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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>

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