{"status": "success", "data": {"description_md": "Let $S$ be the set of ordered triples $(x,y,z)$ of real numbers for which\n\n$$\\log_{10}(x+y) = z \\text{ and } \\log_{10}(x^{2}+y^{2}) = z+1.$$<br>There are real numbers $a$ and $b$ such that for all ordered triples $(x,y,z)$ in $S$ we have $x^{3}+y^{3}=a \\cdot 10^{3z} + b \\cdot 10^{2z}.$ What is the value of $a+b?$\n\n$\\textbf{(A)}\\ \\frac {15}{2} \\qquad <br>\\textbf{(B)}\\ \\frac {29}{2} \\qquad <br>\\textbf{(C)}\\ 15 \\qquad <br>\\textbf{(D)}\\ \\frac {39}{2} \\qquad <br>\\textbf{(E)}\\ 24$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>Let  <span class=\"katex--inline\">S</span>  be the set of ordered triples  <span class=\"katex--inline\">(x,y,z)</span>  of real numbers for which</p>&#10;<p> <span class=\"katex--display\">\\log_{10}(x+y) = z \\text{ and } \\log_{10}(x^{2}+y^{2}) = z+1.</span> <br/>There are real numbers  <span class=\"katex--inline\">a</span>  and  <span class=\"katex--inline\">b</span>  such that for all ordered triples  <span class=\"katex--inline\">(x,y,z)</span>  in  <span class=\"katex--inline\">S</span>  we have  <span class=\"katex--inline\">x^{3}+y^{3}=a \\cdot 10^{3z} + b \\cdot 10^{2z}.</span>  What is the value of  <span class=\"katex--inline\">a+b?</span> </p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ \\frac {15}{2} \\qquad \\textbf{(B)}\\ \\frac {29}{2} \\qquad \\textbf{(C)}\\ 15 \\qquad \\textbf{(D)}\\ \\frac {39}{2} \\qquad \\textbf{(E)}\\ 24</span> </p>&#10;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2005 AMC 12B Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/05_amc12B_p24", "prev": "/problem/05_amc12B_p22"}}