad<span class='latex-bold'>(D)</span>\ 7 \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
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
6
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
7
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
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an
>
None of the preceding
19
1
Hide problems
P19 [Combinatorics] - Turkish NMO 1st Round - 2003
At least how many elements does the set which contains all of the midpoints of segments connecting
2003
2003
2003
different points in a plane have?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
2006
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
4001
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
4003
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
4006
<
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 preceeding
<span class='latex-bold'>(A)</span>\ 2006 \qquad<span class='latex-bold'>(B)</span>\ 4001 \qquad<span class='latex-bold'>(C)</span>\ 4003 \qquad<span class='latex-bold'>(D)</span>\ 4006 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceeding}
<
s
p
an
c
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4001
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None of the preceeding
18
1
Hide problems
P18 [Number Theory] - Turkish NMO 1st Round - 2003
What is the least integer
n
>
2003
n>2003
n
>
2003
such that
5
n
+
n
5
5^n + n^5
5
n
+
n
5
is a multiple of
11
11
11
?
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2010
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(
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)
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>
2011
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2014
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None of the preceding
<span class='latex-bold'>(A)</span>\ 2010 \qquad<span class='latex-bold'>(B)</span>\ 2011 \qquad<span class='latex-bold'>(C)</span>\ 2012 \qquad<span class='latex-bold'>(D)</span>\ 2014 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
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2010
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2011
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2012
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(
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)
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2014
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None of the preceding
17
1
Hide problems
P17 [Geometry] - Turkish NMO 1st Round - 2003
The circle
C
1
C_1
C
1
and the circle
C
2
C_2
C
2
passing through the center of
C
1
C_1
C
1
intersect each other at
A
A
A
and
B
B
B
. The line tangent to
C
2
C_2
C
2
at
B
B
B
meets
C
1
C_1
C
1
at
B
B
B
and
D
D
D
. If the radius of
C
1
C_1
C
1
is
3
\sqrt 3
3
and the radius of
C
2
C_2
C
2
is
2
2
2
, find
∣
A
B
∣
∣
B
D
∣
\dfrac{|AB|}{|BD|}
∣
B
D
∣
∣
A
B
∣
.
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2
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3
2
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3
2
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1
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5
2
<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 {2\sqrt 3}2 \qquad<span class='latex-bold'>(D)</span>\ 1 \qquad<span class='latex-bold'>(E)</span>\ \dfrac {\sqrt 5}2
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1
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3
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2
3
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2
5
16
1
Hide problems
P16 [Algebra] - Turkish NMO 1st Round - 2003
For which of the following values of real number
t
t
t
, the equation
x
4
−
t
x
+
1
t
=
0
x^4-tx+\dfrac 1t = 0
x
4
−
t
x
+
t
1
=
0
has no root on the interval
[
1
,
2
]
[1,2]
[
1
,
2
]
?
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6
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7
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8
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9
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None of the preceding
<span class='latex-bold'>(A)</span>\ 6 \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}
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A
)
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>
6
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(
B
)
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7
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8
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(
D
)
<
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>
9
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None of the preceding
15
1
Hide problems
P15 [Combinatorics] - Turkish NMO 1st Round - 2003
Galatasaray and Fenerbahce have qualified last
16
16
16
in the Europen Champions League. Aftar a random draw, eight matches are regulated in that knock-out phase. The winners of the eight matches will qualify for the next round - round of
8
8
8
. Knock-out phase continues until one team remains. If each team has equal chance to win, what is the propability of having a Galatasaray-Fenerbahce match?
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1
32
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1
16
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8
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1
4
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None of the preceding
<span class='latex-bold'>(A)</span>\ \dfrac {1}{32} \qquad<span class='latex-bold'>(B)</span>\ \dfrac {1}{16} \qquad<span class='latex-bold'>(C)</span>\ \dfrac {1}{8} \qquad<span class='latex-bold'>(D)</span>\ \dfrac {1}{4} \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
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32
1
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16
1
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8
1
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4
1
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None of the preceding
14
1
Hide problems
P14 [Number Theory] - Turkish NMO 1st Round - 2003
How many primes
p
p
p
are there such that
5
p
(
2
p
+
1
−
1
)
5p(2^{p+1}-1)
5
p
(
2
p
+
1
−
1
)
is a perfect square?
