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Soros Olympiad in Mathematics
IV Soros Olympiad 1997 - 98 (Russia)
10.2
sum k!/((x+k)..(x+1))=1 (IV Soros Olympiad 1997-98 Correspondence 10.2)
sum k!/((x+k)..(x+1))=1 (IV Soros Olympiad 1997-98 Correspondence 10.2)
Source:
June 1, 2024
algebra
Problem Statement
Solve the equation
10
x
+
10
+
10
⋅
9
(
x
+
10
)
(
x
+
9
)
+
10
⋅
9
⋅
8
(
x
+
10
)
(
x
+
9
)
(
x
+
8
)
+
.
.
.
+
10
⋅
9
⋅
.
.
.
⋅
2
⋅
1
(
x
+
10
)
(
x
+
9
)
⋅
.
.
.
⋅
(
x
+
1
)
=
11
\frac{10}{x+10}+\frac{10\cdot 9}{(x+10)(x+9)}+\frac{10\cdot 9\cdot 8}{(x+10)(x+9)(x+8)}+ ...+\frac{10\cdot 9\cdot ... \cdot 2 \cdot 1}{(x+10)(x+9)\cdot ... \cdot(x+1)}=11
x
+
10
10
+
(
x
+
10
)
(
x
+
9
)
10
⋅
9
+
(
x
+
10
)
(
x
+
9
)
(
x
+
8
)
10
⋅
9
⋅
8
+
...
+
(
x
+
10
)
(
x
+
9
)
⋅
...
⋅
(
x
+
1
)
10
⋅
9
⋅
...
⋅
2
⋅
1
=
11
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