{"status": "success", "data": {"description_md": "Cities $A$, $B$, $C$, $D$, $W$, and $L$ are located in the country of Portaland. A one-way magic portal exists in each of the cities $A$, $B$, $C$, and $D$. A portal code shows a portal's destinations and their corresponding probability. The format of portal codes is Portal $M$: ($N$, $P$, $Q$), which represents that by taking the portal in city $M$, the probability of travelling to cities $N$, $P$, $Q$ are $\\frac{1}{2}$, $\\frac{1}{3}$, and $\\frac{1}{6}$, respectively. The portal codes of the four portals are as follows:\n\nPortal $A:\\;$ ($B$, $C$, $D$)\nPortal $B:\\;$ ($C$, $W$, $L$)\nPortal $C:\\;$ ($D$, $W$, $L$)\nPortal $D:\\;$ ($A$, $W$, $L$)\n\nIf Mike starts in city $A$ and continually takes portals until he reaches a city without a portal, the probability that he ends up in city $W$ can be expressed as $\\frac{x}{y}$, where $x$ and $y$ are relatively prime positive integers. Compute $x+y$.  ", "description_html": "<p>Cities <span class=\"katex--inline\">A</span>, <span class=\"katex--inline\">B</span>, <span class=\"katex--inline\">C</span>, <span class=\"katex--inline\">D</span>, <span class=\"katex--inline\">W</span>, and <span class=\"katex--inline\">L</span> are located in the country of Portaland. A one-way magic portal exists in each of the cities <span class=\"katex--inline\">A</span>, <span class=\"katex--inline\">B</span>, <span class=\"katex--inline\">C</span>, and <span class=\"katex--inline\">D</span>. A portal code shows a portal&#8217;s destinations and their corresponding probability. The format of portal codes is Portal <span class=\"katex--inline\">M</span>: (<span class=\"katex--inline\">N</span>, <span class=\"katex--inline\">P</span>, <span class=\"katex--inline\">Q</span>), which represents that by taking the portal in city <span class=\"katex--inline\">M</span>, the probability of travelling to cities <span class=\"katex--inline\">N</span>, <span class=\"katex--inline\">P</span>, <span class=\"katex--inline\">Q</span> are <span class=\"katex--inline\">\\frac{1}{2}</span>, <span class=\"katex--inline\">\\frac{1}{3}</span>, and <span class=\"katex--inline\">\\frac{1}{6}</span>, respectively. The portal codes of the four portals are as follows:</p>&#10;<p>Portal <span class=\"katex--inline\">A:\\;</span> (<span class=\"katex--inline\">B</span>, <span class=\"katex--inline\">C</span>, <span class=\"katex--inline\">D</span>)<br/>&#10;Portal <span class=\"katex--inline\">B:\\;</span> (<span class=\"katex--inline\">C</span>, <span class=\"katex--inline\">W</span>, <span class=\"katex--inline\">L</span>)<br/>&#10;Portal <span class=\"katex--inline\">C:\\;</span> (<span class=\"katex--inline\">D</span>, <span class=\"katex--inline\">W</span>, <span class=\"katex--inline\">L</span>)<br/>&#10;Portal <span class=\"katex--inline\">D:\\;</span> (<span class=\"katex--inline\">A</span>, <span class=\"katex--inline\">W</span>, <span class=\"katex--inline\">L</span>)</p>&#10;<p>If Mike starts in city <span class=\"katex--inline\">A</span> and continually takes portals until he reaches a city without a portal, the probability that he ends up in city <span class=\"katex--inline\">W</span> can be expressed as <span class=\"katex--inline\">\\frac{x}{y}</span>, where <span class=\"katex--inline\">x</span> and <span class=\"katex--inline\">y</span> are relatively prime positive integers. Compute <span class=\"katex--inline\">x+y</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "TxO Math Bowl 2024 - Team Contest - Problem 16", "can_next": true, "can_prev": true, "nxt": "/problem/txo2024team-p17", "prev": "/problem/txo2024team-p15"}}