{"status": "success", "data": {"description_md": "On Halloween $31$ children walked into the principal's office asking for candy. They\ncan be classified into three types: Some always lie; some always tell the truth; and\nsome alternately lie and tell the truth. The alternaters arbitrarily choose their first\nresponse, either a lie or the truth, but each subsequent statement has the opposite\ntruth value from its predecessor. The principal asked everyone the same three\nquestions in this order.\n\"Are you a truth-teller?\" The principal gave a piece of candy to each of the $22$\nchildren who answered yes.\n\"Are you an alternater?\" The principal gave a piece of candy to each of the $15$\nchildren who answered yes.\n\"Are you a liar?\" The principal gave a piece of candy to each of the $9$ children who\nanswered yes.\nHow many pieces of candy in all did the principal give to the children who always\ntell the truth?\n\n$\\textbf{(A) } 7 \\qquad \\textbf{(B) } 12 \\qquad \\textbf{(C) } 21 \\qquad \\textbf{(D) } 27 \\qquad \\textbf{(E) } 31$", "description_html": "<p>On Halloween  <span class=\"katex--inline\">31</span>  children walked into the principal&#8217;s office asking for candy. They<br/>\ncan be classified into three types: Some always lie; some always tell the truth; and<br/>\nsome alternately lie and tell the truth. The alternaters arbitrarily choose their first<br/>\nresponse, either a lie or the truth, but each subsequent statement has the opposite<br/>\ntruth value from its predecessor. The principal asked everyone the same three<br/>\nquestions in this order.<br/>\n&#8220;Are you a truth-teller?&#8221; The principal gave a piece of candy to each of the  <span class=\"katex--inline\">22</span> <br/>\nchildren who answered yes.<br/>\n&#8220;Are you an alternater?&#8221; The principal gave a piece of candy to each of the  <span class=\"katex--inline\">15</span> <br/>\nchildren who answered yes.<br/>\n&#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<br/>\nanswered yes.<br/>\nHow many pieces of candy in all did the principal give to the children who always<br/>\ntell the truth?</p>\n<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>\n<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": "2022 AMC 10A Problem 12", "can_next": true, "can_prev": true, "nxt": "/problem/22_amc10A_p13", "prev": "/problem/22_amc10A_p11"}}