[/quote]
,
,
,
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)
such that ![\[ f(x^2 + yf(z)) = xf(x) + zf(y) , \]](http://latex.artofproblemsolving.com/2/b/a/2baee1d8464992fc38a16561b48b18f4b35807bd.png)
for all 
.
steps and has to reach 
is valid but 
is not valid.
,
.
and 
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 positive integers is called central if for every positive integer 
terms of the sequence is equal to 
,
,
,
of positive integers such that for every central sequence 
,
,
,
.
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
The point 
is chosen inside the triangle 
so that 
. Let 
, 
, and 
be the point of intersection of the line 
with the circle circumscribed around 
, different from 
. Let also 
and 
, where 
is a line passing through the point and parallel to 
. The point 
lies on the other side of the line 
than the point 
. The circle circumscribed around 
, the second intersects the line at a point lying inside the triangle . Prove that 
, that is, the angles 
and 
are equal if and only if the segments 
and 
are equal. .
is chosen inside the triangle 
so that 
. Let 
, 
, and 
be the point of intersection of the line 
with the circle circumscribed around 
, different from 
. Let also 
and 
, where 
is a line passing through the point and parallel to 
. The point 
lies on the other side of the line 
than the point 
. The circle circumscribed around 
, the second intersects the line at a point lying inside the triangle . Prove that 
, that is, the angles 
and 
are equal if and only if the segments 
and 
are equal. .This post has been edited 1 time. Last edited by parmenides51, Jul 1, 2021, 2:06 PM
