[/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 
lies on line 
.
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 
intersect on the circumcircle of 
.
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
In the acute triangle 
, with 
, let 
denote the midpoint of the side ![$[AC], A_1$](//latex.artofproblemsolving.com/5/f/9/5f98e8c14ba68deb751bb5071609140baf48ccca.png)
and 
denote the feet of the altitudes drawn from 
and 
, respectively. Let 
be the intersection point of the tangents in and 
to the circumcircle of triangle 
be the intersection point of lines 
and 
and 
be the intersection point of lines 
and .
a) Prove that is the incircle of triangle 
.
b) The circumcircles of triangles and 
meet again at 
. Prove that the orthocenter 
of triangle is on the line 
.
c) Prove that the point lies on the circumcircle of triangle .
, with 
, let 
denote the midpoint of the side ![$[AC], A_1$](http://latex.artofproblemsolving.com/5/f/9/5f98e8c14ba68deb751bb5071609140baf48ccca.png)
and 
denote the feet of the altitudes drawn from 
and 
, respectively. Let 
be the intersection point of the tangents in and 
to the circumcircle of triangle 
be the intersection point of lines 
and 
and 
be the intersection point of lines 
and .a) Prove that
is the incircle of triangle 
.b) The circumcircles of triangles
and 
meet again at 
. Prove that the orthocenter 
of triangle is on the line 
.c) Prove that the point
lies on the circumcircle of triangle .This post has been edited 1 time. Last edited by parmenides51, Jun 21, 2022, 1:51 AM
