Difference between revisions of "Talk:2012 USAMO Problems/Problem 3"
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− | The answer is the set of all integers that are at least 3. | + | The answer is the set of all integers that are at least <math>3</math>. |
− | For composite n where there are two primes p_1 and p_2 such that n | + | For composite <math>n</math> where there are two primes <math>p_1</math> and <math>p_2</math> such that <math>\frac{n}{2}<p_1<p_2<n</math>, here's your construction: |
− | Pick maximal integers j_1 and j_2 such that<math> | + | Pick maximal integers <math>j_1</math> and <math>j_2</math> such that <math>p_1^{j_1}p_2^{j_2}</math> divides <math>i</math>. |
− | Pick a minimal positive integer s such that <math> | + | Pick a minimal positive integer s such that <math>\frac{n(n+1)}{2}+(s-1)p_1 \equiv 0</math> (mod <math>p_2</math>). (You know it exists since <math>p_1</math> and <math>p_2</math> are relatively prime.) |
− | Pick an integer t such that<math> | + | Pick an integer t such that<math> \frac{n(n+1)}{2}+(s-1)p_1+(t-1)p_2=0</math>. (It exists because of how we defined s. It also must be negative.) |
− | Then <math>a_i=(s^ | + | Then <math>a_i=(s^{j_1})(t^{j_2})</math>. |
For n=4: | For n=4: | ||
− | <math>a_i=(-1)^ | + | <math>a_i=(-1)^{j_1+j_2}</math>, where<math>2^{j_1}3^{j_2} </math>divides i. |
For n=6: | For n=6: | ||
− | <math>a_i= | + | <math>a_i=2^{j_1}(-5)^{j_2}</math>, where <math>3^{j_1}5^{j_2}</math>divides i. |
For n=10: | For n=10: | ||
− | <math>a_i= | + | <math>a_i=2^{j_1}(-9)^{j_2}</math>, where <math>5^{j_1}7^{j_2}</math>divides i. |
[I don't know LaTeX, so someone else can input it.] | [I don't know LaTeX, so someone else can input it.] | ||
--[[User:Mage24365|Mage24365]] 09:00, 25 April 2012 (EDT) | --[[User:Mage24365|Mage24365]] 09:00, 25 April 2012 (EDT) |
Latest revision as of 16:22, 3 May 2012
The answer is the set of all integers that are at least .
For composite where there are two primes
and
such that
, here's your construction:
Pick maximal integers and
such that
divides
.
Pick a minimal positive integer s such that (mod
). (You know it exists since
and
are relatively prime.)
Pick an integer t such that. (It exists because of how we defined s. It also must be negative.)
Then .
For n=4:
, where
divides i.
For n=6:
, where
divides i.
For n=10:
, where
divides i.
[I don't know LaTeX, so someone else can input it.]
--Mage24365 09:00, 25 April 2012 (EDT)