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Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2004 National Olympiad First Round
2004 National Olympiad First Round
Part of
National Olympiad First Round
Subcontests
(36)
36
1
Hide problems
P36 [Algebra] - Turkish NMO 1st Round - 2004
If the function
f
f
f
satisfies the equation
f
(
x
)
+
f
(
1
1
−
x
3
3
)
=
x
3
f(x) + f\left ( \dfrac{1}{\sqrt[3]{1-x^3}}\right ) = x^3
f
(
x
)
+
f
(
3
1
−
x
3
1
)
=
x
3
for every real
x
≠
1
x \neq 1
x
=
1
, what is
f
(
−
1
)
f(-1)
f
(
−
1
)
?
<
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a
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−
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o
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(
A
)
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−
1
<
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s
s
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l
a
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x
−
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o
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>
(
B
)
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1
4
<
s
p
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n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
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1
2
<
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l
a
s
s
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′
l
a
t
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x
−
b
o
l
d
′
>
(
D
)
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>
7
4
<
s
p
a
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c
l
a
s
s
=
′
l
a
t
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x
−
b
o
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d
′
>
(
E
)
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None of above
<span class='latex-bold'>(A)</span>\ -1 \qquad<span class='latex-bold'>(B)</span>\ \dfrac 14 \qquad<span class='latex-bold'>(C)</span>\ \dfrac 12 \qquad<span class='latex-bold'>(D)</span>\ \dfrac 74 \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}
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(
A
)
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−
1
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(
B
)
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4
1
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l
a
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x
−
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(
C
)
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2
1
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ss
=
′
l
a
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x
−
b
o
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d
′
>
(
D
)
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>
4
7
<
s
p
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c
l
a
ss
=
′
l
a
t
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x
−
b
o
l
d
′
>
(
E
)
<
/
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p
an
>
None of above
32
1
Hide problems
P32 [Algebra] - Turkish NMO 1st Round - 2004
If
a
a
a
and
b
b
b
are the roots of the equation
x
2
−
2
c
x
−
5
d
=
0
x^2-2cx-5d = 0
x
2
−
2
c
x
−
5
d
=
0
,
c
c
c
and
d
d
d
are the roots of the equation
x
2
−
2
a
x
−
5
b
=
0
x^2-2ax-5b=0
x
2
−
2
a
x
−
5
b
=
0
, where
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
are distinct real numbers, what is
a
+
b
+
c
+
d
a+b+c+d
a
+
b
+
c
+
d
?
<
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s
s
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o
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(
A
)
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10
<
s
p
a
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c
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a
s
s
=
′
l
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x
−
b
o
l
d
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>
(
B
)
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15
<
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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>
20
<
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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>
25
<
s
p
a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
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30
<span class='latex-bold'>(A)</span>\ 10 \qquad<span class='latex-bold'>(B)</span>\ 15 \qquad<span class='latex-bold'>(C)</span>\ 20 \qquad<span class='latex-bold'>(D)</span>\ 25 \qquad<span class='latex-bold'>(E)</span>\ 30
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(
A
)
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10
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
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>
15
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
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>
20
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
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>
25
<
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p
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c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
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an
>
30
28
1
Hide problems
P28 [Algebra] - Turkish NMO 1st Round - 2004
What is the largest possible value of
8
x
2
+
9
x
y
+
18
y
2
+
2
x
+
3
y
8x^2+9xy+18y^2+2x+3y
8
x
2
+
9
x
y
+
18
y
2
+
2
x
+
3
y
such that
4
x
2
+
9
y
2
=
8
4x^2 + 9y^2 = 8
4
x
2
+
9
y
2
=
8
where
x
,
y
x,y
x
,
y
are real numbers?
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(
A
)
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23
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s
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(
B
)
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26
<
s
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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
′
>
(
C
)
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/
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29
<
s
p
a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
31
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
35
<span class='latex-bold'>(A)</span>\ 23 \qquad<span class='latex-bold'>(B)</span>\ 26 \qquad<span class='latex-bold'>(C)</span>\ 29 \qquad<span class='latex-bold'>(D)</span>\ 31 \qquad<span class='latex-bold'>(E)</span>\ 35
<
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(
A
)
<
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>
23
<
s
p
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c
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a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
B
)
<
/
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>
26
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
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>
29
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
31
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
35
20
1
Hide problems
P20 [Algebra] - Turkish NMO 1st Round - 2004
What is the largest real number
C
C
C
that satisfies the inequality
x
2
≥
C
⌊
x
⌋
(
x
−
⌊
x
⌋
)
x^2 \geq C \lfloor x \rfloor (x-\lfloor x \rfloor)
x
2
≥
C
⌊
x
⌋
(
x
−
⌊
x
⌋)
for every real
x
x
x
?
