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Home » Bo6 Season 2 Hype Is UNSTOPPABLE—Are You Ready for the Shocking Twists? - ClickBalance

Bo6 Season 2 Hype Is UNSTOPPABLE—Are You Ready for the Shocking Twists? - ClickBalance

📅 May 19, 2026 👤 scraface

May 19, 2026

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📌 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 $10$?

📌 First, compute the least common multiple (LCM) of $12$ and $18$.

📌 \text{lcm}(12, 18) = \text{lcm}(2^2 \cdot 3, 2 \cdot 3^2) = 2^2 \cdot 3^2 = 36

📌 The smallest prime greater than $10$ is $11$, so its square is $11^2 = 121$.

📌 We now seek the smallest four-digit number divisible by both $36$ and $121$.

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