[/quote]
,
is satisfied. Point 
is taken on the circumcircle of 
such that 
.
which passes from 
and 
respectively at 
and 
.
passes from circumcenter of 
.
rectangles that can be covered without gaps and without overlaps with hooks such that
is the foot of the altitude from 
of triangle 
and 
are marked such that the circumcircles of triangles 
and 
are tangent, call this circles 
and 
respectively. Tangent lines to circles 
.
.
![\[{x^3}{y^2}\left( {2y - x} \right) = {x^2}{y^4} - 36\]](http://latex.artofproblemsolving.com/d/5/6/d5668b2a486f5264e1cf2faeb363ac69bf91e3f0.png)
.
and 
is chosen on 
at 
.
be an acute triangle with 
,
be the center of its circumcircle. Let 
be the reflection of 
with respect to 
.
at 
,
intersects the line through 
.
be a point on the ray 
,
.
.
be an even integer, and let 
be an arbitrary set of 
distinct integers. Define![\[
E(V,n)
\;=\;
\bigl\{(u,v)\in V\times V : u < v,\ u+v = n^k\text{ for some }k\in\mathbb{N}\bigr\}.
\]](http://latex.artofproblemsolving.com/4/b/6/4b65bed83b027596f154741763fb518e2415863d.png)
For each even 
.
be a triangle with side‐lengths 
,
,
.
be the areas of triangles 
,
,
,![\[
\frac{abc}{12R}
\;\le\;
\frac{P_1^2 + P_2^2 + P_3^2}{P}
\;\le\;
\frac{3R^3}{4\sqrt[3]{abc}}.
\]](http://latex.artofproblemsolving.com/1/3/9/139a3e15f08afb78ca0b8301f24d30db8f202207.png)
such that
divides 
for every 
,
we have![\[
f\bigl(f(a)+kb\bigr)\;=\;f\bigl(a + k\,f(b)\bigr).
\]](http://latex.artofproblemsolving.com/c/f/f/cff41352422c8765978b8fe6dc966ba934a9df49.png)
be positive real numbers. Prove the inequality![\[
\frac{x_1 + 3x_2}{x_2 + x_3}
\;+\;
\frac{x_2 + 3x_3}{x_3 + x_4}
\;+\;
\frac{x_3 + 3x_4}{x_4 + x_1}
\;+\;
\frac{x_4 + 3x_1}{x_1 + x_2}
\;\ge\;8.
\]](http://latex.artofproblemsolving.com/5/c/0/5c062a37f6ff120232ae87eb0093a4ce73ee0eb1.png)
.
(measured between centers), provided it does not overlap any other leaf. If the lake is large enough that edge effects never interfere, what is the least number of days required to have 
leaves in the lake?
lie the following points: 
and 
on 
,
on 
,
on 
.![\[
P = AM\cap BN,\quad
R = KM\cap LN,\quad
S = KN\cap LM,
\]](http://latex.artofproblemsolving.com/7/e/a/7ea90a0f1261ed7445bbac58fb0cdd76810259f3.png)
and let the line 
meet 
.
,
are collinear.
such that 
for all 
.
be a natural number, and let 
subdivided into 
unit squares. Determine for which values of 
Y by
Let 
be a given positive integer. Luca, the lazy flea sits on one of the vertices of a regular 
-gon. For each jump, Luca picks an axis of symmetry of the polygon, and reflects herself on the chosen axis of symmetry. Let 
denote the number of different ways Luca can make jumps such that she returns to her original position in the end, and does not pick the same axis twice. (It is possible that Luca's jump does not change her position, however, it still counts as a jump.)
a) Find the value of if is odd.
b) Prove that if is even, then
![\[P(n)=(n-1)!\cdot n!\cdot \sum_{d\mid n}\left(\varphi\left(\frac{n}d\right)\binom{2d}{d}\right).\]](//latex.artofproblemsolving.com/3/0/9/30992e5982bce4a032d5e96e0fb5fce8ab6ff21a.png)
Proposed by Péter Csikvári and Kartal Nagy, Budapest
be a given positive integer. Luca, the lazy flea sits on one of the vertices of a regular 
-gon. For each jump, Luca picks an axis of symmetry of the polygon, and reflects herself on the chosen axis of symmetry. Let 
denote the number of different ways Luca can make jumps such that she returns to her original position in the end, and does not pick the same axis twice. (It is possible that Luca's jump does not change her position, however, it still counts as a jump.)a) Find the value of
if is odd.b) Prove that if
is even, then![\[P(n)=(n-1)!\cdot n!\cdot \sum_{d\mid n}\left(\varphi\left(\frac{n}d\right)\binom{2d}{d}\right).\]](http://latex.artofproblemsolving.com/3/0/9/30992e5982bce4a032d5e96e0fb5fce8ab6ff21a.png)
Proposed by Péter Csikvári and Kartal Nagy, Budapest
