{"status": "success", "data": {"description_md": "Munch has 20 gold coins and 20 silver coins placed in a row in some random order. Define a block as a subsequence of any adjacent coins which are the same. For a value of $k$ ($1 \\le k \\le 40$), Munch performs the following operation: He finds the biggest block which contains the $k$th coin from the left, and moves that block to the start. For example, if the row was $\\text{GGGSSGSS}$, and $k = 5$, then the new row becomes $\\text{SSGGGGSS}$.\n\nJerry Yang saw Munch's brilliant idea, and got jealous. Jerry now has has $n$ gold coins and $n$ silver coins placed in a row in some random order. For a value of $l$ ($1 \\le l \\le 2n$), Jerry performs the following operation: He finds the biggest block which contains the $l$th coin from the left, and takes it out. He then reverses the order of the row, and adds the block back at the start. For example, if the row was $\\text{GGGSSGSS}$, and $l = 5$, then the new row becomes $\\text{SSSSGGGG}$.\n\nLet $M$ be the sum of all values of $k$ such that for all initial arrangement of the coins, after some number of operations, Munch will have all the gold coins in 1 block, and all the silver coins in another block. Let $J$ be the sum of all values of $l$ such that for all initial arrangement of the coins, after some number of operations, Jerry will have all the gold coins in 1 block, and all the silver coins in another block. Find the minimum value of $n$ such that $J \\ge M$.", "description_html": "<p>Munch has 20 gold coins and 20 silver coins placed in a row in some random order. Define a block as a subsequence of any adjacent coins which are the same. For a value of <span class=\"katex--inline\">k</span> (<span class=\"katex--inline\">1 \\le k \\le 40</span>), Munch performs the following operation: He finds the biggest block which contains the <span class=\"katex--inline\">k</span>th coin from the left, and moves that block to the start. For example, if the row was <span class=\"katex--inline\">\\text{GGGSSGSS}</span>, and <span class=\"katex--inline\">k = 5</span>, then the new row becomes <span class=\"katex--inline\">\\text{SSGGGGSS}</span>.</p>&#10;<p>Jerry Yang saw Munch&#8217;s brilliant idea, and got jealous. Jerry now has has <span class=\"katex--inline\">n</span> gold coins and <span class=\"katex--inline\">n</span> silver coins placed in a row in some random order. For a value of <span class=\"katex--inline\">l</span> (<span class=\"katex--inline\">1 \\le l \\le 2n</span>), Jerry performs the following operation: He finds the biggest block which contains the <span class=\"katex--inline\">l</span>th coin from the left, and takes it out. He then reverses the order of the row, and adds the block back at the start. For example, if the row was <span class=\"katex--inline\">\\text{GGGSSGSS}</span>, and <span class=\"katex--inline\">l = 5</span>, then the new row becomes <span class=\"katex--inline\">\\text{SSSSGGGG}</span>.</p>&#10;<p>Let <span class=\"katex--inline\">M</span> be the sum of all values of <span class=\"katex--inline\">k</span> such that for all initial arrangement of the coins, after some number of operations, Munch will have all the gold coins in 1 block, and all the silver coins in another block. Let <span class=\"katex--inline\">J</span> be the sum of all values of <span class=\"katex--inline\">l</span> such that for all initial arrangement of the coins, after some number of operations, Jerry will have all the gold coins in 1 block, and all the silver coins in another block. Find the minimum value of <span class=\"katex--inline\">n</span> such that <span class=\"katex--inline\">J \\ge M</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 6, "problem_name": "AMC Practice #1 - Problem 15", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/amc1-p14"}}