Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2022 USAJMO Problems/Problem 1"

(Created page with "We claim that <math>m</math> satisfies the given conditions if and only if <math>m</math> is squareful. To begin, we let the common difference be <math>d</math> and the commo...")
 
Line 1: Line 1:
 +
==Problem==
 +
For which positive integers <math>m</math> does there exist an infinite arithmetic sequence of integers <math>a_1,a_2,\cdots</math> and an infinite geometric sequence of integers <math>g_1,g_2,\cdots</math> satisfying the following properties?
 +
 +
<math>\bullet</math> <math>a_n-g_n</math> is divisible by <math>m</math> for all integers <math>n>1</math>;
 +
 +
<math>\bullet</math> <math>a_2-a_1</math> is not divisible by <math>m</math>.
 +
 +
==Solution 1==
 +
 
We claim that <math>m</math> satisfies the given conditions if and only if <math>m</math> is squareful.
 
We claim that <math>m</math> satisfies the given conditions if and only if <math>m</math> is squareful.
  
Line 15: Line 24:
 
<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)\iff (3),(4)</math>, and the conditions <math>g_{l-1}(r-l)\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 not squareful.
+
Whee! Restating, <math>(1),(2)\iff (3),(4)</math>, and the conditions <math>g_{l-1}(r-l)\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 20:33, 19 April 2022

Problem

For which positive integers $m$ does there exist an infinite arithmetic sequence of integers $a_1,a_2,\cdots$ and an infinite geometric sequence of integers $g_1,g_2,\cdots$ satisfying the following properties?

$\bullet$ $a_n-g_n$ is divisible by $m$ for all integers $n>1$;

$\bullet$ $a_2-a_1$ is not divisible by $m$.

Solution 1

We claim that $m$ satisfies the given conditions if and only if $m$ is squareful.

To begin, we let the common difference be $d$ and the common ratio be $r$. Then, rewriting the conditions modulo $m$ gives: \[a_2-a_1=d\not\equiv 0\pmod{m}\text{ (1)}\] \[a_n\equiv g_n\pmod{m}\text{ (2)}\]

Condition $(1)$ holds iff no consecutive terms in $a_i$ are equivalent modulo $m$, which is the same thing as never having consecutive, equal, terms, in $a_i\pmod{m}$. By Condition $(2)$, this is also the same as never having equal, consecutive, terms in $g_i\pmod{m}$:

\[(1)\iff g_l\not\equiv g_{l-1}\pmod{m}\text{ for any integer }l>1\] \[\iff g_{l-1}(r-l)\not\equiv 0\pmod{m}.\text{ (3)}\]


Also, Condition $(2)$ holds iff \[g_{l+1}-g_l\equiv g_l-g_{l-1}\pmod{m}\] \[g_{l-1}(r-1)^2\equiv0\pmod{m}\text{ (4)}.\]

Whee! Restating, $(1),(2)\iff (3),(4)$, and the conditions $g_{l-1}(r-l)\not\equiv 0\pmod{m}$ and $g_{l-1}(r-1)^2\equiv0\pmod{m}$ hold if and only if $m$ is squareful.

[will finish that step here]

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2022_USAJMO_Problems/Problem_1&oldid=173488"