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Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2008 National Olympiad First Round
16
16
Part of
2008 National Olympiad First Round
Problems
(1)
Turkey NMO 2008 1st Round - P16 (Combinatorics)
Source:
8/25/2012
A class of
50
50
50
students took an exam with
4
4
4
questions. At least
1
1
1
of any
40
40
40
students gave exactly
3
3
3
, at least
2
2
2
of any
40
40
40
gave exactly
2
2
2
, and at least
3
3
3
of any
40
40
40
gave exactly
1
1
1
correct answers. At least
4
4
4
of any
40
40
40
students gave exactly
4
4
4
wrong answers. What is the least number of students who gave an odd number of correct answers?
<
s
p
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(
A
)
<
/
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a
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>
18
<
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x
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>
(
B
)
<
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24
<
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x
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>
(
C
)
<
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p
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>
26
<
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c
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a
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s
=
′
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x
−
b
o
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>
(
D
)
<
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p
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>
28
<
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c
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a
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=
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x
−
b
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>
(
E
)
<
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None of the above
<span class='latex-bold'>(A)</span>\ 18 \qquad<span class='latex-bold'>(B)</span>\ 24 \qquad<span class='latex-bold'>(C)</span>\ 26 \qquad<span class='latex-bold'>(D)</span>\ 28 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
<
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an
c
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−
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>
(
A
)
<
/
s
p
an
>
18
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
24
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
26
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
28
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the above
inequalities