2023 AIME II Problems/Problem 8
Solution 1
For any , we have, The second and the fifth equalities follow from the property that .
Therefore,
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Solution 2 (Moduli)
Because the answer must be a positive integer, it is just equal to the modulus of the product. Define .
Then, our product is equal to
, and we may observe that and are conjugates for any , meaning that their magnitudes are the same. Thus, our product is
Let us simplify the first term. Expanding, we obtain
Rearranging and cancelling, we obtain
By the cosine subtraction formula, we have .
See also
2023 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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