such that there exists a unique integer 
such that 
with the following property:![\[ n!\mid a^n + 1
\]](http://latex.artofproblemsolving.com/e/2/3/e23d39aab685888993d355d8c93d4e6475f53487.png)
(use in a problem for junior) to practice. Can anyone share to me?
side length and with center 
.
is drawn where point 
lies on line 
.
Y by Mango247, Mango247, Mango247
In a convex quadrilateral 
, the diagonals 
and 
are perpendicular, with 
, where 
and 
are the midpoints of sides 
and 
, respectively. Find the length of the diagonal and the angle between straight lines and 
.
, the diagonals 
and 
are perpendicular, with 
, where 
and 
are the midpoints of sides 
and 
, respectively. Find the length of the diagonal and the angle between straight lines and 
.![[asy]
real a=15;
real b=-26;
real c=-20;
real d=51;
pair A=(0,a); pair B=(b,0); pair C=(0,c); pair D=(d,0);
draw(A--B--C--D--A); draw(A--C, blue); draw(B--D, blue);
pair K=(A+B)/2; pair L=(C+D)/2; pair M=(B+C)/2; pair N=(A+D)/2;
draw(K--M--L--N--K, dashed);
draw(K--L, magenta);
label("$A$", A, dir(90));
label("$B$", B, W);
label("$C$", C, S);
label("$D$", D, E);
label("$K$", K, dir(135));
label("$L$", L, dir(-45));
label("$M$", M, dir(-135));
label("$N$", N, dir(45));
[/asy]](http://latex.artofproblemsolving.com/1/f/d/1fd1b57ec8b8a5ceb6aa458a7eb9f8b25051cae5.png)
and 
be the midpoints of 
and 
,
is the Varignon parallelogram of quadrilateral 
,
.
.
be the acute angle between 
and 
.
,
.
,
