{"status": "success", "data": {"description_md": "On Halloween $31$ children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order:\n- \"Are you a truth-teller?\" The principal gave a piece of candy to each of the $22$ children who answered yes.\n- \"Are you an alternater?\" The principal gave a piece of candy to each of the $15$<br>children who answered yes.\n- \"Are you a liar?\" The principal gave a piece of candy to each of the $9$ children who answered yes.\n\nHow many pieces of candy in all did the principal give to the children who always<br>tell the truth?\n\n$\\textbf{(A) } 7 \\qquad \\textbf{(B) } 12 \\qquad \\textbf{(C) } 21 \\qquad \\textbf{(D) } 27 \\qquad \\textbf{(E) } 31$\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>On Halloween <span class=\"katex--inline\">31</span> children walked into the principal&#8217;s office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order:</p>&#10;<ul>&#10;<li>&#8220;Are you a truth-teller?&#8221; The principal gave a piece of candy to each of the <span class=\"katex--inline\">22</span> children who answered yes.</li>&#10;<li>&#8220;Are you an alternater?&#8221; The principal gave a piece of candy to each of the <span class=\"katex--inline\">15</span><br/>children who answered yes.</li>&#10;<li>&#8220;Are you a liar?&#8221; The principal gave a piece of candy to each of the <span class=\"katex--inline\">9</span> children who answered yes.</li>&#10;</ul>&#10;<p>How many pieces of candy in all did the principal give to the children who always<br/>tell the truth?</p>&#10;<p><span class=\"katex--inline\">\\textbf{(A) } 7 \\qquad \\textbf{(B) } 12 \\qquad \\textbf{(C) } 21 \\qquad \\textbf{(D) } 27 \\qquad \\textbf{(E) } 31</span></p>&#10;<hr/>&#10;<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>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 2, "problem_name": "2022 AMC 12A Problem 9", "can_next": true, "can_prev": true, "nxt": "/problem/22_amc12A_p10", "prev": "/problem/22_amc12A_p08"}}