AoPS Academy
, 
, and 
Point 
is chosen on the circle with radius , point 
is chosen on the circle with radius , and point 
is chosen on the circle with radius Given that 
, find the maximum and minimum values of 
in terms of , , and
. Each step the pig has probability 
of moving 
unit in the positive direction and probability 
of moving unit in the negative direction. Find the expected number of steps until the yellow pig visits 
.
, 
, 
, and 
. Find 
.
, let 
. Find all convex sets 
such that![\[\mathcal{E}(B \cup C) = B \cup C.\]](http://latex.artofproblemsolving.com/e/c/3/ec34115c9e6e169c19c739220065f647934a1a5f.png)
and ![$f:[a,b]\rightarrow\mathbb{R}$](http://latex.artofproblemsolving.com/6/7/c/67c320754a36aaec178e61f04bfcbac41c0bf675.png)
be a strictly monotonous function such that 
. Show that 
.
for which there exists a scrictly monotonous function 
such that 
, prove that a convex closed set with a nonempty interior that contains exactly one extreme point must contain a ray. (SOLVED), find all convex closed sets with a nonempty interior that contain exactly one extreme point.
, ∀x∈[0,1].
a distance 
from the start. What is the maximum time it could have taken to travel the distance ?
and 
. Prove that
.
, then![\[
a+b+a^2+b^2 \geq 1 \geq 28-6\sqrt{21}.
\]](http://latex.artofproblemsolving.com/6/7/7/677ab9d63defffddb7a8c986c2e2b4ddb4e8267e.png)
, then![\[
\frac{1}{b^2-b+1} \leq \frac{1}{a^2-a+1}, a^2 + a + 1 \geq b^2 + b + 1
\]](http://latex.artofproblemsolving.com/e/8/f/e8fdd29170a984431f76f172d52feb9724f7edfc.png)
which implies that![\[
\frac{a^2+a+1}{a^2-a+1}+\frac{b^2+b+1}{b^2-b+1} \geq \frac{a^2+a+1}{b^2-b+1}+\frac{b^2+b+1}{a^2-a+1} \geq 3.
\]](http://latex.artofproblemsolving.com/8/a/3/8a3d11ba00dc8246113ff3c5881989a12be3f4c9.png)
![\[
3 \geq 6-\left(\frac{a^2+a+1}{a^2-a+1}+\frac{b^2+b+1}{b^2-b+1}\right)
=\frac{2(a-1)^2}{a^2-a+1}+\frac{2(b-1)^2}{b^2-b+1}
\geq \frac{2(1-a+1-b)^2}{a^2-a+1+b^2-b+1}.
\]](http://latex.artofproblemsolving.com/2/2/2/222d49414291c3fd180471f92d561d4d0f69b171.png)
and 
. Then,![\[
3\left(u^2-u+1-2v\right) \geq 2\,(2-u)^2.
\]](http://latex.artofproblemsolving.com/2/a/4/2a4022474a7acca1c57c06b05a98c40e269e5c80.png)
![\[
u^2-4v \geq \frac{u^2-10u+4}{3}.
\]](http://latex.artofproblemsolving.com/c/2/4/c24df229451c119fb2ac4cac1352030302fa6f9a.png)
![\[
a+b+a^2+b^2 = u^2-2v+u = \frac{u^2-4v}{2}+\frac{u^2}{2}+u,
\]](http://latex.artofproblemsolving.com/4/9/4/49466166b20f3199205c898f5de4e53481919827.png)
if 
, then![\[
\frac{u^2-4v}{2}+\frac{u^2}{2}+u \geq \frac{u^2-10u+4}{6}+\frac{u^2}{2}+u
=\frac{2}{3}\,(u^2-u+1).
\]](http://latex.artofproblemsolving.com/d/7/2/d72ef57efa4ba4f14b344a02649dd4c961867fd7.png)
is monotonically decreasing on ![\( u\in \left(0,5-\sqrt{21}\right] \)](http://latex.artofproblemsolving.com/2/b/d/2bde58e2769d4081e0740823e379e1ed883a7ca5.png)
, it follows that![\[
\frac{2}{3}\,(u^2-u+1) \geq \frac{2}{3}\Bigl[(5-\sqrt{21})^2-(5-\sqrt{21})+1\Bigr]
= 28-6\sqrt{21}.
\]](http://latex.artofproblemsolving.com/d/b/c/dbc2c7413768ebff082b089c4c28ffa2197d6a03.png)
, then since 
, we have![\[
\frac{u^2-4v}{2}+\frac{u^2}{2}+u \geq \frac{u^2}{2}+u \geq \frac{(5-\sqrt{21})^2}{2}+(5-\sqrt{21})
= 28-6\sqrt{21}.
\]](http://latex.artofproblemsolving.com/b/4/7/b4785f87dac778351daf902c356f8c2409418183.png)
![\[
a+b+a^2+b^2 \geq 28-6\sqrt{21}.
\]](http://latex.artofproblemsolving.com/2/8/1/28126ac3dc744ba5841d2065f3965fb53db64d91.png)
(mod 
)
.
.
Let 
.
Let 
.
.
Let 
.