[/quote]
,
,![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](http://latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
be triangle, inscribed in parabola. Tangents in points 
forms triangle 
.
)
such that the equality 
holds for all pairs of real numbers 
.
is an obvious solution. Now, let 
.
or 
.
.
,
:
,
be an integer. Find all sequences 
satisfying the following conditions:![\[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 + n;
\]](http://latex.artofproblemsolving.com/3/c/5/3c509ec2e9013e8d3be492c8eb44a7c33841b74e.png)
![\[ \text{ (b) } a_{i + 1} + a_{i + 2} + \ldots + a_{i + n} < a_{i + n + 1} + a_{i + n + 2} + \ldots + a_{i + 2n} \text{ for all } 0 \leq i \leq n^2 - n.
\]](http://latex.artofproblemsolving.com/9/7/d/97d2a467d1c0dc8594ec024c3bb9b8c87ee85b19.png)
.
and let 
.
be the foot of the altitude from 
and let 
be the centroid of the triangle 
be a circle through 
.
and 
are collinear.
![$P\left(\sqrt[3]5+\sqrt[3]{25}\right)=5+\sqrt[3]5$](http://latex.artofproblemsolving.com/5/8/e/58ec2a921cd988bd5ae017a8653544ddd758225f.png)
.
-
.
,
in 
such that the sets ![\[ A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100 \]](http://latex.artofproblemsolving.com/d/5/c/d5c2a7b943e8c7200361698f3eb6f83700ad68d6.png)
are pairwise disjoint.
such that for every index 
,
,
is a multiple of 
.
and 
be the points in which the median and the angle bisector, respectively, at 
and 
meets 
and 
,
the point in which the perpendicular at 
produced.
is perpendicular to 
and 
such that for all 
if 
then 
Find the smallest of sum of all elements of 
such that there exist positive integers 
for which the following holds 
for all 
.
Y by
Let 
be a point inside the triangle 
. 
intersects 
at 
. Points 
and 
are defined similarly. Line 
intersects 
at points 
, 
. The tangents at these points to intersect BC at points 
, 
. Points 
, 
, 
, 
are defined similarly. Prove that points , , , , , lie on one conic
be a point inside the triangle 
. 
intersects 
at 
. Points 
and 
are defined similarly. Line 
intersects 
at points 
, 
. The tangents at these points to intersect BC at points 
, 
. Points 
, 
, 
, 
are defined similarly. Prove that points , , , , , lie on one conicThis post has been edited 3 times. Last edited by moony_, Apr 11, 2025, 9:44 AM