<
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a
s
s
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a
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x
−
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>
(
A
)
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n
>
0
<
s
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a
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s
=
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x
−
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o
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′
>
(
B
)
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1
<
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a
s
s
=
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−
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>
(
C
)
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>
4
<
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a
s
s
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−
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o
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>
(
D
)
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>
9
<
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s
s
=
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o
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>
(
E
)
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25
<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>\ 9 \qquad<span class='latex-bold'>(E)</span>\ 25
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−
b
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d
′
>
(
A
)
<
/
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>
0
<
s
p
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c
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a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
B
)
<
/
s
p
an
>
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
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>
4
<
s
p
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c
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a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
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>
9
<
s
p
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c
l
a
ss
=
′
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a
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e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
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>
25
24
1
Hide problems
P24 [Algebra] - Turkish NMO 1st Round - 2004
What is the sum of cubes of real roots of the equation
x
3
−
2
x
2
−
x
+
1
=
0
x^3-2x^2-x+1=0
x
3
−
2
x
2
−
x
+
1
=
0
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
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>
−
6
<
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c
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a
s
s
=
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l
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t
e
x
−
b
o
l
d
′
>
(
B
)
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2
<
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a
s
s
=
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−
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>
(
C
)
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8
<
s
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a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
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a
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>
11
<
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
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>
None of above
<span class='latex-bold'>(A)</span>\ -6 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 8 \qquad<span class='latex-bold'>(D)</span>\ 11 \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}
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(
A
)
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−
6
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p
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=
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l
a
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x
−
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o
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d
′
>
(
B
)
<
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2
<
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p
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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
′
>
(
C
)
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>
8
<
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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
′
>
(
D
)
<
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s
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>
11
<
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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
l
d
′
>
(
E
)
<
/
s
p
an
>
None of above
16
1
Hide problems
P16 [Algebra] - Turkish NMO 1st Round - 2004
What is the sum of real roots of the equation
x
4
−
4
x
3
+
5
x
2
−
4
x
+
1
=
0
x^4-4x^3+5x^2-4x+1 = 0
x
4
−
4
x
3
+
5
x
2
−
4
x
+
1
=
0
?
<
s
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a
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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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′
>
(
A
)
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>
5
<
s
p
a
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c
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a
s
s
=
′
l
a
t
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x
−
b
o
l
d
′
>
(
B
)
<
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s
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a
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>
4
<
s
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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>
3
<
s
p
a
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c
l
a
s
s
=
′
l
a
t
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x
−
b
o
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d
′
>
(
D
)
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/
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>
2
<
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a
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s
=
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−
b
o
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′
>
(
E
)
<
/
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a
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>
1
<span class='latex-bold'>(A)</span>\ 5 \qquad<span class='latex-bold'>(B)</span>\ 4 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 2 \qquad<span class='latex-bold'>(E)</span>\ 1
<
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o
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(
A
)
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/
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>
5
<
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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
′
>
(
B
)
<
/
s
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>
4
<
s
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
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d
′
>
(
C
)
<
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an
>
3
<
s
p
an
c
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a
ss
=
′
l
a
t
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x
−
b
o
l
d
′
>
(
D
)
<
/
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>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
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>
1
12
1
Hide problems
P12 [Algebra] - Turkish NMO 1st Round - 2004
What is the least value of
(
x
−
1
)
(
x
−
2
)
(
x
−
3
)
(
x
−
4
)
(x-1)(x-2)(x-3)(x-4)
(
x
−
1
)
(
x
−
2
)
(
x
−
3
)
(
x
−
4
)
where
x
x
x
is a real number?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
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′
>
(
A
)
<
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s
p
a
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>
−
1
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
−
1
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
−
1
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
−
1
<
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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′
>
(
E
)
<
/
s
p
a
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>
−
2
<span class='latex-bold'>(A)</span>\ -\dfrac 14 \qquad<span class='latex-bold'>(B)</span>\ - \dfrac 13 \qquad<span class='latex-bold'>(C)</span>\ -\dfrac 12 \qquad<span class='latex-bold'>(D)</span>\ -1 \qquad<span class='latex-bold'>(E)</span>\ -2
<
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=
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a
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x
−
b
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d
′
>
(
A
)
<
/
s
p
an
>
−
4
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
−
3
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
−
2
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
−
1
<
s
p
an
c
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a
ss
=
′
l
a
t
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x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
−
2
8
1
Hide problems
P08 [Algebra] - Turkish NMO 1st Round - 2004
For how many triples of positive integers
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
, there exists a positive integer
n
n
n
such that
x
n
=
y
n
+
1
=
z
n
+
2
\dfrac{x}{n} = \dfrac{y}{n+1} = \dfrac{z}{n+2}
n
x
=
n
+
1
y
=
n
+
2
z
where
x
+
y
+
z
=
90
x+y+z=90
x
+
y
+
z
=
90
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
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x
−
b
o
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′
>
(
A
)
<
/
s
p
a
n
>
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
5
<
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
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a
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>
9
<span class='latex-bold'>(A)</span>\ 4 \qquad<span class='latex-bold'>(B)</span>\ 5 \qquad<span class='latex-bold'>(C)</span>\ 6 \qquad<span class='latex-bold'>(D)</span>\ 7 \qquad<span class='latex-bold'>(E)</span>\ 9
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
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
)
<
/
s
p
an
>
9
4
1
Hide problems
P04 [Algebra] - Turkish NMO 1st Round - 2004
What is the difference between the maximum value and the minimum value of the sum
a
1
+
2
a
2
+
3
a
3
+
4
a
4
+
5
a
5
a_1 + 2a_2 + 3a_3 + 4a_4 + 5a_5
a
1
+
2
a
2
+
3
a
3
+
4
a
4
+
5
a
5
where
{
a
1
,
a
2
,
a
3
,
a
4
,
a
5
}
=
{
1
,
2
,
3
,
4
,
5
}
\{a_1,a_2,a_3,a_4,a_5\} = \{1,2,3,4,5\}
{
a
1
,
a
2
,
a
3
,
a
4
,
a
5
}
=
{
1
,
2
,
3
,
4
,
5
}
?
