[/quote]
,
such that ![\[ f(x^2 + yf(z)) = xf(x) + zf(y) , \]](http://latex.artofproblemsolving.com/2/b/a/2baee1d8464992fc38a16561b48b18f4b35807bd.png)
for all 
.
,
,
be an infinite sequence of positive integers. Suppose that there is an integer 
such that, for each 
,
is an integer. Prove that there is a positive integer 
such that 
for all 
.
,
,
be positive real numbers such that 
.![$$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$](http://latex.artofproblemsolving.com/2/0/7/207934bca53047858a3820f37705171ff0abfd45.png)
and 
and 
Prove that![\[\frac{a}{(b+c-a)^2} + \frac{b}{(c+a-b)^2} + \frac{c}{(a+b-c)^2} \geq \frac{3}{(abc)^2}.\]](http://latex.artofproblemsolving.com/0/a/b/0ab4b4e6c28a1ae67788525642661dbcd04d0bd8.png)
steps and has to reach 
is valid but 
is not valid.
,
lie on a line in this order and satisfy 
.
be the midpoints of 
and 
,
is acute, and let 
be its orthocentre. Points 
and 
,
and 
and 
and 
such for any 
.
satisfy,
Where 
}
be an integer. We are given a balance and 
weights of weight 
.
to satisfy that the equation 
has at least 1 roots 
of the equation![\[n^3=p^2-p-1\]](http://latex.artofproblemsolving.com/f/f/8/ff88093bb0aa93afcfdff3ae25149ec5fbacb61f.png)
is a prime number and 
?
be an integer. In a configuration of an 
board, each of the 
cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate 
counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of 
Y by Adventure10
A non-equilateral triangle 
of perimeter 
is inscribed in circle 
.Points 
and 
are arc midpoints of arcs 
and 
, respectively. Tangent to at 
intersects line 
at 
.
It turns out that the midpoint of segment
lies on line 
. Find the length of the segment .
(А. Кузнецов)
of perimeter 
is inscribed in circle 
.Points 
and 
are arc midpoints of arcs 
and 
, respectively. Tangent to at 
intersects line 
at 
.It turns out that the midpoint of segment
lies on line 
. Find the length of the segment .(А. Кузнецов)
This post has been edited 1 time. Last edited by XbenX, Apr 14, 2019, 8:50 AM
and 
be the incenter, 
with 
Clearly, 
is isosceles, where 
is the intersection of 
we have that 
hence 
Because 
we have that 
is the Miquel point of 
hence by some angle chasing, 
is the reflection of 
is perpendicular of 
which is known to be equivalent with 
and 
trivially follows.
,
the bisector of 
.
,
is a median in the right-angled triangle 
,
.
,
intersects 
.
and 
be the midpoints of 
and 
,
,
which means that the triangle 
is isosceles with 
.
,
