2021 JMPSC Invitationals Problems/Problem 11
Problem
For some , the arithmetic progression
has exactly
perfect squares. Find the maximum possible value of
Solution
First note that the integers in the given arithmetic progression are precisely the integers which leave a remainder of when divided by
.
Suppose a perfect square is in this arithmetic progression. Observe that the remainders when
,
,
,
, and
are divided by
are
,
,
,
, and
, respectively. Furthermore, for any integer
,
and so
and
leave the same remainder when divided by
. It follows that the perfect squares in this arithmetic progression are exactly the numbers of the form
and
, respectively.
Finally, the sequence of such squares is
In particular, the first and second such squares are associated with , the third and fourth are associated with
, and so on. It follows that the
such number, which is associated with
, is
Therefore the arithmetic progression must not reach . This means the desired answer is
~djmathman
See also
- Other 2021 JMPSC Invitationals Problems
- 2021 JMPSC Invitationals Answer Key
- All JMPSC Problems and Solutions
The problems on this page are copyrighted by the Junior Mathematicians' Problem Solving Competition.