2022 AIME II Problems/Problem 10
Find the remainder whenis divided by .
Video Solution by OmegaLearn
To solve this problem, we need to use the following result:
Now, we use this result to solve this problem.
Therefore, modulo 1000, .
~Steven Chen (www.professorchenedu.com)
Solution 2 (similar to solution 1)
Doing simple algebra calculation will give the following equation:
Next, by using Hockey-Stick Identity, we have:
Since seems like a completely arbitrary number, we can use Engineer's Induction by listing out the first few sums. These are, in the order of how many terms there are starting from term: , , , , , and . Notice that these are just , , , , , . It's clear that this pattern continues up to terms, noticing that the "indexing" starts with instead of . Thus, the value of the sum is .
As in solution 1, obtain Write this as
We can safely write this expression as , since plugging and into both equal meaning they won't contribute to the sum.
Use the sum of powers formulae. We obtain
We can factor the following expression as and simplifying, we have
Substituting and simplifying gets so we would like to find To do this, get Next,
Solution 5 (Telescoping)
For the last step, see Solution 1.
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