{"status": "success", "data": {"description_md": "A teacher was leading a class of four perfectly logical students. The teacher chose a set $S$ of four integers and gave a different number in $S$ to each student. Then the teacher announced to the class that the numbers in $S$ were four consecutive two-digit positive integers, that some number in $S$ was divisible by $6$, and a different number in $S$ was divisible by $7$. The teacher then asked if any of the students could deduce what $S$ is, but in unison, all of the students replied no.<br><br>However, upon hearing that all four students replied no, each student was able to determine the elements of $S$. Find the sum of all possible values of the greatest element of $S$.\n___\nLeading zeroes must be inputted, so if your answer is `34`, then input `034`. Full 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>A teacher was leading a class of four perfectly logical students. The teacher chose a set <span class=\"katex--inline\">S</span> of four integers and gave a different number in <span class=\"katex--inline\">S</span> to each student. Then the teacher announced to the class that the numbers in <span class=\"katex--inline\">S</span> were four consecutive two-digit positive integers, that some number in <span class=\"katex--inline\">S</span> was divisible by <span class=\"katex--inline\">6</span>, and a different number in <span class=\"katex--inline\">S</span> was divisible by <span class=\"katex--inline\">7</span>. The teacher then asked if any of the students could deduce what <span class=\"katex--inline\">S</span> is, but in unison, all of the students replied no.<br/><br/>However, upon hearing that all four students replied no, each student was able to determine the elements of <span class=\"katex--inline\">S</span>. Find the sum of all possible values of the greatest element of <span class=\"katex--inline\">S</span>.</p>&#10;<hr><p>Leading zeroes must be inputted, so if your answer is <code>34</code>, then input <code>034</code>. 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": "2021 AIME II Problem 11", "can_next": true, "can_prev": true, "nxt": "/problem/21_aime_II_p12", "prev": "/problem/21_aime_II_p10"}}