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.![\[{x^3}{y^2}\left( {2y - x} \right) = {x^2}{y^4} - 36\]](http://latex.artofproblemsolving.com/d/5/6/d5668b2a486f5264e1cf2faeb363ac69bf91e3f0.png)
, where 
is a positive integer.
with exactly 
elements such that the sum of its elements is a prime number.
+1 w
satisfies the following condition.
, ![$\sum_{k=1}^{n}\frac{1}{2}\left(1 - (-1)^{\left[\frac{n}{k}\right]}\right)a_k=1$](http://latex.artofproblemsolving.com/3/1/0/3109e3f024dc59c2cdad60efa270494d6095c330.png)
holds.
, compute 
.
such that 
for all 
.
. 
amount of people stand on coordinates 
where 
. Every person got a water cup and two people are considered to be neighbour if the distance between them is 
. At the first minute, the person standing on coordinates 
got litres of water, and the other 
people's water cup is empty. Every minute, two neighbouring people are chosen that does not have the same amount of water in their water cups, and they equalize the amount of water in their water cups. will not have more than 
litres of water.
be positive real numbers such that 
. Prove that 
has centroid 
. A line parallel to 
passing through intersects the circumcircle of at a point 
. Let lines 
and intersect at 
. Suppose a point 
is chosen on such that the tangent of the circumcircle of 
at , the tangent of the circumcircle of at 
and concur. Prove that 
. is called beautiful if, for every integer 
, the base-
representation of contains the consecutive digits 
, 
, , 
(in this order, from left to right). Determine whether the set of all beautiful integers is finite.
such that for all positive integers 
,![\[
f(mn) = \mathrm{lcm}(m, n) \cdot \gcd(f(m), f(n)),
\]](http://latex.artofproblemsolving.com/6/0/7/6071692e2716901e88cc1cfc19ed60388c10701d.png)
where 
and 
denote the least common multiple and the greatest common divisor of 
and 
, respectively. be a function satisfying the following conditions for all 
:![\[
\begin{cases}
f(n+1) > f(n) \\
f(f(n)) = 2n + 2
\end{cases}
\]](http://latex.artofproblemsolving.com/0/4/7/047e28c9d899d7de443ae40ac2c756fda851fafb.png)
Find the value of 
.
differents subjects, each consist of 
students. Given that for any different subjects, there exists a student compete in both subjects. Prove that there exists a student who compete in at least 
different subjects.
diagonals. A regular pentagon has diagonals. is the smallest positive integer such that a regular -gon has greater than or equal to 
diagonals. What is the sum of the digits of ?![\[\mathrm a. ~10\qquad \mathrm b. ~11 \qquad \mathrm c. ~12 \qquad\mathrm d. ~13\qquad\mathrm e. ~14\]](http://latex.artofproblemsolving.com/c/4/c/c4cf4ad27f6cab816910e560c00d7b2204709716.png)
-gon is to consider the 
possible line segments between any points. Of these will be the side-lengths of the polygon and not an interior diagonal, and so we consider 
:
Testing some cases above 
we find that 
is the minimum such solution, giving 
as the solution. possible starting vertices there are 
options for the next non-adjacent vertex to complete the diagonal. Dividing by two for the double-counting of points to lines gives 
as before.
(as the sides are not counted in the diagonals), so works iff 
. Multiplying both sides by gives an equivalence to ![\[n(n-1) - 2n \ge 4048 \iff n^2 - 3n \ge 4048 \iff 4n^2 - 12n \ge 16192 \iff (2n-3)^2 \ge 16201\]](http://latex.artofproblemsolving.com/c/b/b/cbbb0d43a739059ab760b0a178ea0356f6126d6e.png)
, it is equivalent to 
, which for positive implies works iff 
so the answer is 
.
such that 
coprime, 
and 
.
You will get 
and so sum of digits is 
.