Difference between revisions of "2022 USAJMO Problems/Problem 1"
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<cmath>a_n\equiv g_n\pmod{m}\text{ (2)}</cmath> | <cmath>a_n\equiv g_n\pmod{m}\text{ (2)}</cmath> | ||
− | Condition <math>(1)</math> holds | + | Condition <math>(1)</math> holds if no consecutive terms in <math>a_i</math> are equivalent modulo <math>m</math>, which is the same thing as never having consecutive, equal, terms, in <math>a_i\pmod{m}</math>. By Condition <math>(2)</math>, this is also the same as never having equal, consecutive, terms in <math>g_i\pmod{m}</math>: |
<cmath>(1)\iff g_l\not\equiv g_{l-1}\pmod{m}\text{ for any integer }l>1</cmath> | <cmath>(1)\iff g_l\not\equiv g_{l-1}\pmod{m}\text{ for any integer }l>1</cmath> | ||
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− | Also, Condition <math>(2)</math> holds | + | Also, Condition <math>(2)</math> holds if |
<cmath>g_{l+1}-g_l\equiv g_l-g_{l-1}\pmod{m}</cmath> | <cmath>g_{l+1}-g_l\equiv g_l-g_{l-1}\pmod{m}</cmath> | ||
<cmath>g_{l-1}(r-1)^2\equiv0\pmod{m}\text{ (4)}.</cmath> | <cmath>g_{l-1}(r-1)^2\equiv0\pmod{m}\text{ (4)}.</cmath> | ||
− | Whee! Restating, <math>(1),(2)\ | + | Whee! Restating, <math>(1),(2)\if (3),(4)</math>, and the conditions <math>g_{l-1}(r-1)\not\equiv 0\pmod{m}</math> and <math>g_{l-1}(r-1)^2\equiv0\pmod{m}</math> hold if and only if <math>m</math> is squareful. |
[will finish that step here] | [will finish that step here] |
Revision as of 19:01, 12 November 2023
Problem
For which positive integers does there exist an infinite arithmetic sequence of integers
and an infinite geometric sequence of integers
satisfying the following properties?
is divisible by
for all integers
;
is not divisible by
.
Solution 1
We claim that satisfies the given conditions if and only if
is squareful.
To begin, we let the common difference be and the common ratio be
. Then, rewriting the conditions modulo
gives:
Condition holds if no consecutive terms in
are equivalent modulo
, which is the same thing as never having consecutive, equal, terms, in
. By Condition
, this is also the same as never having equal, consecutive, terms in
:
Also, Condition holds if
Whee! Restating, $(1),(2)\if (3),(4)$ (Error compiling LaTeX. Unknown error_msg), and the conditions and
hold if and only if
is squareful.
[will finish that step here]
See Also
2022 USAJMO (Problems • Resources) | ||
Preceded by First Question |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.