[/quote]
,
be pairwise different positive real numbers such that![\[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\]](http://latex.artofproblemsolving.com/8/a/e/8ae0bd1f55eb2fb0b7209b62739a9b98f462ee56.png)
is an integer for every 
Prove that 
is a perfect square.
and 
touches each other externally at a point 
,
is inscribed in 
,
are tangent to 
and 
respectively. Line 
bisects 
and meets segment 
at 
.
meets the arc 
(not passing through the point 
)
different from 
.
.
on the 
-
.
the distance between 
and the line 
.
.
in terms of 
.
moves in the range of 
,
.
,
such that the inequality :![\[f(x) \leq f(x)^3-2f(x)^2+2\]](http://latex.artofproblemsolving.com/1/2/c/12c4c4c850c1c2886f002dfcb0bead075e2ab679.png)
.
of positive integers satisfying both the following properties:
,
(where 
)
for which ![\[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\]](http://latex.artofproblemsolving.com/4/b/6/4b66fe235e9a56cc646dbd38aae72f8ddfeb77de.png)
[/list]
such that, for every positive integer 
the integer
is a multiple of 
(Note that 
denotes the greatest integer less than or equal to 
For example, 
and 
)
be an acute triangle with circumcenter 
and orthocenter 
,
.
in 
be the midpoint of 
.
.
and 
Prove that
and 
.
grid lies a coin with its tail side up. A move consists of the following steps.
square in the grid;
for which it is possible that every coin shows head-side up after a finite number of moves.
and 
.![\[P=\frac{1}{5-2a}+\frac{1}{5-2b}+\frac{1}{5-2c}.\]](http://latex.artofproblemsolving.com/c/3/7/c3708768a84c2bbb1e4fae502a9bb1e92cb13d02.png)
be the incentre of acute-angled triangle 
,
at 
,
respectively. Let line 
and 
,
.
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
