[/quote]
,
satisfying 
,![\[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]](http://latex.artofproblemsolving.com/f/e/8/fe85cf52b883c214806ab837a2eb25cebb511d7b.png)
such that there exists a unique integer 
such that 
with the following property:![\[ n!\mid a^n + 1
\]](http://latex.artofproblemsolving.com/e/2/3/e23d39aab685888993d355d8c93d4e6475f53487.png)
(use in a problem for junior) to practice. Can anyone share to me?
be a square with 
side length and with center 
.
is drawn where point 
.
be an odd integer, 
.
integers 
such that for any 
,
and 
.
for which satisfy 
for some 
,
is greater than 
.
such that![\[f(x+yf(x))+f(xf(y)-y)=f(x)-f(y)+2xy\]](http://latex.artofproblemsolving.com/e/b/c/ebccae7e765d9fc3c8605e58a7399a70127b39b1.png)
, such that 
Prove that :
equality cases : ?
distinct positive integers such that adding the product of any two of them to 
yields a perfect square?
be a sequence of positive integers with 
,
and![\[a_n = a_{n-1}^{a_{n-1}a_{n-2}}-1\]](http://latex.artofproblemsolving.com/1/b/f/1bf1a88979e7d2b33530de979d66a4e81032953d.png)
for all 
.
is a prime less than 
for some positive integer 
,
such that 
.
for which
be an acute angled scalene triangle with circumcircle 
.
and 
respectively. Let 
and 
be points on segments 
and 
Prove that 
and 
.
grid with 
rows and 
columns, and each square of the grid is filled with a non-negative integer. Let 
be the average of all the numbers in the grid. Prove that if 
the there exist two identical rows (meaning same numbers in the same order).
Y by PavelMath, Adventure10
We are given an acute triangle 
with 
and orthocenter 
. The point 
lies symmetric to 
with respect to the altitude 
. Let 
be the intersection of the lines 
and 
. Prove that the circumcenter of the triangle 
lies on the line 
.
(Karl Czakler)
with 
and orthocenter 
. The point 
lies symmetric to 
with respect to the altitude 
. Let 
be the intersection of the lines 
and 
. Prove that the circumcenter of the triangle 
lies on the line 
.(Karl Czakler)
centered at the point 
and with radius 
and center of 
,
the point symmetric to 
and put 
and 
.
.
Now we prove that the points 
,
Finally, let us prove that 
.
.
.
and 
is cyclic.
is symmetric to the circle 
be the foot of altitude from 
.
Also that 
So 
which means that 
which is what we desired 
line 
is both altitude and median, so we have
.
are collinear. 
