[/quote]
,
.
be a point on the perpendicular bisector of 
(see figure) and 
,
be the intersections of the perpendicular bisectors of 
and 
.
and 
and the point 
is given. Let 
be set of all 
segments of real line of type ![$[i, j]$](http://latex.artofproblemsolving.com/a/6/4/a644edd67caa59eaded765c417b6d9822f782ec2.png)
,
and 
are integers, 
.
is said to be valuable if the intersection of any two segments from 
is either empty, or is a segment of nonzero length belonging to 
rows of coins. There is 1 coin in the first row, 2 coins in the second row, 3 coins in the third row, and so on, up to 
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 
.
Y by Mango247
Points 
and 
are marked on the sides 
and 
of a triangle 
, respectively. Let 
be the intersection point of the segments 
and 
. Let 
and 
be the areas of the triangles 
and 
, respectively (see the fig.). Prove that :
a)
b)
and 
are marked on the sides 
and 
of a triangle 
, respectively. Let 
be the intersection point of the segments 
and 
. Let 
and 
be the areas of the triangles 
and 
, respectively (see the fig.). Prove that :a)
b)
This post has been edited 1 time. Last edited by parmenides51, Dec 25, 2020, 4:14 PM
and 
With 
and 
where 
are positive reals less than 
we get 
and 
Therefore, 
![$xz/y^2>9\Longleftrightarrow 9(k\mu)^2-[9(k+\mu)-10]k\mu-(k+\mu)<0.$](http://latex.artofproblemsolving.com/b/7/b/b7bfda2112771ea2c9a6cafa90074fbe59f34fd3.png)
The discriminant of the quadratic is ![$\Delta=[9(k+\mu)-10]^2+36(k+\mu)$](http://latex.artofproblemsolving.com/a/d/9/ad9b18035f31559d74f6398db2bf75cc2c5637da.png)
and the solutions are 
It is now easy to show that 
and that 
and therefore this inequality holds.
then 
with equality for 
![$xz/y=[ABC],$](http://latex.artofproblemsolving.com/b/8/2/b82102c6870144910acbd3179b9251aed1ed56ed.png)
which can be found in this ![$\frac{y}{[ABC]}\le \frac{5\sqrt{5}-11}{2}.$](http://latex.artofproblemsolving.com/3/3/6/336d472a9312b8216f7f04d9bd4e0e113fdd126b.png)
For comparison note that for an equilateral triangle where 
are the midpoints of the sides 
respectively, we have ![$\frac{y}{[ABC]}=\frac{1}{12}.$](http://latex.artofproblemsolving.com/0/f/9/0f9f2996a514615f4dfdeec7c7dc55cef1943ab1.png)
I don't know why, but I find this to be interesting 
.
