AoPS Academy
be triangle, inscribed in parabola. Tangents in points 
forms triangle 
. Prove that 
. (
is area of triangle 
).
, in which 
: 
, but the proof is important.
+2 w
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)
be an acute triangle with circumcircle 
. Let 
be the midpoint of 
and let 
be the midpoint of 
. Let 
be the foot of the altitude from 
and let 
be the centroid of the triangle . Let 
be a circle through and that is tangent to the circle at a point 
. Prove that the points 
and 
are collinear.
with integer coefficients such that ![$P\left(\sqrt[3]5+\sqrt[3]{25}\right)=5+\sqrt[3]5$](http://latex.artofproblemsolving.com/5/8/e/58ec2a921cd988bd5ae017a8653544ddd758225f.png)
.
be a 
-element subset of the set 
. Prove that there exist numbers 
, 
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.
positive integers 
such that for every index 
, with 
, 
is a multiple of 
. Determine the greatest possible value of the largest of the numbers.
be a triangle. Let 
and 
be the points in which the median and the angle bisector, respectively, at meet the side 
. Let 
and be the points in which the perpendicular at to 
meets 
and 
, respectively. And 
the point in which the perpendicular at to meets 
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 
.
be an equilateral triangle, be a point inside it, and 
be the intersections of 
with the sides of . If 
are the midpoints of , 
, , show that there is a triangle with sides 
, 
and 
.
be an equilateral triangle, be a point inside it, and be the intersections of with the sides of . If are the midpoints of , , , show that there is a triangle with sides , and .
to be on the three medians of the triangle.
. By Ceva's theorem, if 
and 
, then 
(
, 
, 
). Let the side length be 
, then![\[A'A''=\left|BA'-BA''\right|=\left|\frac b{b+1}-\frac12\right|=\frac{|b-1|}{2(b+1)}.\]](http://latex.artofproblemsolving.com/f/6/3/f639ba63d9b8b789471e9073da82530d4382d2bc.png)
Similarly we obtain 
and 
. Multiplying by 
, we need to show that![\[\frac{|a-1|}{a+1}+\frac{|b-1|}{b+1}=\frac{(a+1)|b-1|+(b+1)|a-1|}{(a+1)(b+1)}>\frac{|a-b|}{a+b}.\tag{*}\]](http://latex.artofproblemsolving.com/b/5/8/b580ae87663620b5fc0f193a4773ebb952c1a37e.png)
On the one hand,
where
On the other hand,
where![\[(2)\Leftrightarrow2(a+b)>(a+1)(b+1)\Leftrightarrow(a-1)(b-1)<0.\]](http://latex.artofproblemsolving.com/2/5/7/25716af9dc6ff71bcb238991b68fe4e78108c64b.png)
At least one of 
and 
holds, so we are done.
?
