Difference between revisions of "2021 JMPSC Sprint Problems/Problem 11"
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Now apply <math>f(x)=x-2</math> to get | Now apply <math>f(x)=x-2</math> to get | ||
<cmath>S_2=\{1,2,3,\cdots,48\}</cmath> | <cmath>S_2=\{1,2,3,\cdots,48\}</cmath> | ||
| − | Which clearly has cardinality <math>\boxed{ | + | Which clearly has cardinality <math>\boxed{48}</math>. |
~yofro | ~yofro | ||
Latest revision as of 18:55, 7 September 2021
Problem
How many numbers are in the finite sequence of consecutive perfect squares ![\[9, 16, 25, \ldots , 2500?\]](http://latex.artofproblemsolving.com/0/1/f/01f9f1e62863b82d5548bc8d8a32e611c94449ca.png)
Solution
The perfect squares are from 
to 
. Therefore, the answer is the amount of positive integers between 
and 
, inclusive. This is just 
.
Solution 2 (General Method)
The set is
![\[S=\{3^2,4^2,5^2,\cdots,50^2\}\]](http://latex.artofproblemsolving.com/a/e/9/ae9b3741279b2ae9404f386ce2d7dff0a48fc797.png)
Notice that the cardinality of
![\[\{a_1,a_2,\cdots,a_n\}\]](http://latex.artofproblemsolving.com/9/0/d/90d5c46dd980b3c511c9a6bcbcdb8b55975114a9.png)
is equal to the cardinality of
![\[\{f(a_1),f(a_2),\cdots, f(a_n)\}\]](http://latex.artofproblemsolving.com/6/b/8/6b85b228b9cf42b9192652ead773eda4cf7d9b72.png)
For all functions 
with domain containing 
.
In our case, apply 
to get
![\[S_1=\{3,4,5,\cdots,50\}\]](http://latex.artofproblemsolving.com/3/3/6/3361045d0f611194925b6692560f262500e7d68c.png)
Now apply 
to get
![\[S_2=\{1,2,3,\cdots,48\}\]](http://latex.artofproblemsolving.com/b/c/3/bc333d6c06a8b6783bf6b1bf441eefd788c5ccfc.png)
Which clearly has cardinality 
.
~yofro
See also
The problems on this page are copyrighted by the Junior Mathematicians' Problem Solving Competition. 
