answer:In the context of solving a mystery, relevant and irrelevant information can be nuanced terms. As you pointed out, all the information provided was connected to the case, making it technically relevant. However, when it comes to making progress in solving the mystery, some pieces of information are more crucial than others. Relevant information, in this context, refers to details that directly contribute to identifying the perpetrator, motive, or key events surrounding the crime. These pieces of information have a direct impact on the investigation and can be used to build a theory or make connections between suspects, evidence, and the crime scene. In the provided scenario, examples of highly relevant information include: * The torn piece of fabric caught in the window lock * The fact that the spouse's black suit jacket is missing * The lack of forced entry or struggle, suggesting the killer was someone the victim knew These details are crucial because they: * Provide physical evidence that can be tied to a suspect * Create a suspicious circumstance that implicates a suspect * Establish a key aspect of the crime scene that helps to narrow down the list of potential perpetrators On the other hand, less relevant or background information might include: * The fact that the victim was shot (while important, this detail doesn't directly point to a specific suspect) * The former employee's claim that their black suit jacket is at home (while potentially relevant, this information requires verification and doesn't immediately contribute to the investigation) * The general relationships between the suspects and the victim (while important for understanding motives and potential conflicts, these details don't directly tie to the physical evidence or the crime scene) This type of information is not necessarily irrelevant, but it's less critical in the immediate context of identifying the perpetrator. These details might become more relevant later in the investigation, as they can help to establish motives, alibis, or other connections between suspects and the crime. In general, the most useful information for solving a mystery is that which: * Provides direct evidence of a crime or a suspect's involvement * Creates a clear connection between a suspect and the crime scene * Establishes a suspicious circumstance that can be investigated further * Helps to rule out or confirm specific theories or suspects By prioritizing the most relevant information and using it to inform the investigation, detectives can make progress in solving the mystery and ultimately uncover the truth.

answer:1. **Define ( Q(X) ):** Define the polynomial ( Q(X) ) such that ( Q(X) = P(X) - n ). Given ( P(k_i) = n ) for ( 1 leq i leq n ), it follows that: [ Q(k_i) = P(k_i) - n = n - n = 0 ] Therefore, ( Q ) has ( n ) roots ( k_1, k_2, ldots, k_n ), and ( text{deg}(Q) = n ). 2. **Expression for ( Q(X) ):** Since ( Q ) is a polynomial of degree ( n ) with ( n ) roots, there exists an integer ( c ) such that: [ Q(X) = c (X - k_1)(X - k_2) cdots (X - k_n) ] Here, ( c ) is an integer because ( P ) has integer coefficients. 3. **Evaluate ( Q(0) ):** Evaluate ( Q ) at ( X = 0 ): [ Q(0) = P(0) - n = 0 - n = -n ] On the other hand: [ Q(0) = c(-k_1)(-k_2) cdots (-k_n) = c (-1)^n k_1 k_2 cdots k_n ] Therefore, we obtain: [ -n = c (-1)^n k_1 k_2 cdots k_n ] 4. **Determine bounds on ( |Q(0)| ):** Without loss of generality, ( k_i neq 0 ) and are distinct integers. For simplicity, since ( n geq 1 ), at least ( n-2 ) of the ( k_i ) must be of absolute value at least 2 (because at most two ( k_i ) can be ( pm 1 )): [ |k_1 k_2 cdots k_n| geq 1 cdot 1 cdot 2 cdot 2 cdots 2 = 2^{n-2} ] Thus: [ |c(-1)^n k_1 k_2 cdots k_n| geq 2^{n-2} ] 5. **Solve the inequality for ( n ):** Hence: [ n geq 2^{n-2} ] 6. **Verify by induction that ( n leq 4 ):** We show by induction that ( 2^{m-2} > m ) for ( m geq 5 ): - For the base case, ( m = 5 ): [ 2^{5-2} = 2^3 = 8 quad text{and} quad 8 > 5 ] - Inductive step: Assume ( 2^{m-2} > m ). We need to show ( 2^{m-1} > m + 1 ): [ 2^{m-1} = 2 cdot 2^{m-2} > 2m ] Given ( 2m > m+1 ) for ( m geq 2 ), it follows that: [ 2^{m-1} > m + 1 ] From this, the inequality ( 2^{n-2} geq n ) shows that for ( n geq 5 ) the inequality ( n leq 2^{n-2} ) is not satisfied, therefore ( n geq 5 ) is impossible. 7. **Verify ( n = 1, 2, 3, 4 ):** - For ( n = 1 ): [ P(X) = X quad (text{with } k_1 = 1) ] - For ( n = 2 ): [ P(X) = (X-1)(X-2) + 2 = X^2 - 3X + 2 quad (text{with } k_1 = 1, k_2 = 2) ] - For ( n = 3 ): [ P(X) = (X-1)(X+1)(X-3) + 3 = X^3 - 3X quad (text{with } k_1 = 1, k_2 = -1, k_3 = 3) ] - For ( n = 4 ): [ P(X) = (X-1)(X+1)(X+2)(X-2) + 4 = X^4 - 5X^2 + 4 quad (text{with } k_1 = 1, k_2 = -1, k_3 = 2, k_4 = -2) ] 8. **Conclusion:** The integers ( n ) that satisfy the conditions are: [ boxed{1, 2, 3, 4} ]

answer:Michael is enthusiastic about collaborating on a project focused on different ecosystems each month, including hands-on projects and social media.

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