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Home » Swag Bag FOMO: The Ultimate Style Statement That All Your Friends Will Steal! - Crankk.io

Swag Bag FOMO: The Ultimate Style Statement That All Your Friends Will Steal! - Crankk.io

📅 February 28, 2026 👤 scraface

Feb 28, 2026

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📌 Question: Determine the number of real solutions to the equation \(\sin(2x) = \frac{1}{2}$ for \(x \in [0, 2\pi]\).

📌 Solution: The equation \(\sin(2x) = \frac{1}{2}\) has solutions when \(2x = \frac{\pi}{6} + 2\pi n\) or \(2x = \frac{5\pi}{6} + 2\pi n\) for integer \(n\). Solving for \(x\) in \([0, 2\pi]\):

📌 \(x = \frac{\pi}{12} + \pi n\): Solutions are \(\frac{\pi}{12}, \frac{13\pi}{12}\).

📌 \(x = \frac{5\pi}{12} + \pi n\): Solutions are \(\frac{5\pi}{12}, \frac{17\pi}{12}\).

📌 Total of 4 solutions.

📌 Question: If \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) are unit vectors, find the maximum value of \(\mathbf{u} \cdot (\mathbf{v} + \mathbf{w})\).

📌 Solution: The dot product \(\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}\). Since \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) are unit vectors, each dot product is at most 1. The maximum occurs when \(\mathbf{u}\) aligns with both \(\mathbf{v}\) and \(\mathbf{w}\), i.e., \(\mathbf{v} = \mathbf{w} = \mathbf{u}\). Then \(\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = 1 + 1 = 2\).

📌 \boxed{2}Question: What is the smallest four-digit number that is a multiple of the least common multiple of the numbers $12$ and $18$, and also divisible by the square of the smallest prime greater than $10$?

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