<
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a
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c
l
a
s
s
=
′
l
a
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x
−
b
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′
>
(
A
)
<
/
s
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a
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>
20
<
s
p
a
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c
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a
s
s
=
′
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a
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e
x
−
b
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′
>
(
B
)
<
/
s
p
a
n
>
15
<
s
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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>
10
<
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
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c
l
a
s
s
=
′
l
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t
e
x
−
b
o
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d
′
>
(
E
)
<
/
s
p
a
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>
0
<span class='latex-bold'>(A)</span>\ 20 \qquad<span class='latex-bold'>(B)</span>\ 15 \qquad<span class='latex-bold'>(C)</span>\ 10 \qquad<span class='latex-bold'>(D)</span>\ 5 \qquad<span class='latex-bold'>(E)</span>\ 0
<
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p
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a
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=
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−
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(
A
)
<
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>
20
<
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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>
15
<
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
>
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
0
35
1
Hide problems
P35 [Combinatorics] - Turkish NMO 1st Round - 2004
We are placing
n
n
n
integers whose sum is
94
94
94
over a circle such that each number is equal to the absolute value of the difference of (clockwise) next two numbers. What is the largest
n
n
n
that makes such placing possible?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
188
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
186
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
141
<
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
>
47
<span class='latex-bold'>(A)</span>\ 188 \qquad<span class='latex-bold'>(B)</span>\ 186 \qquad<span class='latex-bold'>(C)</span>\ 141 \qquad<span class='latex-bold'>(D)</span>\ 100 \qquad<span class='latex-bold'>(E)</span>\ 47
<
s
p
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c
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a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
188
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
186
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
141
<
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
>
47
31
1
Hide problems
P31 [Combinatorics] - Turkish NMO 1st Round - 2004
For how many different values of integer
n
n
n
, one can find
n
n
n
different lines in the plane such that each line intersects with exacly
2004
2004
2004
of other lines?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
12
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
11
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
9
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
6
<
s
p
a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
1
<span class='latex-bold'>(A)</span>\ 12 \qquad<span class='latex-bold'>(B)</span>\ 11 \qquad<span class='latex-bold'>(C)</span>\ 9 \qquad<span class='latex-bold'>(D)</span>\ 6 \qquad<span class='latex-bold'>(E)</span>\ 1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
12
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
B
)
<
/
s
p
an
>
11
<
s
p
an
c
l
a
ss
=
′
l
a
t
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x
−
b
o
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d
′
>
(
C
)
<
/
s
p
an
>
9
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
6
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
1
27
1
Hide problems
P27 [Combinatorics] - Turkish NMO 1st Round - 2004
We have
31
31
31
pieces where
1
1
1
is written on two of them,
2
2
2
is written on eight of them,
3
3
3
is written on twelve of them,
4
4
4
is written on four of them, and
5
5
5
is written on five of them. We place
30
30
30
of them into a
5
×
6
5\times 6
5
×
6
chessboard such that the sum of numbers on any row is equal to a fixed number and the sum of numbers on any column is equal to a fixed number. What is the number written on the piece which is not placed?
<
s
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−
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(
A
)
<
/
s
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a
n
>
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
4
<
s
p
a
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c
l
a
s
s
=
′
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x
−
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o
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d
′
>
(
E
)
<
/
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a
n
>
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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p
an
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
)
<
/
s
p
an
>
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
B
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
5
23
1
Hide problems
P23 [Combinatorics] - Turkish NMO 1st Round - 2004
What is the maximal possible value of
n
n
n
such that no matter how
25
25
25
squares are selected in an infinite chessboard one can find
n
n
n
squares in which none of them share a common corner?
<
s
p
a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
A
)
<
/
s
p
a
n
>
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
>
9
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
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x
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(
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10
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11
<span class='latex-bold'>(A)</span>\ 7 \qquad<span class='latex-bold'>(B)</span>\ 8 \qquad<span class='latex-bold'>(C)</span>\ 9 \qquad<span class='latex-bold'>(D)</span>\ 10 \qquad<span class='latex-bold'>(E)</span>\ 11
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(
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7
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(
B
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8
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(
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9
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10
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11
19
1
Hide problems
P19 [Combinatorics] - Turkish NMO 1st Round - 2004
If we have a number
x
x
x
at a certain step, then at the next step we have
x
+
1
x+1
x
+
1
or
−
1
x
-\frac 1x
−
x
1
. If we start with the number
1
1
1
, which of the following cannot be got after a finite number of steps?
