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In this exercise, you're given two algorithms, A and B, which have time complexities represented by functions \( T_A(n) \) and \( T_B(n) \), respectively. Specifically:

- \( T_A(n) = 0.1n^2 \log_{10} n \) (microseconds) for algorithm A.
- \( T_B(n) = 2.5n^2 \) (microseconds) for algorithm B.

You need to:

1. **Compare the two algorithms in the Big-O notation**: Big-O notation helps classify the growth rates of functions, so you're being asked to determine which algorithm grows slower for large \( n \), meaning it performs better for large problem sizes.

2. **Find a problem size \( n_0 \)** where one algorithm begins to outperform the other**: For \( n > n_0 \), you want to identify at what point one of the algorithms starts to consistently outperform the other.

3. **Give a recommendation**: If your problems have sizes \( n > 10^9 \), which algorithm (A or B) should you choose based on the analysis.

### What you need to do:

- First, compare the terms \( 0.1n^2 \log_{10} n \) and \( 2.5n^2 \) to see how they scale as \( n \) increases. Since both have \( n^2 \), the comparison comes down to how \( \log_{10} n \) behaves compared to the constant coefficient in \( T_B(n) \).
  
- Solve for \( n_0 \) by setting \( T_A(n) = T_B(n) \) and solve for \( n \) to determine when one algorithm overtakes the other.

- Use the analysis to recommend which algorithm to choose for \( n > 10^9 \).

Would you like me to walk through these steps in more detail, including solving for \( n_0 \)?

Algorithms A and B spend exactly T_A(n) = 0.1n^2 log_10 n and T_B(n) = 2.5n^2 microseconds, respectively, for a problem of size n. Choose the algorithm which is better in the Big-Oh sense, and find out a problem size n_0 such that for any larger size n > n_0, the chosen algorithm outperforms the other. If your problems are of the size n > 10^9, which algorithm will you recommend to use?

This problem involves comparing two algorithms, A and B, based on their time complexities: T_A(n) = 0.1n^2 log_10 n and T_B(n) = 2.5n^2. Learn how to determine the problem size n_0, where one algorithm outperforms the other, and find out which algorithm to choose for problem sizes n > 10^9.

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