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Brazil National Olympiad
1985 Brazil National Olympiad
1
1
Part of
1985 Brazil National Olympiad
Problems
(1)
1/1 \cdot 4 + 1/4\cdot 7 + 1/7\cdot 10+ ...+1/2998 \cdot 3001, sum with hint
Source: Brazilian Mathematical Olympiad 1985 P1
7/27/2018
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
are integers with
a
d
≠
b
c
ad \ne bc
a
d
=
b
c
. Show that
1
/
(
(
a
x
+
b
)
(
c
x
+
d
)
)
1/((ax+b)(cx+d))
1/
((
a
x
+
b
)
(
c
x
+
d
))
can be written in the form
r
/
(
a
x
+
b
)
+
s
/
(
c
x
+
d
)
r/(ax+b) + s/(cx+d)
r
/
(
a
x
+
b
)
+
s
/
(
c
x
+
d
)
. Find the sum
1
/
1
⋅
4
+
1
/
4
⋅
7
+
1
/
7
⋅
10
+
.
.
.
+
1
/
2998
⋅
3001
1/1\cdot 4 + 1/4\cdot 7 + 1/7\cdot 10 + ... + 1/2998 \cdot 3001
1/1
⋅
4
+
1/4
⋅
7
+
1/7
⋅
10
+
...
+
1/2998
⋅
3001
.
algebra
Sum