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−
2
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(
B
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1
2
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(
C
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>
5
3
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(
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7
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None of above
<span class='latex-bold'>(A)</span>\ -2 \qquad<span class='latex-bold'>(B)</span>\ \dfrac 12 \qquad<span class='latex-bold'>(C)</span>\ \dfrac 53 \qquad<span class='latex-bold'>(D)</span>\ 7 \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}
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−
2
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(
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2
1
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C
)
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3
5
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(
D
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7
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None of above
15
1
Hide problems
P15 [Combinatorics] - Turkish NMO 1st Round - 2004
How many
10
10
10
-digit positive integers can be written by using four
0
0
0
s, five
1
1
1
s, and one
2
2
2
?
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(
A
)
<
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>
1260
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(
B
)
<
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a
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>
1134
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(
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>
756
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D
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630
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None of above
<span class='latex-bold'>(A)</span>\ 1260 \qquad<span class='latex-bold'>(B)</span>\ 1134 \qquad<span class='latex-bold'>(C)</span>\ 756 \qquad<span class='latex-bold'>(D)</span>\ 630 \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}
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1260
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(
B
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1134
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756
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(
D
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630
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None of above
11
1
Hide problems
P11 [Combinatorics] - Turkish NMO 1st Round - 2004
We write one of the numbers
0
0
0
and
1
1
1
into each unit square of a chessboard with
40
40
40
rows and
7
7
7
columns. If any two rows have different sequences, at most how many
1
1
1
s can be written into the unit squares?
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198
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(
B
)
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>
128
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(
C
)
<
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>
82
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(
D
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40
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(
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None of above
<span class='latex-bold'>(A)</span>\ 198 \qquad<span class='latex-bold'>(B)</span>\ 128 \qquad<span class='latex-bold'>(C)</span>\ 82 \qquad<span class='latex-bold'>(D)</span>\ 40 \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}
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198
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(
B
)
<
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128
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(
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82
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(
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)
<
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>
40
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(
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None of above
7
1
Hide problems
P07 [Combinatorics] - Turkish NMO 1st Round - 2004
At least how many weighings of a balanced scale are needed to order four stones with distinct weights from the lightest to the heaviest?
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4
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5
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6
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7
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8
<span class='latex-bold'>(A)</span>\ 4 \qquad<span class='latex-bold'>(B)</span>\ 5 \qquad<span class='latex-bold'>(C)</span>\ 6 \qquad<span class='latex-bold'>(D)</span>\ 7 \qquad<span class='latex-bold'>(E)</span>\ 8
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4
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(
B
)
<
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5
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(
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6
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(
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7
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8
3
1
Hide problems
P03 [Combinatorics] - Turkish NMO 1st Round - 2004
At most how many elements does a set have such that all elements are less than
102
102
102
and it doesn't contain the sum of any two elements?
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49
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50
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51
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54
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62
<span class='latex-bold'>(A)</span>\ 49 \qquad<span class='latex-bold'>(B)</span>\ 50 \qquad<span class='latex-bold'>(C)</span>\ 51 \qquad<span class='latex-bold'>(D)</span>\ 54 \qquad<span class='latex-bold'>(E)</span>\ 62
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49
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(
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50
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51
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54
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(
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)
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62
34
1
Hide problems
P34 [Number Theory] - Turkish NMO 1st Round - 2004
How many positive integers which divide
5
n
11
−
2
n
5
−
3
n
5n^{11}-2n^5-3n
5
n
11
−
2
n
5
−
3
n
for all positive integers
n
n
n
are there?
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(
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2
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(
B
)
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5
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)
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6
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>
(
D
)
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>
12
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(
E
)
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18
<span class='latex-bold'>(A)</span>\ 2 \qquad<span class='latex-bold'>(B)</span>\ 5 \qquad<span class='latex-bold'>(C)</span>\ 6 \qquad<span class='latex-bold'>(D)</span>\ 12 \qquad<span class='latex-bold'>(E)</span>\ 18
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(
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2
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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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6
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(
D
)
<
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>
12
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>
(
E
)
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18
30
1
Hide problems
P30 [Number Theory] - Turkish NMO 1st Round - 2004
How many primes
p
p
p
are there such that the number of positive divisors of
p
2
+
23
p^2+23
p
2
+
23
is equal to
14
14
14
?
