Key Concepts and Formulas

• Cauchy Functional Equation for Natural Numbers: If $f: \mathbf{N} \to \mathbf{R}$ satisfies $f(m+n) = f(m) + f(n)$ for all $m, n \in \mathbf{N}$, then $f(x) = cx$ for some constant $c$.
• Sum of First $n$ Natural Numbers: The sum of the first $n$ natural numbers is given by the formula $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$.
• Properties of Summation: The sum of a constant over a range is the constant multiplied by the number of terms: $\sum_{k=1}^{N} c = Nc$. Also, $\sum_{k=1}^{N} (a_k + b_k) = \sum_{k=1}^{N} a_k + \sum_{k=1}^{N} b_k$.

Step-by-Step Solution

Step 1: Determine the form of the function $f(x)$. The problem states that $f$ satisfies $f(\mathrm{~m}+\mathrm{n})=f(\mathrm{~m})+f(\mathrm{n})$ for all $\mathrm{m}, \mathrm{n} \in \mathbf{N}$, and $f(1)=1$. This is a form of the Cauchy functional equation restricted to natural numbers. For this domain, the functional equation implies that $f(x)$ must be a linear function of the form $f(x) = cx$. We can prove this by induction: Base case: For $x=1$, $f(1) = c \cdot 1 = c$. Given $f(1)=1$, we have $c=1$. Inductive hypothesis: Assume $f(k) = k$ for some natural number $k$. Inductive step: We want to show $f(k+1) = k+1$. Using the functional equation, $f(k+1) = f(k) + f(1)$. By the inductive hypothesis, $f(k) = k$. We are given $f(1)=1$. So, $f(k+1) = k + 1$. Therefore, by mathematical induction, $f(x) = x$ for all $x \in \mathbf{N}$.

• Why this step is taken: Identifying the explicit form of $f(x)$ is essential to evaluate the summation.

Step 2: Substitute $f(x)=x$ into the summation. The given inequality involves the summation \sum_\limits{\mathrm{k}=1}^{2022} f(\lambda+\mathrm{k}). Since $f(x)=x$, we have $f(\lambda+k) = \lambda+k$. Substituting this into the summation, we get: $\sum_{k=1}^{2022} (\lambda+k)$

• Why this step is taken: This step transforms the problem from one involving a function into a problem involving a standard summation.

Step 3: Evaluate the summation. We can split the summation into two parts using the linearity of summation: $\sum_{k=1}^{2022} (\lambda+k) = \sum_{k=1}^{2022} \lambda + \sum_{k=1}^{2022} k$ The first part is the sum of a constant $\lambda$ for 2022 terms: $\sum_{k=1}^{2022} \lambda = 2022 \lambda$ The second part is the sum of the first 2022 natural numbers, which can be calculated using the formula $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$ with $n=2022$: $\sum_{k=1}^{2022} k = \frac{2022(2022+1)}{2} = \frac{2022 \times 2023}{2}$ Combining these two parts, the summation evaluates to: $2022 \lambda + \frac{2022 \times 2023}{2}$

• Why this step is taken: This simplifies the summation into an algebraic expression involving $\lambda$.

Step 4: Set up and solve the inequality. The problem states that \sum_\limits{\mathrm{k}=1}^{2022} f(\lambda+\mathrm{k}) \leq(2022)^2. Substituting the evaluated sum: $2022 \lambda + \frac{2022 \times 2023}{2} \leq (2022)^2$ To simplify, divide the entire inequality by 2022 (which is positive, so the inequality direction is preserved): $\frac{2022 \lambda}{2022} + \frac{2022 \times 2023}{2 \times 2022} \leq \frac{(2022)^2}{2022}$ $\lambda + \frac{2023}{2} \leq 2022$ Now, isolate $\lambda$: $\lambda \leq 2022 - \frac{2023}{2}$ Find a common denominator for the right side: $\lambda \leq \frac{2 \times 2022}{2} - \frac{2023}{2}$ $\lambda \leq \frac{4044 - 2023}{2}$ $\lambda \leq \frac{2021}{2}$ $\lambda \leq 1010.5$

• Why this step is taken: This step establishes the upper bound for the possible values of $\lambda$.

Step 5: Determine the largest natural number $\lambda$. The inequality $\lambda \leq 1010.5$ tells us that $\lambda$ must be less than or equal to $1010.5$. The problem asks for the largest natural number $\lambda$. Natural numbers are positive integers ($1, 2, 3, \dots$). The largest natural number that satisfies $\lambda \leq 1010.5$ is $1010$.

• Why this step is taken: This step applies the constraint that $\lambda$ must be a natural number to find the specific answer.

Common Mistakes & Tips

• Misinterpreting the Functional Equation: While $f(x)=cx$ is the solution for real numbers, for natural numbers, it can be derived directly from $f(1)=1$ and the additive property.
• Forgetting the Domain: Ensure that the final answer for $\lambda$ is a natural number, as specified in the question.
• Algebraic Errors in Inequality: Double-check the arithmetic when solving the inequality, especially when dealing with fractions.

Summary

The problem requires understanding a specific functional equation, $f(m+n) = f(m) + f(n)$ with $f(1)=1$, which leads to the function $f(x)=x$. This function is then substituted into a summation, which is evaluated using the formula for the sum of the first $n$ natural numbers. The resulting expression is used to form an inequality in terms of $\lambda$. Solving this inequality gives an upper bound for $\lambda$, and the largest natural number satisfying this bound is identified as the final answer.

The final answer is \boxed{1010}.