{"status": "success", "data": {"description_md": "For positive integers $a$, $b$, and $c$ with $a < b < c$, consider collections of postage stamps in denominations $a$, $b$, and $c$ cents that contain at least one stamp of each denomination. If there exists such a collection that contains sub-collections worth every whole number of cents up to $1000$ cents, let $f(a, b, c)$ be the minimum number of stamps in such a collection. Find the sum of the three least values of $c$ such that $f(a, b, c) = 97$ for some choice of $a$ and $b$.\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>For positive integers <span class=\"katex--inline\">a</span>, <span class=\"katex--inline\">b</span>, and <span class=\"katex--inline\">c</span> with <span class=\"katex--inline\">a &lt; b &lt; c</span>, consider collections of postage stamps in denominations <span class=\"katex--inline\">a</span>, <span class=\"katex--inline\">b</span>, and <span class=\"katex--inline\">c</span> cents that contain at least one stamp of each denomination. If there exists such a collection that contains sub-collections worth every whole number of cents up to <span class=\"katex--inline\">1000</span> cents, let <span class=\"katex--inline\">f(a, b, c)</span> be the minimum number of stamps in such a collection. Find the sum of the three least values of <span class=\"katex--inline\">c</span> such that <span class=\"katex--inline\">f(a, b, c) = 97</span> for some choice of <span class=\"katex--inline\">a</span> and <span class=\"katex--inline\">b</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": 6, "problem_name": "2022 AIME II Problem 14", "can_next": true, "can_prev": true, "nxt": "/problem/22_aime_II_p15", "prev": "/problem/22_aime_II_p13"}}