<
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(
A
)
<
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0
<
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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
)
<
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s
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a
n
>
2
<
s
p
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n
c
l
a
s
s
=
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l
a
t
e
x
−
b
o
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d
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>
(
D
)
<
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a
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>
3
<
s
p
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n
c
l
a
s
s
=
′
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a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
None of above
<span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ 3 \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}
<
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p
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c
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a
ss
=
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a
t
e
x
−
b
o
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d
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(
A
)
<
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s
p
an
>
0
<
s
p
an
c
l
a
ss
=
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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
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an
>
3
<
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 above
26
1
Hide problems
P26 [Number Theory] - Turkish NMO 1st Round - 2004
What is the last two digits of base-
3
3
3
representation of
200
5
200
3
2004
+
3
2005^{2003^{2004}+3}
200
5
200
3
2004
+
3
?
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(
A
)
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21
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(
B
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>
01
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(
C
)
<
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>
11
<
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a
s
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=
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(
D
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02
<
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(
E
)
<
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>
22
<span class='latex-bold'>(A)</span>\ 21 \qquad<span class='latex-bold'>(B)</span>\ 01 \qquad<span class='latex-bold'>(C)</span>\ 11 \qquad<span class='latex-bold'>(D)</span>\ 02 \qquad<span class='latex-bold'>(E)</span>\ 22
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a
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(
A
)
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>
21
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a
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=
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a
t
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x
−
b
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(
B
)
<
/
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>
01
<
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p
an
c
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a
ss
=
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a
t
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−
b
o
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d
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(
C
)
<
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>
11
<
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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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−
b
o
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(
D
)
<
/
s
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an
>
02
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a
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=
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t
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x
−
b
o
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d
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>
(
E
)
<
/
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an
>
22
22
1
Hide problems
P22 [Number Theory] - Turkish NMO 1st Round - 2004
For which of the following expressions, there exists an integer
x
x
x
such that the expression is divisble by
25
25
25
?
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−
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>
(
A
)
<
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a
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>
x
3
−
3
x
2
+
8
x
−
1
<
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l
a
s
s
=
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a
t
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x
−
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o
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>
(
B
)
<
/
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a
n
>
x
3
+
3
x
2
−
2
x
+
1
<
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p
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c
l
a
s
s
=
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a
t
e
x
−
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o
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>
(
C
)
<
/
s
p
a
n
>
x
3
+
14
x
2
+
3
x
−
8
<
s
p
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c
l
a
s
s
=
′
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a
t
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x
−
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>
(
D
)
<
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s
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a
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>
x
3
−
5
x
2
+
x
+
1
<
s
p
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a
s
s
=
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−
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>
(
E
)
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a
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>
None of above
<span class='latex-bold'>(A)</span>\ x^3-3x^2+8x-1 \\ \qquad<span class='latex-bold'>(B)</span>\ x^3+3x^2-2x+1 \\ \qquad<span class='latex-bold'>(C)</span>\ x^3+14x^2+3x-8 \\ \qquad<span class='latex-bold'>(D)</span>\ x^3-5x^2+x+1 \\ \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}
<
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>
(
A
)
<
/
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an
>
x
3
−
3
x
2
+
8
x
−
1
<
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p
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a
ss
=
′
l
a
t
e
x
−
b
o
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d
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>
(
B
)
<
/
s
p
an
>
x
3
+
3
x
2
−
2
x
+
1
<
s
p
an
c
l
a
ss
=
′
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a
t
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x
−
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o
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>
(
C
)
<
/
s
p
an
>
x
3
+
14
x
2
+
3
x
−
8
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
x
3
−
5
x
2
+
x
+
1
<
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 above
18
1
Hide problems
P18 [Number Theory] - Turkish NMO 1st Round - 2004
How many consequtive numbers are there in the set of positive integers in which powers of all prime factors in their prime factorizations are odd numbers?
<
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(
A
)
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>
3
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a
s
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=
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t
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−
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>
(
B
)
<
/
s
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a
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>
7
<
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a
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c
l
a
s
s
=
′
l
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x
−
b
o
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>
(
C
)
<
/
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a
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>
8
<
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a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
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a
n
>
10
<
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a
s
s
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−
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>
(
E
)
<
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a
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>
15
<span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ 7 \qquad<span class='latex-bold'>(C)</span>\ 8 \qquad<span class='latex-bold'>(D)</span>\ 10 \qquad<span class='latex-bold'>(E)</span>\ 15
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−
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(
A
)
<
/
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>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
B
)
<
/
s
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>
7
<
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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−
b
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d
′
>
(
C
)
<
/
s
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>
8
<
s
p
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c
l
a
ss
=
′
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a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
10
<
s
p
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c
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a
ss
=
′
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a
t
e
x
−
b
o
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d
′
>
(
E
)
<
/
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an
>
15
14
1
Hide problems
P14 [Number Theory] - Turkish NMO 1st Round - 2004
What is
o
−
w
o-w
o
−
w
, if
g
u
n
2
=
w
o
w
g
u
n
gun^2 = wowgun
gu
n
2
=
w
o
w
gu
n
where
g
,
n
,
o
,
u
,
w
∈
{
0
,
1
,
2
,
…
,
9
}
g,n,o,u,w \in \{0,1,2,\dots, 9\}
g
,
n
,
o
,
u
,
w
∈
{
0
,
1
,
2
,
…
,
9
}
?