<
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(
A
)
<
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a
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>
0
<
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p
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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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>
(
B
)
<
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a
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>
1
<
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c
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a
s
s
=
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−
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)
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>
2
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=
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−
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>
(
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)
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>
3
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(
E
)
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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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(
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)
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0
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p
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=
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x
−
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>
(
B
)
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1
<
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p
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=
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(
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)
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2
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=
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(
D
)
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3
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None of the preceding
12
1
Hide problems
P12 [Algebra] - Turkish NMO 1st Round - 2003
How many real triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
are there such that
4
x
2
1
+
4
x
2
=
y
\dfrac{4x^2}{1+4x^2}=y
1
+
4
x
2
4
x
2
=
y
,
4
y
2
1
+
4
y
2
=
z
\dfrac{4y^2}{1+4y^2}=z
1
+
4
y
2
4
y
2
=
z
,
4
z
2
1
+
4
z
2
=
x
\dfrac{4z^2}{1+4z^2}=x
1
+
4
z
2
4
z
2
=
x
?
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)
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2
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(
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)
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4
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−
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)
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6
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s
=
′
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a
t
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x
−
b
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>
(
D
)
<
/
s
p
a
n
>
Infinitely many
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None of the preceding
<span class='latex-bold'>(A)</span>\ 2 \qquad<span class='latex-bold'>(B)</span>\ 4 \qquad<span class='latex-bold'>(C)</span>\ 6 \qquad<span class='latex-bold'>(D)</span>\ \text{Infinitely many} \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
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=
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−
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(
B
)
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>
4
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′
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e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
6
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
Infinitely many
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the preceding
11
1
Hide problems
P11 [Combinatorics] - Turkish NMO 1st Round - 2003
What is the probability of having no
B
B
B
before the first
A
A
A
in a random permutation of the word
ABRAKADABRA
\text{ABRAKADABRA}
ABRAKADABRA
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
2
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
7
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
5
6
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
6
7
<
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>\ \dfrac 23 \qquad<span class='latex-bold'>(B)</span>\ \dfrac 57 \qquad<span class='latex-bold'>(C)</span>\ \dfrac 56 \qquad<span class='latex-bold'>(D)</span>\ \dfrac 67 \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
>
3
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
7
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
6
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
7
6
<
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 [Number Theory] - Turkish NMO 1st Round - 2003
Which of the followings is congruent (in
m
o
d
25
\bmod{25}
mod
25
) to the sum in of integers
0
≤
x
<
25
0\leq x < 25
0
≤
x
<
25
such that
x
3
+
3
x
2
−
2
x
+
4
≡
0
(
m
o
d
25
)
x^3+3x^2-2x+4 \equiv 0 \pmod{25}
x
3
+
3
x
2
−
2
x
+
4
≡
0
(
mod
25
)
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
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
>
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
17
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
22
<
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>\ 4 \qquad<span class='latex-bold'>(C)</span>\ 17 \qquad<span class='latex-bold'>(D)</span>\ 22 \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
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
17
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
22
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the preceding
9
1
Hide problems
P09 [Geometry] - Turkish NMO 1st Round - 2003
How many integer triangles are there with inradius
1
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
>
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
>
Infinite
<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{Infinite}
<
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
>
Infinite
8
1
Hide problems
P08 [Algebra] - Turkish NMO 1st Round - 2003
Let
P
P
P
be a polynomial such that
(
x
−
4
)
P
(
2
x
)
=
4
(
x
−
1
)
P
(
x
)
(x-4)P(2x) = 4(x-1)P(x)
(
x
−
4
)
P
(
2
x
)
=
4
(
x
−
1
)
P
(
x
)
, for every real
x
x
x
. If
P
(
0
)
≠
0
P(0) \neq 0
P
(
0
)
=
0
, what is the degree of
P
P
P
?
<
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
7
1
Hide problems
P07 [Combinatorics] - Turkish NMO 1st Round - 2003
Starting with the sequence
AAAIEE
\text{AAAIEE}
AAAIEE
, we replace
AIE
\text{AIE}
AIE
with
EA
\text{EA}
EA
,
AE
\text{AE}
AE
with
IE
\text{IE}
IE
,
E
\text{E}
E
with
AI
\text{AI}
AI
. After repeating replace operations many times, which of the following cannot be got?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
AIAIIAI
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
AIAIAI
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
AIAAA
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
AIAA
<
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>\ \text{AIAIIAI} \qquad<span class='latex-bold'>(B)</span>\ \text{AIAIAI} \qquad<span class='latex-bold'>(C)</span>\ \text{AIAAA} \qquad<span class='latex-bold'>(D)</span>\ \text{AIAA} \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
>
AIAIIAI
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
AIAIAI
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
AIAAA
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
AIAA
<
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 [Number Theory] - Turkish NMO 1st Round - 2003
How many
0
0
0
s are there at the end of the decimal representation of
2000
!
2000!
2000
!
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
222
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
499
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
625
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
999
<
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>\ 222 \qquad<span class='latex-bold'>(B)</span>\ 499 \qquad<span class='latex-bold'>(C)</span>\ 625 \qquad<span class='latex-bold'>(D)</span>\ 999 \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
>
222
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
499
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
625
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
999
<
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 - 2003
Let
A
B
C
ABC
A
BC
be a triangle and
D
D
D
be the foot of the altitude from
C
C
C
to
A
B
AB
A
B
. If
∣
C
H
∣
=
∣
H
D
∣
|CH|=|HD|
∣
C
H
∣
=
∣
HD
∣
where
H
H
H
is the orthocenter, what is
tan
A
^
⋅
tan
B
^
\tan \widehat {A} \cdot \tan \widehat{B}
tan
A
⋅
tan
B
?
<
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
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n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
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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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>
3
/
2
<
s
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a
n
c
l
a
s
s
=
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a
t
e
x
−
b
o
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>
(
D
)
<
/
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a
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>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
E
)
<
/
s
p
a
n
>
None of the preceding
<span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ \sqrt 2 \qquad<span class='latex-bold'>(C)</span>\ 3/2 \qquad<span class='latex-bold'>(D)</span>\ \sqrt 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
−
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o
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d
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>
(
A
)
<
/
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an
>
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
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an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
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an
>
3/2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
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(
D
)
<
/
s
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>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
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an
>
None of the preceding
4
1
Hide problems
P04 [Algebra] - Turkish NMO 1st Round - 2003
How many pairs of positive integers
(
a
,
b
)
(a,b)
(
a
,
b
)
are there such that the roots of polynomial
x
2
−
a
x
−
b
x^2-ax-b
x
2
−
a
x
−
b
are not greater than
5
5
5
?
<
s
p
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n
c
l
a
s
s
=
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a
t
e
x
−
b
o
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>
(
A
)
<
/
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a
n
>
40
<
s
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n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
B
)
<
/
s
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a
n
>
50
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
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s
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a
n
>
65
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
75
<
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>\ 40 \qquad<span class='latex-bold'>(B)</span>\ 50 \qquad<span class='latex-bold'>(C)</span>\ 65 \qquad<span class='latex-bold'>(D)</span>\ 75 \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
>
40
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
50
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
65
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
75
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
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an
>
None of the preceding
3
1
Hide problems
P03 [Combinatorics] - Turkish NMO 1st Round - 2003
At most how many positive integers less than
51
51
51
are there such that no one is triple of another one?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
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o
l
d
′
>
(
A
)
<
/
s
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a
n
>
17
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
36
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
38
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
39
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
E
)
<
/
s
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a
n
>
None of the preceding
<span class='latex-bold'>(A)</span>\ 17 \qquad<span class='latex-bold'>(B)</span>\ 36 \qquad<span class='latex-bold'>(C)</span>\ 38 \qquad<span class='latex-bold'>(D)</span>\ 39 \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
>
17
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
36
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
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an
>
38
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
39
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
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an
>
None of the preceding
2
1
Hide problems
P02 [Number Theory] - Turkish NMO 1st Round - 2003
How many prime divisors does the number
1
⋅
2003
+
2
⋅
2002
+
3
⋅
2001
+
⋯
+
2001
⋅
3
+
2002
⋅
2
+
2003
⋅
1
1\cdot 2003 + 2\cdot 2002 + 3\cdot 2001 + \cdots + 2001 \cdot 3 + 2002 \cdot 2 + 2003 \cdot 1
1
⋅
2003
+
2
⋅
2002
+
3
⋅
2001
+
⋯
+
2001
⋅
3
+
2002
⋅
2
+
2003
⋅
1
have?
