Difference between revisions of "2023 AMC 12A Problems/Problem 25"
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− | Note that <math>\tan{kx} = \frac{\binom{k}{1}\tan{x} - \binom{k}{3}\tan^{3}{x} + \cdots \pm \binom{k}{k}\tan^{k}{x}}{\binom{k}{0}\tan^{0}{x} - \binom{k}{2}\tan^{2}{x} + \cdots + \binom{k}{k-1}\tan^{k-1}{x}}</math>, where k is odd and the sign of each term alternates between positive and negative. To realize this during the test, you should know the formulas of <math>\tan{2x}, \tan{3x},</math> and <math>\tan{4x}</math>, and can notice the pattern from that. The expression given essentially matches the formula of <math>\tan{kx}</math> exactly. <math>a_{2023}</math> is evidently equivalent to <math>\pm\binom{2023}{2023}</math>, or 1. However, it could be positive or negative. Notice that in the numerator, whenever the exponent of the tangent term is congruent to 1 mod 4, the term is positive. Whenever the exponent of the tangent term is 3 mod 4, the term is negative. 2023, which is assigned to k, is congruent to 3 mod 4. This means that the term of <math>\binom{k}{k}\tan^{k}</math> is <math>\boxed{-1}</math>. | + | Note that <math>\tan{kx} = \frac{\binom{k}{1}\tan{x} - \binom{k}{3}\tan^{3}{x} + \cdots \pm \binom{k}{k}\tan^{k}{x}}{\binom{k}{0}\tan^{0}{x} - \binom{k}{2}\tan^{2}{x} + \cdots + \binom{k}{k-1}\tan^{k-1}{x}}</math>, where k is odd and the sign of each term alternates between positive and negative. To realize this during the test, you should know the formulas of <math>\tan{2x}, \tan{3x},</math> and <math>\tan{4x}</math>, and can notice the pattern from that. The expression given essentially matches the formula of <math>\tan{kx}</math> exactly. <math>a_{2023}</math> is evidently equivalent to <math>\pm\binom{2023}{2023}</math>, or 1. However, it could be positive or negative. Notice that in the numerator, whenever the exponent of the tangent term is congruent to 1 mod 4, the term is positive. Whenever the exponent of the tangent term is 3 mod 4, the term is negative. 2023, which is assigned to k, is congruent to 3 mod 4. This means that the term of <math>\binom{k}{k}\tan^{k}{x}</math> is <math>\boxed{-1}</math>. |
~lprado | ~lprado |
Revision as of 00:25, 10 November 2023
Problem
There is a unique sequence of integers such that
whenever
is defined. What is
Solution
Note that , where k is odd and the sign of each term alternates between positive and negative. To realize this during the test, you should know the formulas of
and
, and can notice the pattern from that. The expression given essentially matches the formula of
exactly.
is evidently equivalent to
, or 1. However, it could be positive or negative. Notice that in the numerator, whenever the exponent of the tangent term is congruent to 1 mod 4, the term is positive. Whenever the exponent of the tangent term is 3 mod 4, the term is negative. 2023, which is assigned to k, is congruent to 3 mod 4. This means that the term of
is
.
~lprado
See Also
2023 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 24 |
Followed by Last Problem |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.