Difference between revisions of "1993 USAMO Problems/Problem 4"
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Note that if there is some <math>f_n=f_{n-1}</math> for any <math>n</math>, then <math>\frac{f_{n}+f_{n-1}}{2}=f_n</math>, which is odd. Thus, <math>f_{n+1}=f_n=f_{n-1}</math> and by induction, all <math>f_p</math> is constant for <math>p\ge n</math>. | Note that if there is some <math>f_n=f_{n-1}</math> for any <math>n</math>, then <math>\frac{f_{n}+f_{n-1}}{2}=f_n</math>, which is odd. Thus, <math>f_{n+1}=f_n=f_{n-1}</math> and by induction, all <math>f_p</math> is constant for <math>p\ge n</math>. | ||
− | Also note that <math>f_n | + | Also note that <math>f_n>0</math> since average of <math>2</math> positive number is always positive. |
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Then, <math>f_{n+1}\le\frac{f_{n}+f_{n-1}}{2}< \max (f_{n},f_{n-1})</math>, <math>f_{n+2}\le\frac{f_{n+1}+f_{n}}{2}< \max (f_{n},f_{n+1})\le\max (f_{n},f_{n-1})</math> | Then, <math>f_{n+1}\le\frac{f_{n}+f_{n-1}}{2}< \max (f_{n},f_{n-1})</math>, <math>f_{n+2}\le\frac{f_{n+1}+f_{n}}{2}< \max (f_{n},f_{n+1})\le\max (f_{n},f_{n-1})</math> | ||
− | Thus, <math>\max (f_{n},f{n-1})> \max (f_{n+1},f_{n+2})</math> and that means that <math>\max (f_{2n},f_{2n+1})</math> is a strictly decreasing function and it must reach <math>0</math> as <math>n\rightarrow\infty</math>, which contradict with the fact that <math>f_n | + | Thus, <math>\max (f_{n},f{n-1})> \max (f_{n+1},f_{n+2})</math> and that means that <math>\max (f_{2n},f_{2n+1})</math> is a strictly decreasing function and it must reach <math>0</math> as <math>n\rightarrow\infty</math>, which contradict with the fact that <math>f_n>0</math>. |
<br/><b>Part 1</b> proven. | <br/><b>Part 1</b> proven. |
Revision as of 19:53, 22 April 2010
Problem 4
Let ,
be odd positive integers. Define the sequence
by putting
,
, and by letting fn for
be the greatest odd divisor of
.
Show that
is constant for
sufficiently large and determine the eventual
value as a function of
and
.
Solution
Part 1) Prove that is constant for sufficiently large
.
Note that if there is some for any
, then
, which is odd. Thus,
and by induction, all
is constant for
.
Also note that since average of
positive number is always positive.
Thus, assume for contradiction, ,
.
Then, ,
Thus, and that means that
is a strictly decreasing function and it must reach
as
, which contradict with the fact that
.
Part 1 proven.
Part 2) Show that the constant is .
For any where
.
for
with the same property except with
and
.
Therefore, if I prove that the constant for any with relatively prime
,
is
, then I have shown that part 2 is true.
Lemma) If , then
.
Assume for contradiction that , since both
and
are odd,
is not divisible by
.
for some
such that
is odd.
, where
and
is another integer.
Thus, is divisible by
which contradicts with the assumption that
.
Lemma proven
By induction, since
.
Since there must exist some where
(part 1),
.
Resources
1993 USAMO (Problems • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |