{"status": "success", "data": {"description_md": "Isosceles $\\triangle ABC$ has a right angle at $C$. Point $P$ is inside $\\triangle ABC$, such that $PA=11$, $PB=7$, and $PC=6$. Legs $\\overline{AC}$ and $\\overline{BC}$ have length $s=\\sqrt{a+b\\sqrt{2}}$, where $a$ and $b$ are positive integers. What is $a+b$?
[asy] pathpen = linewidth(0.7); pen f = fontsize(10); size(5cm); pair B = (0,sqrt(85+42*sqrt(2))); pair A = (B.y,0); pair C = (0,0); pair P = IP(arc(B,7,180,360),arc(C,6,0,90)); D(A--B--C--cycle); D(P--A); D(P--B); D(P--C); MP(\"A\",D(A),plain.E,f); MP(\"B\",D(B),plain.N,f); MP(\"C\",D(C),plain.SW,f); MP(\"P\",D(P),plain.NE,f); [/asy]
\n\n$\\mathrm{(A)}\\ 85
\\qquad
\\mathrm{(B)}\\ 91
\\qquad
\\mathrm{(C)}\\ 108
\\qquad
\\mathrm{(D)}\\ 121
\\qquad
\\mathrm{(E)}\\ 127$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "

Isosceles \\triangle ABC has a right angle at C . Point P is inside \\triangle ABC , such that PA=11 , PB=7 , and PC=6 . Legs \\overline{AC} and \\overline{BC} have length s=\\sqrt{a+b\\sqrt{2}} , where a and b are positive integers. What is a+b ?

\"[asy]

\\mathrm{(A)}\\ 85\\qquad\\mathrm{(B)}\\ 91\\qquad\\mathrm{(C)}\\ 108\\qquad\\mathrm{(D)}\\ 121\\qquad\\mathrm{(E)}\\ 127

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2006 AMC 12B Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/06_amc12B_p24", "prev": "/problem/06_amc12B_p22"}}