2009 USAMO Problems/Problem 6
Problem
Let be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that
Suppose that
is also an infinite, nonconstant sequence of rational numbers with the property that
is an integer for all
and
. Prove that there exists a rational number
such that
and
are integers for all
and
.
Solution
Suppose the can be represented as
for every
, and suppose
can be represented as
. Let's start with only the first two terms in the two sequences,
and
for sequence
and
and
for sequence
. Then by the conditions of the problem, we have
is an integer, or
r = \frac{b_1 b_2}{d_1 d_2}
s_2 - s_1
b_1 b_2
t_2 - t_1
d_1 d_2
\frac{b_1 b_2}{d_1 d_2}$will always give an integer.
Now suppose we kept adding$ (Error compiling LaTeX. Unknown error_msg)s_it_i
s_m = \frac{a_m}{b_m}
s
t_m = \frac{c_m}{d_m}
t
m
r
\frac{\prod_{n=1}^{m}b_n}{\prod_{n=1}^{m}d_n}
m
s_i
t_i
r
\frac{\prod_{n=1}^{m}b_n}{\prod_{n=1}^{m}d_n}$, and we are done.
See Also
2009 USAMO (Problems • Resources) | ||
Preceded by Problem 5 |
Followed by Last question | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
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