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AMC 12/AHSME
2018 AMC 12/AHSME
9
Gauss Sum for High Schoolers
Gauss Sum for High Schoolers
Source: 2018 AMC 12B #9
February 16, 2018
Gauss
2018 AMC 12B
2018 AMC
AMC 12
Problem Statement
What is
∑
i
=
1
100
∑
j
=
1
100
(
i
+
j
)
?
\sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) ?
i
=
1
∑
100
j
=
1
∑
100
(
i
+
j
)?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
100
,
100
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
500
,
500
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
505
,
000
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
1
,
001
,
000
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
1
,
010
,
000
<span class='latex-bold'>(A) </span>100,100 \qquad <span class='latex-bold'>(B) </span>500,500\qquad <span class='latex-bold'>(C) </span>505,000 \qquad <span class='latex-bold'>(D) </span>1,001,000 \qquad <span class='latex-bold'>(E) </span>1,010,000 \qquad
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
100
,
100
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
500
,
500
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
505
,
000
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
1
,
001
,
000
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
1
,
010
,
000
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