[/quote]
,
is an integer. Prove that 
are integers.
(
well). i want to study so if you guys have some easy or normal problems please send me
and 
.
.
of degree 
.![$[a;b]$](http://latex.artofproblemsolving.com/0/0/6/0066f4b2714f88cdaf883b00f877aa1355b7f9ef.png)
and Maxim will give in response the maximum value of 
![\[
v_p(n!) \leq \frac{n}{p - 1}
\]](http://latex.artofproblemsolving.com/5/7/c/57ca4d0a6532fb4b97687a97b9c1fb35ad02b9f2.png)
and 
Prove that
Let 
and 
Prove that
not isosceles, the incircle 
.
is the intersection of 
.
is the projection of 
onto 
cut 
.
is the projection of 
onto 
.
cut 
respectively. Prove 
are collinear
be the set of non-negative integers and 
be the set of positive real numbers. Let 
be a function such that 
and 
for all integers 
,
for all integers 
.
.
which satisfy 
![$ \frac{a+b+c}{3} \ge \sqrt[10]{\frac{a^3+b^3+c^3}{3}}$](http://latex.artofproblemsolving.com/c/d/a/cdae05ad34239366b0dd3921243522c711b6ae58.png)
method, I can assum 
![$ \frac{2x+\frac{1}{x^2}}{3} \ge \sqrt[10]{ \frac{2x^3+ \frac{1}{x^6}}{3}} $](http://latex.artofproblemsolving.com/c/d/4/cd481672bee1e87d8c97ac623436e0ec75cfa73a.png)
are given such that 
.
in natural numbers m and n.
denote the smallest possible value of 
where 
are sets such that 
and 
whenever 
.
and 
intersect at 
of the two circles touches 
at 
respectively. Let 
be the symmetric points of 
intersects circles 
respectively which are distinct from 
and 
are equal.
Y by
An acute triangle is a triangle that has all angles less that 
( is a Right Angle). Let 
be a right-angled triangle with 
Let 
be the altitude from 
to 
and let 
be the intersection of the angle bisector of 
with 
Let 
be the altitude from to 
Prove that the circumcircle of 
passes through the midpoint of 
( is a Right Angle). Let 
be a right-angled triangle with 
Let 
be the altitude from 
to 
and let 
be the intersection of the angle bisector of 
with 
Let 
be the altitude from to 
Prove that the circumcircle of 
passes through the midpoint of 
This post has been edited 1 time. Last edited by PEKKA, Jan 19, 2024, 4:17 PM
Reason: latex
Reason: latex
are concyclic. Let 
and let 
Then angle chasing tells us that 
and we're done
be 
.
the angle 
.
.
.
.
,
.
,
is cyclic. 
,
(angles sum in right 
)
because of (1)
is isosceles with base 
cyclic
as 
)