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−
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(
A
)
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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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>
(
B
)
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a
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>
2
<
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c
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a
s
s
=
′
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a
t
e
x
−
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′
>
(
C
)
<
/
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a
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>
3
<
s
p
a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
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a
n
>
5
<
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s
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−
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>
(
E
)
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>
None of above
<span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 5 \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}
<
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p
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c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
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>
(
A
)
<
/
s
p
an
>
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
5
<
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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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>
None of above
10
1
Hide problems
P10 [Number Theory] - Turkish NMO 1st Round - 2004
Let
a
1
=
7
a_1 = \sqrt 7
a
1
=
7
and
b
i
=
⌊
a
i
⌋
b_i = \lfloor a_i \rfloor
b
i
=
⌊
a
i
⌋
,
a
i
+
1
=
1
b
i
−
⌊
b
i
⌋
a_{i+1} = \dfrac{1}{b_i - \lfloor b_i \rfloor}
a
i
+
1
=
b
i
−
⌊
b
i
⌋
1
for each
i
≥
i
i\geq i
i
≥
i
. What is the smallest integer
n
n
n
greater than
2004
2004
2004
such that
b
n
b_n
b
n
is divisible by
4
4
4
? (
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
denotes the largest integer less than or equal to
x
x
x
)
<
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(
A
)
<
/
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>
2005
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c
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a
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s
=
′
l
a
t
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x
−
b
o
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d
′
>
(
B
)
<
/
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a
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>
2006
<
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c
l
a
s
s
=
′
l
a
t
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x
−
b
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>
(
C
)
<
/
s
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a
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>
2007
<
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p
a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
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>
(
D
)
<
/
s
p
a
n
>
2008
<
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a
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c
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a
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s
=
′
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e
x
−
b
o
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d
′
>
(
E
)
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>
None of above
<span class='latex-bold'>(A)</span>\ 2005 \qquad<span class='latex-bold'>(B)</span>\ 2006 \qquad<span class='latex-bold'>(C)</span>\ 2007 \qquad<span class='latex-bold'>(D)</span>\ 2008 \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}
<
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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
′
>
(
A
)
<
/
s
p
an
>
2005
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2006
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2007
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
2008
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of above
6
1
Hide problems
P06 [Number Theory] - Turkish NMO 1st Round - 2004
For which of the following value of
n
n
n
, there exists integers
a
,
b
a,b
a
,
b
such that
a
2
+
a
b
−
6
b
2
=
n
a^2 + ab-6b^2 = n
a
2
+
ab
−
6
b
2
=
n
?
<
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>
(
A
)
<
/
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a
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>
17
<
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p
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
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a
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>
19
<
s
p
a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
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a
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>
29
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
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o
l
d
′
>
(
D
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31
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37
<span class='latex-bold'>(A)</span>\ 17 \qquad<span class='latex-bold'>(B)</span>\ 19 \qquad<span class='latex-bold'>(C)</span>\ 29 \qquad<span class='latex-bold'>(D)</span>\ 31 \qquad<span class='latex-bold'>(E)</span>\ 37
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19
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29
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p
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o
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31
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E
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37
2
1
Hide problems
P02 [Number Theory] - Turkish NMO 1st Round - 2004
How many pairs of integers
(
x
,
y
)
(x,y)
(
x
,
y
)
are there such that
2
x
+
5
y
=
x
y
−
1
2x+5y=xy-1
2
x
+
5
y
=
x
y
−
1
?
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(
A
)
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1
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(
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3
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a
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s
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4
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a
s
s
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(
D
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6
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12
<span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 3 \qquad<span class='latex-bold'>(C)</span>\ 4 \qquad<span class='latex-bold'>(D)</span>\ 6 \qquad<span class='latex-bold'>(E)</span>\ 12
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1
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−
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(
B
)
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/
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3
<
s
p
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c
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a
ss
=
′
l
a
t
e
x
−
b
o
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d
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>
(
C
)
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4
<
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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6
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−
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E
)
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12
33
1
Hide problems
P33 [Geometry] - Turkish NMO 1st Round - 2004
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid such that
∣
A
B
∣
=
9
|AB|=9
∣
A
B
∣
=
9
,
∣
C
D
∣
=
5
|CD|=5
∣
C
D
∣
=
5
and
B
C
∥
A
D
BC\parallel AD
BC
∥
A
D
. Let the internal angle bisector of angle
D
D
D
meet the internal angle bisectors of angles
A
A
A
and
C
C
C
at
M
M
M
and
N
N
N
, respectively. Let the internal angle bisector of angle
B
B
B
meet the internal angle bisectors of angles
A
A
A
and
C
C
C
at
L
L
L
and
K
K
K
, respectively. If
K
K
K
is on
[
A
D
]
[AD]
[
A
D
]
and
∣
L
M
∣
∣
K
N
∣
=
3
7
\dfrac{|LM|}{|KN|} = \dfrac 37
∣
K
N
∣
∣
L
M
∣
=
7
3
, what is
∣
M
N
∣
∣
K
L
∣
\dfrac{|MN|}{|KL|}
∣
K
L
∣
∣
MN
∣
?
