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<p>To solve the problem, we need to evaluate the sum <span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.519em; padding-right: 0.06em;">f</span></span><span class="mjx-mrow MJXc-space1"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.256em; padding-bottom: 1.256em;">(</span></span><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.2em; top: -1.368em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span><span class="mjx-denominator" style="width: 2.2em; bottom: -0.778em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2023</span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 2.2em;" class="mjx-line"></span></span><span style="height: 2.146em; vertical-align: -0.778em;" class="mjx-vsize"></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.256em; padding-bottom: 1.256em;">)</span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.519em; padding-right: 0.06em;">f</span></span><span class="mjx-mrow MJXc-space1"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.256em; padding-bottom: 1.256em;">(</span></span><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.2em; top: -1.368em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span><span class="mjx-denominator" style="width: 2.2em; bottom: -0.778em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2023</span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 2.2em;" class="mjx-line"></span></span><span style="height: 2.146em; vertical-align: -0.778em;" class="mjx-vsize"></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.256em; padding-bottom: 1.256em;">)</span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.144em; padding-bottom: 0.372em;">…</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.519em; padding-right: 0.06em;">f</span></span><span class="mjx-mrow MJXc-space1"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.256em; padding-bottom: 1.256em;">(</span></span><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.2em; top: -1.368em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2022</span></span></span><span class="mjx-denominator" style="width: 2.2em; bottom: -0.778em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2023</span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 2.2em;" class="mjx-line"></span></span><span style="height: 2.146em; vertical-align: -0.778em;" class="mjx-vsize"></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.256em; padding-bottom: 1.256em;">)</span></span></span></span></span></span> where <span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.519em; padding-right: 0.06em;">f</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 3.216em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.216em; top: -1.583em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span></span></span></span><span class="mjx-denominator" style="width: 3.216em; bottom: -1.054em;"><span class="mjx-mrow"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 3.216em;" class="mjx-line"></span></span><span style="height: 2.637em; vertical-align: -1.054em;" class="mjx-vsize"></span></span></span></span></span>.</p><p><strong>Step 1: Simplify the function \( f(x) \)</strong></p><p>We start with the function: \( f(x) = \frac{2^{2x}}{2^{2x} + 2} \)</p><p>We can rewrite \( 2^{2x} \) as \( 4^x \): \( f(x) = \frac{4^x}{4^x + 2} \)</p><p><strong>Step 2: Find \( f(1-x) \)</strong></p><p>Next, we will find \( f(1-x) \): \( f(1-x) = \frac{4^{1-x}}{4^{1-x} + 2} = \frac{\frac{4}{4^x}}{\frac{4}{4^x} + 2} = \frac{4}{4 + 2 \cdot 4^x} \)</p><p><strong>Step 3: Compute \( f(x) + f(1-x) \)</strong></p><p>Now we will compute \( f(x) + f(1-x) \): \( f(x) + f(1-x) = \frac{4^x}{4^x + 2} + \frac{4}{4 + 2 \cdot 4^x} \)</p><p>To combine these fractions, we find a common denominator: \( = \frac{4^x(4 + 2 \cdot 4^x) + 4(4^x + 2)}{(4^x + 2)(4 + 2 \cdot 4^x)} \)</p><p><strong>Step 4: Simplify the numerator</strong></p><p>Expanding the numerator: \( = \frac{4^{x+1} + 2 \cdot 4^{2x} + 4^{x+1} + 8}{(4^x + 2)(4 + 2 \cdot 4^x)} \) \( = \frac{2 \cdot 4^{x+1} + 2 \cdot 4^{2x} + 8}{(4^x + 2)(4 + 2 \cdot 4^x)} \) Factoring out 2: \( = \frac{2(4^{x+1} + 4^{2x} + 4)}{(4^x + 2)(4 + 2 \cdot 4^x)} \)</p><p><strong>Step 5: Evaluate \( f(x) + f(1-x) \)</strong></p><p>Notice that \( f(x) + f(1-x) = 1 \): \( f(x) + f(1-x) = 1 \)</p><p><strong>Step 6: Sum \( f\left(\frac{k}{2023}\right) \)</strong></p><p>Now, we can compute the sum: \( f\left(\frac{1}{2023}\right) + f\left(\frac{2}{2023}\right) + \ldots + f\left(\frac{2022}{2023}\right) \)</p><p>We can pair the terms: \( f\left(\frac{k}{2023}\right) + f\left(\frac{2023-k}{2023}\right) = 1 \)</p><p>Since there are 2022 terms, we can pair them up: \( \text{Number of pairs} = \frac{2022}{2} = 1011 \)</p><p><strong>Step 7: Final Result</strong></p><p>Thus, the total sum is: \( 1011 \cdot 1 = 1011 \)</p><p><strong>Conclusion</strong></p><p>The final answer is: \( \boxed{1011} \)</p>

In this Spanish name, the first or paternal surname is Wallace and the second or maternal family name is del Barrio.

In this Spanish name, the first or paternal surname is Wallace and the second or maternal family name is del Barrio.