[/quote]
,
satisfying
for all 
.
with![\[2x^3zf(z)+yf(y) \ge 3yz^2f(x)\]](http://latex.artofproblemsolving.com/a/b/f/abf10d7e679f6f04db556424a0a0eda26f371fd4.png)
for all 
.
,
for all 
.
are primes, which satisfy:
is an integer
.
![$a,b,c\in[0,1]$](http://latex.artofproblemsolving.com/8/b/d/8bd8f441b63556fdf318ad140bbd3e8e358e9e1d.png)
,
,![\[f(a,b,c)=(1-a^2)^2+(1-b^2)^2+(1-c^2)^2.\]](http://latex.artofproblemsolving.com/7/c/1/7c1611d9b80f5f694afed1894acd57a93aa5b31c.png)
,
be the orthocenter of 
.
at 
.
be the midpoint of 
,
meet 
.
at 
is tangent to 
is positive integer and 
are primes solve the system: 
and 
be an integer. The sequence 
is defined via 
,
and![\[ a_{n+2} = ka_{n+1} + a_n \]](http://latex.artofproblemsolving.com/b/5/e/b5e3e6f9f0b0709113498266e41b372299a2986f.png)
for any positive integer 
of positive integers such that 
.
such that 
and there exists an integer 
such that 
is a multiple of 
.
be reals such that 
Prove that
as 
fir 
.
for all 
.
,s.t :
and orthocenter 
intersect 
respectively. Lines through 
intersects 
at 
respectively. Point 
is a parallelogram. Prove that lines 
and 
are concurrent at a point on 
.
Y by
In a triangle 
angle 
. Point 
is the midpoint of 
, and 
is the foot of altitude from point 
. Points 
and 
are marked such that 
is equilateral, and 
and this points lie in the same half-plane with respect to , not in the same as .
Prove that the circumcircles of
and 
are tangent.
Ivan Korshunau
angle 
. Point 
is the midpoint of 
, and 
is the foot of altitude from point 
. Points 
and 
are marked such that 
is equilateral, and 
and this points lie in the same half-plane with respect to , not in the same as .Prove that the circumcircles of
and 
are tangent.Ivan Korshunau
This post has been edited 2 times. Last edited by nAalniaOMliO, Apr 4, 2025, 4:47 PM
,
