AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
, then prove that the following inequality holds:![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](http://latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
be triangle, inscribed in parabola. Tangents in points 
forms triangle 
. Prove that 
. (
is area of triangle 
).
such that the equality 
holds for all pairs of real numbers 
.
, 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 
.
such that: 
. Find the maximum: 
such that: . Find the maximum: 
such that: . Find the maximum: 
so we only need to study the inequality for negative numbers and the function is convex for 
.
and the maximum value is equal to 
.
so we only need to study the inequality for negative numbers and the function is convex for .
, we get:
Therefore, it is enough to study the function for 
. But the function 
is convex for 
with convex constraints, so it is enough to check the borders where we get a maximum at equal to .
, we get:Therefore, it is enough to study the function for . But the function is convex for with convex constraints, so it is enough to check the borders where we get a maximum at equal to .
we obtain a greater value. Why don't we get a greater value with a condition 
?
we obtain a greater value. Why don't we get a greater value with a condition ? as it does not verify the condition !!!! I know you are a very smart guy 
, but not sure what do you mean with your suggestion.
be real numbers such as 
and . Prove that:
we obtain a greater value. Why don't we get a greater value with a condition ? as it does not verify the condition !!!! is convex on ![$[-2,0]$](http://latex.artofproblemsolving.com/5/5/a/55a2deada78acc1831c348210e10b1e6a6ede5a9.png)
and without condition you can get a maximal value by checking 
.? If for any you'll obtain that the condition indeed occurs, so your reasoning is true. Otherwise, your reasoning is total wrong, I think.
is convex on and without condition you can get a maximal value by checking .? If for any you'll obtain that the condition indeed occurs, so your reasoning is true. Otherwise, your reasoning is total wrong, I think. is essential to finding not just the border conditions. Check my example:
and 
.. But the function is convex for with convex constraints, so it is enough to check the borders where we get a maximum at equal to .
is an obvious solution. Now, let 
.
or 
.
.
Prove that