Nice problem
2021 AMC 10A (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF
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Instructions
- This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
- You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer.
- No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the SAT if before 2006. No problems on the test will require the use of a calculator).
- Figures are not necessarily drawn to scale.
- You will have 75 minutes working time to complete the test.
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A number 
is said to be special if
![\[\lim_{q \to\infty} \frac{\cos \left( \sum_{k=0}^{\epsilon} \sqrt{ \frac{\epsilon ^ {k \epsilon + q}}{\epsilon \cdot \pi \cdot x} } \right)}{(2x+4) \lfloor qx \rfloor}\]](//latex.artofproblemsolving.com/a/c/9/ac98c42cf35422faca742c3b49a024475e5bdfd3.png)
is true for all 
.
Find all special numbers.
is said to be special if
![\[\lim_{q \to\infty} \frac{\cos \left( \sum_{k=0}^{\epsilon} \sqrt{ \frac{\epsilon ^ {k \epsilon + q}}{\epsilon \cdot \pi \cdot x} } \right)}{(2x+4) \lfloor qx \rfloor}\]](http://latex.artofproblemsolving.com/a/c/9/ac98c42cf35422faca742c3b49a024475e5bdfd3.png)
is true for all 
.
Find all special numbers.
