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.
, such that 
and 
.
such that 
is prime, 
with 
and ?
be a function such that for every regular 
-gon 
we have 
. Prove that 
for all reals 
.
be a triangle (
). 
are the altitudes of the triangle. The bisector of 
intersects 
at 
. Let 
be a point such that 
and 
. Prove that 
passes through the midpoint of 
.
then prove \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{2}\cdot\frac{abc}{ab+bc+ca}\ge \frac{5}{3}.$$
with 
and at most one of them being zero, show that![\[\frac{1}{\max(a^2,b)}+\frac{1}{\max(b^2,c)}+\frac{1}{\max(c^2,a)}\geq\frac{27}{4}\]](http://latex.artofproblemsolving.com/9/e/1/9e14a9ef2d1a0a98006bcf716fa33c3d9ea29bd5.png)
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 
. Prove that
Equality holds when 
or 
and 
, Ivica and Marica play the following game: Given two piles of cookies, on each turn a player takes 
cookies from one of the piles, of which he eats 
and puts of is arbitrary in every move. Players take turns alternatively, with Ivica going
for which Marica has a winning strategy.
and 
Prove that
Let and 
Prove that
Let and 
Prove that
Let and 
Prove that
m and four small circles with radii 
m. It is to be painted using the three shown colours. What is the cost of painting the figure?
m and four small circles with radii m. It is to be painted using the three shown colours. What is the cost of painting the figure?
m and four small circles with radii m. It is to be painted using the three shown colours. What is the cost of painting the figure?
& 
.
![$$[\text {dark grey region}]=[\text {large circle}] - 4[\text {small circle}] + [\text {light grey region}]$$](http://latex.artofproblemsolving.com/0/3/9/0390a9d95d74e3e3dc280268036ea185bdaf144b.png)
Note that we have to add the grey region back because the 4 circles are overlapping. Therefore if we subtract the area of the 4 circles then we will subtract too much of the area. Now we can substitute the values of the large circle and the small circle since we know the radii of these:![$$[\text {dark grey region}]= 2^2\pi - 4(1^2\pi) + [\text {light grey region}]$$](http://latex.artofproblemsolving.com/c/d/0/cd0d621e6e38411d4a0d67d70abd24070496fa31.png)
![$$[\text {dark grey region}]=4\pi - 4\pi + [\text {light grey region}]$$](http://latex.artofproblemsolving.com/6/8/3/683468e58cf7e21a8bdd22d9e5b6c0cdbaf08071.png)
![$$[\text {dark grey region}]=[\text {light grey region}]$$](http://latex.artofproblemsolving.com/8/f/4/8f4fcbdb9e74a4ae76c2894677e9c41b1722a149.png)
by drawing radii 
,
,
,
.
such that it makes a diagonal of the square.![$[\text {Area of sector AOB}]$](http://latex.artofproblemsolving.com/b/3/2/b32e67e52dc91db0a0f5da15ad1e46d448f65fae.png)
and subtract ![$[\triangle AOB]$](http://latex.artofproblemsolving.com/f/a/9/fa9efa06a95ec088e40bbd9c57dbb7349be50b77.png)
from that. In other words the area of the light grey region is:![$$\frac {1}{4}[\text {area of light grey region}] = 2 ([\text {Area of sector AOB}] - [\triangle AOB]).$$](http://latex.artofproblemsolving.com/0/c/8/0c8dfd26f2e3c0fd7dd6e0f964839b63d4a3b2f0.png)
![$$\frac {1}{4}[\text {area of light grey region}] =2((90^\circ/360^\circ)(1^2)\pi - (1)(1)/2$$](http://latex.artofproblemsolving.com/a/c/5/ac55c49c0a5bcb736a46258fce99e037c3c24c6d.png)
![$$\frac {1}{4}[\text {area of light grey region}] =2(\frac {1}{4} \pi - \frac {1}{2})$$](http://latex.artofproblemsolving.com/0/6/7/067c2cbf7681ca740c2ed25551c6db36e4020763.png)
![$$\frac {1}{4}[\text {area of light grey region}] =\frac {1}{2} \pi - 1$$](http://latex.artofproblemsolving.com/5/7/e/57ec9e15f336b621c278726fb05a0fc58aedc9c1.png)
![$$[\text {area of light grey region}]=4(\frac {1}{2} \pi - 1) = 2\pi - 4$$](http://latex.artofproblemsolving.com/e/f/6/ef68d74e1a6226de70a819b398229308ef941f12.png)
We can say that the area of the light grey region equals the area of the dark grey region because we proved that above:![$$[\text {area of light grey region}] =2\pi - 4=[\text {area of dark grey region}].$$](http://latex.artofproblemsolving.com/6/1/c/61c2a6e2d300e8de970a455eb8bc5f1e2481fd25.png)
so the area of all the 4 small circles are 
.![$$[\text {area of medium grey region}]=4\pi - (2\pi - 4) =2\pi +4$$](http://latex.artofproblemsolving.com/5/6/b/56bf82b5525bd7f0dd0bef11e488a1f03074cf8a.png)
dollars.
,
,
be the areas of the dark gray, light gray, and white regions, respectively.
However, we can get that 
.
,
Now, the answer is just
However, this is just
Krona
as the lowest and highest have same area
supposed to be?
We have 
of these sections, so the area would be 
For the 
smaller circles, each has an area of 
but the white sections are each counted twice. That makes the light gray region 
Finally, the dark gray region is the large circle minus the areas of the white and light gray sections. This goes to
Then apply the costs and we get
,
.
... I think I found my mistake![$$[\text {area of medium grey region}]=4\pi - 2(2\pi - 4) =8$$](http://latex.artofproblemsolving.com/9/6/e/96e7e382bbd7fdfa1904f11afa26a3cd6f0ce718.png)