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A
)
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62
63
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B
)
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27
35
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s
s
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C
)
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2
3
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a
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5
21
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24
63
<span class='latex-bold'>(A)</span>\ \dfrac{62}{63} \qquad<span class='latex-bold'>(B)</span>\ \dfrac{27}{35} \qquad<span class='latex-bold'>(C)</span>\ \dfrac{2}{3} \qquad<span class='latex-bold'>(D)</span>\ \dfrac{5}{21} \qquad<span class='latex-bold'>(E)</span>\ \dfrac{24}{63}
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63
62
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B
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27
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C
)
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2
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21
5
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63
24
29
1
Hide problems
P29 [Geometry] - Turkish NMO 1st Round - 2004
Let
M
M
M
be the intersection of the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
of cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
. If
∣
A
B
∣
=
5
|AB|=5
∣
A
B
∣
=
5
,
∣
C
D
∣
=
3
|CD|=3
∣
C
D
∣
=
3
, and
m
(
A
M
B
^
)
=
6
0
∘
m(\widehat{AMB}) = 60^\circ
m
(
A
MB
)
=
6
0
∘
, what is the circumradius of the quadrilateral?
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3
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B
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7
3
3
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6
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D
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4
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34
<span class='latex-bold'>(A)</span>\ 5\sqrt 3 \qquad<span class='latex-bold'>(B)</span>\ \dfrac {7\sqrt 3}{3} \qquad<span class='latex-bold'>(C)</span>\ 6 \qquad<span class='latex-bold'>(D)</span>\ 4 \qquad<span class='latex-bold'>(E)</span>\ \sqrt{34}
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3
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B
)
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3
7
3
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a
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x
−
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6
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c
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a
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a
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(
D
)
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4
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a
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=
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−
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d
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>
(
E
)
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34
25
1
Hide problems
P25 [Geometry] - Turkish NMO 1st Round - 2004
Let
D
D
D
be the foot of the internal angle bisector of the angle
A
A
A
of a triangle
A
B
C
ABC
A
BC
. Let
E
E
E
be a point on side
[
A
C
]
[AC]
[
A
C
]
such that
∣
C
E
∣
=
∣
C
D
∣
|CE|= |CD|
∣
CE
∣
=
∣
C
D
∣
and
∣
A
E
∣
=
6
5
|AE|=6\sqrt 5
∣
A
E
∣
=
6
5
; let
F
F
F
be a point on the ray
[
A
B
[AB
[
A
B
such that
∣
D
B
∣
=
∣
B
F
∣
|DB|=|BF|
∣
D
B
∣
=
∣
BF
∣
and
∣
A
B
∣
<
∣
A
F
∣
=
8
5
|AB|<|AF| = 8\sqrt 5
∣
A
B
∣
<
∣
A
F
∣
=
8
5
. What is
∣
A
D
∣
|AD|
∣
A
D
∣
?
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10
5
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o
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B
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8
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4
15
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5
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None of above
<span class='latex-bold'>(A)</span>\ 10\sqrt 5 \qquad<span class='latex-bold'>(B)</span>\ 8 \qquad<span class='latex-bold'>(C)</span>\ 4\sqrt{15} \qquad<span class='latex-bold'>(D)</span>\ 7\sqrt 5 \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}
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5
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(
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)
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8
<
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a
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a
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x
−
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C
)
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4
15
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p
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c
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a
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=
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a
t
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x
−
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o
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d
′
>
(
D
)
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7
5
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c
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a
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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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None of above
21
1
Hide problems
P21 [Geometry] - Turkish NMO 1st Round - 2004
Let the circles
S
1
S_1
S
1
and
S
2
S_2
S
2
meet at the points
A
A
A
and
B
B
B
. A line through
B
B
B
meets
S
1
S_1
S
1
at a point
D
D
D
other than
B
B
B
and meets
S
2
S_2
S
2
at a point
C
C
C
other than
B
B
B
. The tangent to
S
1
S_1
S
1
through
D
D
D
and the tangent to
S
2
S_2
S
2
through
C
C
C
meet at
E
E
E
. If
∣
A
D
∣
=
15
|AD|=15
∣
A
D
∣
=
15
,
∣
A
C
∣
=
16
|AC|=16
∣
A
C
∣
=
16
,
∣
A
B
∣
=
10
|AB|=10
∣
A
B
∣
=
10
, what is
∣
A
E
∣
|AE|
∣
A
E
∣
?
