{"status": "success", "data": {"description_md": "Two numbers $a$ and $b$ with $0 \\le a \\le 1$ and $0 \\le b \\le 1$ are chosen at random. The number $c$ is defined by $c = 2a + 2b$. The numbers $a, b$ and $c$ are each rounded to the nearest integer to give $A, B$ and $C$, respectively. (For example, if $a = 0.432$ and $b = 0.5$, then $c = 1.864$, and so $A = 0$, $B = 1$, and $C = 2$.) What is the probability that $2A + 2B = C$?\n\n$\\text{(A)}\\ \\frac{15}{32}\\qquad\\text{(B)}\\ \\frac{3}{8}\\qquad\\text{(C)}\\ \\frac{1}{2}\\qquad\\text{(D)}\\ \\frac{7}{16}\\qquad\\text{(E)}\\ \\frac{3}{4}$\n\n___\n\n**[Cayley](https://www.cemc.uwaterloo.ca/contests/past_contests.html#pcf), by [CEMC](https://cemc.uwaterloo.ca/), is licensed under the [CC BY-NC 3.0 DEED](https://creativecommons.org/licenses/by-nc/3.0/). They are not affiliated with or endorse TopsOJ in any way.**\n", "description_html": "<p>Two numbers <span class=\"katex--inline\">a</span> and <span class=\"katex--inline\">b</span> with <span class=\"katex--inline\">0 \\le a \\le 1</span> and <span class=\"katex--inline\">0 \\le b \\le 1</span> are chosen at random. The number <span class=\"katex--inline\">c</span> is defined by <span class=\"katex--inline\">c = 2a + 2b</span>. The numbers <span class=\"katex--inline\">a, b</span> and <span class=\"katex--inline\">c</span> are each rounded to the nearest integer to give <span class=\"katex--inline\">A, B</span> and <span class=\"katex--inline\">C</span>, respectively. (For example, if <span class=\"katex--inline\">a = 0.432</span> and <span class=\"katex--inline\">b = 0.5</span>, then <span class=\"katex--inline\">c = 1.864</span>, and so <span class=\"katex--inline\">A = 0</span>, <span class=\"katex--inline\">B = 1</span>, and <span class=\"katex--inline\">C = 2</span>.) What is the probability that <span class=\"katex--inline\">2A + 2B = C</span>?</p>\n<p><span class=\"katex--inline\">\\text{(A)}\\ \\frac{15}{32}\\qquad\\text{(B)}\\ \\frac{3}{8}\\qquad\\text{(C)}\\ \\frac{1}{2}\\qquad\\text{(D)}\\ \\frac{7}{16}\\qquad\\text{(E)}\\ \\frac{3}{4}</span></p>\n<hr>\n<p><strong><a href=\"https://www.cemc.uwaterloo.ca/contests/past_contests.html#pcf\">Cayley</a>, by <a href=\"https://cemc.uwaterloo.ca/\">CEMC</a>, is licensed under the <a href=\"https://creativecommons.org/licenses/by-nc/3.0/\">CC BY-NC 3.0 DEED</a>. They are not affiliated with or endorse TopsOJ in any way.</strong></p>\n", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2011 Cayley Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/11_cayley_p24"}}