Difference between revisions of "1993 USAMO Problems/Problem 4"
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Let <math>a</math>, <math>b</math> be odd positive integers. Define the sequence <math>(f_n)</math> by putting <math>f_1 = a</math>, | Let <math>a</math>, <math>b</math> be odd positive integers. Define the sequence <math>(f_n)</math> by putting <math>f_1 = a</math>, | ||
− | <math>f_2 = b</math>, and by letting | + | <math>f_2 = b</math>, and by letting <math>f_n</math> for <math>n\ge3</math> be the greatest odd divisor of <math>f_{n-1} + f_{n-2}</math>. |
Show that <math>f_n</math> is constant for <math>n</math> sufficiently large and determine the eventual | Show that <math>f_n</math> is constant for <math>n</math> sufficiently large and determine the eventual | ||
value as a function of <math>a</math> and <math>b</math>. | value as a function of <math>a</math> and <math>b</math>. | ||
− | |||
== Solution == | == Solution == | ||
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<P align="right"><math>\mathbb{Q.E.D}</math></P> | <P align="right"><math>\mathbb{Q.E.D}</math></P> | ||
− | == | + | == See Also == |
{{USAMO box|year=1993|num-b=3|num-a=5}} | {{USAMO box|year=1993|num-b=3|num-a=5}} | ||
* [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=356413#p356413 Discussion on AoPS/MathLinks] | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=356413#p356413 Discussion on AoPS/MathLinks] | ||
+ | {{MAA Notice}} | ||
+ | |||
+ | [[Category:Olympiad Number Theory Problems]] |
Latest revision as of 06:56, 19 July 2016
Problem 4
Let , be odd positive integers. Define the sequence by putting , , and by letting 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), .
See Also
1993 USAMO (Problems • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.