<
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)
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20
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B
)
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24
<
s
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a
n
c
l
a
s
s
=
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l
a
t
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x
−
b
o
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d
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>
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C
)
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25
<
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c
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a
s
s
=
′
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a
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−
b
o
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(
D
)
<
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26
<
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=
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)
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31
<span class='latex-bold'>(A)</span>\ 20 \qquad<span class='latex-bold'>(B)</span>\ 24 \qquad<span class='latex-bold'>(C)</span>\ 25 \qquad<span class='latex-bold'>(D)</span>\ 26 \qquad<span class='latex-bold'>(E)</span>\ 31
<
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(
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)
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20
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p
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a
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=
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a
t
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x
−
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o
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d
′
>
(
B
)
<
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24
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p
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c
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a
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=
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a
t
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x
−
b
o
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d
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(
C
)
<
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25
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
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(
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31
17
1
Hide problems
P17 [Geometry] - Turkish NMO 1st Round - 2004
Let
R
R
R
and
T
T
T
be points respectively on sides
[
B
C
]
[BC]
[
BC
]
and
[
C
D
]
[CD]
[
C
D
]
of a square
A
B
C
D
ABCD
A
BC
D
with side length
6
6
6
such that
∣
C
R
∣
+
∣
R
T
∣
+
∣
T
C
∣
=
12
|CR|+|RT|+|TC|=12
∣
CR
∣
+
∣
RT
∣
+
∣
TC
∣
=
12
. What is
tan
(
R
A
T
^
)
\tan (\widehat{RAT})
tan
(
R
A
T
)
<
s
p
a
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c
l
a
s
s
=
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l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
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a
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>
2
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
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>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
1
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
1
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
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a
n
>
1
<span class='latex-bold'>(A)</span>\ 2\sqrt 3 \qquad<span class='latex-bold'>(B)</span>\ \sqrt 3 \qquad<span class='latex-bold'>(C)</span>\ \dfrac 13 \qquad<span class='latex-bold'>(D)</span>\ \dfrac 12 \qquad<span class='latex-bold'>(E)</span>\ 1
<
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p
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c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
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an
>
2
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
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>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
3
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
2
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
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x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
1
13
1
Hide problems
P13 [Geometry] - Turkish NMO 1st Round - 2004
If the tangents of all interior angles of a triangle are integers, what is the sum of these integers?
<
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p
a
n
c
l
a
s
s
=
′
l
a
t
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x
−
b
o
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d
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>
(
A
)
<
/
s
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a
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>
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
5
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
6
<
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
)
<
/
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a
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>
None of above
<span class='latex-bold'>(A)</span>\ 4 \qquad<span class='latex-bold'>(B)</span>\ 5 \qquad<span class='latex-bold'>(C)</span>\ 6 \qquad<span class='latex-bold'>(D)</span>\ 9 \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
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>
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
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>
6
<
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 above
9
1
Hide problems
P09 [Geometry] - Turkish NMO 1st Round - 2004
What is the area of the region determined by the points outside a triangle with perimeter length
π
\pi
π
where none of these points has a distance greater than
1
1
1
to any corner of the triangle?
<
s
p
a
n
c
l
a
s
s
=
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a
t
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x
−
b
o
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′
>
(
A
)
<
/
s
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a
n
>
4
π
<
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p
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l
a
s
s
=
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a
t
e
x
−
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o
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>
(
B
)
<
/
s
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a
n
>
3
π
<
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
>
5
π
2
<
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c
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a
s
s
=
′
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a
t
e
x
−
b
o
l
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′
>
(
D
)
<
/
s
p
a
n
>
2
π
<
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p
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s
=
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a
t
e
x
−
b
o
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>
(
E
)
<
/
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>
3
π
2
<span class='latex-bold'>(A)</span>\ 4\pi \qquad<span class='latex-bold'>(B)</span>\ 3\pi \qquad<span class='latex-bold'>(C)</span>\ \dfrac{5\pi}2 \qquad<span class='latex-bold'>(D)</span>\ 2\pi \qquad<span class='latex-bold'>(E)</span>\ \dfrac{3\pi}2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
4
π
<
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p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
3
π
<
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ss
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a
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x
−
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(
C
)
<
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>
2
5
π
<
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a
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l
a
t
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x
−
b
o
l
d
′
>
(
D
)
<
/
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an
>
2
π
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(
E
)
<
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2
3
π
5
1
Hide problems
P05 [Geometry] - Turkish NMO 1st Round - 2004
If a triangle has side lengths
a
,
b
,
c
a,b,c
a
,
b
,
c
where
a
≤
2
≤
b
≤
3
a\leq 2 \leq b \leq 3
a
≤
2
≤
b
≤
3
, what is the largest possible value of its area?
<
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a
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−
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(
A
)
<
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s
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a
n
>
3
<
s
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a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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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
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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
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
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a
n
>
None of above
<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>\ \text{None of above}
<
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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
−
b
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d
′
>
(
A
)
<
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>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
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an
>
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
)
<
/
s
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an
>
None of above
1
1
Hide problems
P01 [Geometry] - Turkish NMO 1st Round - 2004
If the circumradius of a regular
n
n
n
-gon is
1
1
1
and the ratio of its perimeter over its area is
4
3
3
\dfrac{4\sqrt 3}{3}
3
4
3
, what is
n
n
n
?
<
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x
−
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(
A
)
<
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a
n
>
3
<
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p
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c
l
a
s
s
=
′
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a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
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
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
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a
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>
8
<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>\ 8
<
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p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
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s
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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
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
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an
>
8