<
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l
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s
s
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−
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(
A
)
<
/
s
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a
n
>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
B
)
<
/
s
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a
n
>
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
5
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
6
<
s
p
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l
a
s
s
=
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−
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>
(
E
)
<
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a
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>
7
<span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ 4 \qquad<span class='latex-bold'>(C)</span>\ 5 \qquad<span class='latex-bold'>(D)</span>\ 6 \qquad<span class='latex-bold'>(E)</span>\ 7
<
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c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
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>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
6
<
s
p
an
c
l
a
ss
=
′
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a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
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an
>
7
1
1
Hide problems
P01 [Geometry] - Turkish NMO 1st Round - 2003
Let
A
B
C
ABC
A
BC
be a triangle such that
∣
A
B
∣
=
7
|AB|=7
∣
A
B
∣
=
7
,
∣
B
C
∣
=
8
|BC|=8
∣
BC
∣
=
8
,
∣
A
C
∣
=
6
|AC|=6
∣
A
C
∣
=
6
. Let
D
D
D
be the midpoint of side
[
B
C
]
[BC]
[
BC
]
. If the circle through
A
A
A
,
B
B
B
and
D
D
D
cuts
A
C
AC
A
C
at
A
A
A
and
E
E
E
, what is
∣
A
E
∣
|AE|
∣
A
E
∣
?
<
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−
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(
A
)
<
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a
n
>
2
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
B
)
<
/
s
p
a
n
>
1
<
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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
)
<
/
s
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a
n
>
3
2
<
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a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
D
)
<
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s
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a
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>
2
<
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p
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a
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s
=
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a
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−
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o
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>
(
E
)
<
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a
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>
3
<span class='latex-bold'>(A)</span>\ \dfrac 23 \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ \dfrac 32 \qquad<span class='latex-bold'>(D)</span>\ 2 \qquad<span class='latex-bold'>(E)</span>\ 3
<
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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
′
>
(
A
)
<
/
s
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>
3
2
<
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
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>
2
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
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x
−
b
o
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d
′
>
(
D
)
<
/
s
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an
>
2
<
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p
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c
l
a
ss
=
′
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a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
3
13
1
Hide problems
P13 [Geometry] - Turkish NMO 1st Round - 2003
Let
A
B
C
ABC
A
BC
be a triangle such that
∣
A
B
∣
=
8
|AB|=8
∣
A
B
∣
=
8
and
∣
A
C
∣
=
2
∣
B
C
∣
|AC|=2|BC|
∣
A
C
∣
=
2∣
BC
∣
. What is the largest value of altitude from side
[
A
B
]
[AB]
[
A
B
]
?
<
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p
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c
l
a
s
s
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x
−
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o
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>
(
A
)
<
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s
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a
n
>
3
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
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
5
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
16
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
E
)
<
/
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a
n
>
6
<span class='latex-bold'>(A)</span>\ 3\sqrt 2 \qquad<span class='latex-bold'>(B)</span>\ 3\sqrt 3 \qquad<span class='latex-bold'>(C)</span>\ 5 \qquad<span class='latex-bold'>(D)</span>\ \dfrac {16}3 \qquad<span class='latex-bold'>(E)</span>\ 6
<
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p
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c
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a
ss
=
′
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a
t
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x
−
b
o
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d
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>
(
A
)
<
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s
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>
3
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
3
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
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an
>
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
3
16
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
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>
6