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, 
, and 
. Points 
and 
are located on 
and 
respectively, such that 
and 
. Find the ratio of 
to 
.
be the circumcircle of a triangle 
. Denote by 
and 
the midpoints of the sides 
and 
, respectively, and denote by 
the midpoint of the arc 
of not containing 
. The circumcircles of the triangles 
and 
intersect the perpendicular bisectors of and at points 
and 
, respectively; assume that and lie inside the triangle . The lines 
and 
intersect at 
. Prove that 
.
which satisfy the following relationship for all real numbers 
and 
![\[f(xf(y)-f(x))=2f(x)+xy.\]](http://latex.artofproblemsolving.com/d/0/e/d0e9e56267aa6241a0323c7b2245e4752c338785.png)
and 
. Prove that
Equality holds when 
Equality holds when 
Equality holds when 
such that:
.
, solve the functional equation 
.
and for all 
, 
Determine the smallest 
such that 
.
, so 
. Summing from 
to 
, we get![\[n!a_n=1!a_1+\sum_{k=1}^{n-1}k!\implies a_n=\frac{2025+\sum\limits_{k=1}^{n-1}k!}{n!}.\]](http://latex.artofproblemsolving.com/b/d/9/bd99b2f350454dbab326def02e1bfde29c059bec.png)
Clearly 
, 
. For 
we have![\[a_n>\frac{(n-1)!+(n-2)!}{n!}=\frac1n+\frac1{(n-1)n}=\frac1{n-1}\ge\frac1{2025},\]](http://latex.artofproblemsolving.com/7/5/2/752e750aa744d1cf297f7df5489b22eaeed1f84b.png)
so it does not work.
, we have![\[\sum_{k=1}^{n-3}k!<(n-2)!\sum_{k=1}^{n-3}\frac1{2^{n-2-k}}<(n-2)!,\]](http://latex.artofproblemsolving.com/9/d/7/9d7a50882ea2f0901d4884de88202b2d8a676140.png)
so
So the answer is 
.
So we want to find the smallest such that 
. Obviously 
. Then the rest is just routine checking (i included it in the script) to get 
.
: is the smallest number such that 
,
: is the smallest number such that ,![\[
\begin{aligned}
&\frac{1}{2025} < \frac{1}{2026} \\[10pt]
&\frac{a_{k-1} + 1}{k} > \frac{\frac{1}{2025} + 1}{k} > \frac{\frac{1}{2026} + 1}{k} = \frac{2027}{2026k} \\[10pt]
&\frac{2027}{2026k} < \frac{1}{2025} \\[10pt]
&2025 \cdot \frac{2027}{2026k} = 2,025.999 < k \\[10pt]
&k = 2026.
\end{aligned}
\]](http://latex.artofproblemsolving.com/9/a/7/9a7b6ad626ab3b6c69b7c7827befd34df0c2b63d.png)
and 
.
and 
Prove that
