{"status": "success", "data": {"description_md": "Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points.
[asy] size(350); defaultpen(linewidth(0.8)+fontsize(11)); real theta = aTan(1.25/2); pair A = 2.5*dir(180+theta), B = (3.35,0), C = -A, D = -B, P = foot(A,B,D), Q = -P, R = foot(B,A,C), S = -R; draw(A--B--C--D--A^^B--D^^R--S^^rightanglemark(A,P,D,6)^^rightanglemark(C,Q,D,6)); draw(B--R^^C--Q^^A--P^^D--S,linetype(\"4 4\")); dot(\"$A$\",A,dir(270)); dot(\"$B$\",B,E); dot(\"$C$\",C,N); dot(\"$D$\",D,W); dot(\"$P$\",P,SE); dot(\"$Q$\",Q,NE); dot(\"$R$\",R,N); dot(\"$S$\",S,dir(270)); [/asy]

Suppose $PQ=6$ and $RS=8,$ and let $d$ denote the length of $\\overline{BD},$ the longer diagonal of $ABCD.$ Then $d^2$ can be written in the form $m+n\\sqrt p,$ where $m,n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m+n+p?$\n\n$\\textbf{(A) }81 \\qquad \\textbf{(B) }89 \\qquad \\textbf{(C) }97\\qquad \\textbf{(D) }105 \\qquad \\textbf{(E) }113$\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": "

Let ABCD be a parallelogram with area 15 . Points P and Q are the projections of A and C, respectively, onto the line BD; and points R and S are the projections of B and D, respectively, onto the line AC. See the figure, which also shows the relative locations of these points.

\"[asy]

Suppose PQ=6 and RS=8, and let d denote the length of \\overline{BD}, the longer diagonal of ABCD. Then d^2 can be written in the form m+n\\sqrt p, where m,n, and p are positive integers and p is not divisible by the square of any prime. What is m+n+p?

\\textbf{(A) }81 \\qquad \\textbf{(B) }89 \\qquad \\textbf{(C) }97\\qquad \\textbf{(D) }105 \\qquad \\textbf{(E) }113

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": "2021 AMC 12B Problem 24", "can_next": true, "can_prev": true, "nxt": "/problem/21_amc12B_p25", "prev": "/problem/21_amc12B_p23"}}