{"status": "success", "data": {"description_md": "Select numbers $a$ and $b$ between $0$ and $1$ independently and at random, and let $c$ be their sum. Let $A, B$ and $C$ be the results when $a, b$ and $c$, respectively, are rounded to the nearest integer. What is the probability that $A + B = C$?\n\n$\\text {(A) } \\frac14 \\qquad \\text {(B) } \\frac13 \\qquad \\text {(C) } \\frac12 \\qquad \\text {(D) } \\frac23 \\qquad \\text {(E) }\\frac34$\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>Select numbers  <span class=\"katex--inline\">a</span>  and  <span class=\"katex--inline\">b</span>  between  <span class=\"katex--inline\">0</span>  and  <span class=\"katex--inline\">1</span>  independently and at random, and let  <span class=\"katex--inline\">c</span>  be their sum. Let  <span class=\"katex--inline\">A, B</span>  and  <span class=\"katex--inline\">C</span>  be the results when  <span class=\"katex--inline\">a, b</span>  and  <span class=\"katex--inline\">c</span> , respectively, are rounded to the nearest integer. What is the probability that  <span class=\"katex--inline\">A + B = C</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\text {(A) } \\frac14 \\qquad \\text {(B) } \\frac13 \\qquad \\text {(C) } \\frac12 \\qquad \\text {(D) } \\frac23 \\qquad \\text {(E) }\\frac34</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": 3, "problem_name": "2004 AMC 12A Problem 20", "can_next": true, "can_prev": true, "nxt": "/problem/04_amc12A_p21", "prev": "/problem/04_amc12A_p19"}}