Found problems: 3
2005 Korea - Final Round, 5
Find all positive integers $m$ and $n$ such that both $3^{m}+1$ and $3^{n}+1$ are divisible by $mn$.
1971 AMC 12/AHSME, 31
[asy]
size(2.5inch);
pair A = (-2,0), B = 2dir(150), D = (2,0), C;
draw(A..(0,2)..D--cycle);
C = intersectionpoint(A..(0,2)..D,Circle(B,arclength(A--B)));
draw(A--B--C--D--cycle);
label("$A$",A,W);
label("$B$",B,NW);
label("$C$",C,N);
label("$D$",D,E);
label("$4$",A--D,S);
label("$1$",A--B,E);
label("$1$",B--C,SE);
//Credit to chezbgone2 for the diagram[/asy]
Quadrilateral $ABCD$ is inscribed in a circle with side $AD$, a diameter of length $4$. If sides $AB$ and $BC$ each have length $1$, then side $CD$ has length
$\textbf{(A) }\frac{7}{2}\qquad\textbf{(B) }\frac{5\sqrt{2}}{2}\qquad\textbf{(C) }\sqrt{11}\qquad\textbf{(D) }\sqrt{13}\qquad \textbf{(E) }2\sqrt{3}$
2010 AIME Problems, 14
In right triangle $ ABC$ with right angle at $ C$, $ \angle BAC < 45$ degrees and $ AB \equal{} 4$. Point $ P$ on $ AB$ is chosen such that $ \angle APC \equal{} 2\angle ACP$ and $ CP \equal{} 1$. The ratio $ \frac{AP}{BP}$ can be represented in the form $ p \plus{} q\sqrt{r}$, where $ p,q,r$ are positive integers and $ r$ is not divisible by the square of any prime. Find $ p\plus{}q\plus{}r$.