[/quote]
,
such that the following equality holds for any real numbers 
and 
.
such that for all 
,![\[xf(x+f(y))=(y-x)f(f(x)).\]](http://latex.artofproblemsolving.com/e/4/a/e4a3bbb8b91d2aa62d699c24df342fa59be71915.png)
be positive real numbers. Prove that
,
What is the measure of angle 
?
Expanding gives
Regroup terms: 
Since 
,
,
this becomes
so 
Thus 
or 
To prove 
then 
so 
and 
which implies 
contradicting the first equation. Hence, 
and 
.
of real numbers such that the following conditions hold for all 
:
![$x,y\in[0,\frac{\pi}{2}]$](http://latex.artofproblemsolving.com/1/6/8/1682b58d11bfb7d88fb826a857bb2bf8a8ac9969.png)
satisfy 
and 
.
are acute, we have: ![\[\cos x = \sqrt{1-\sin^2 x} = \sqrt{1-(5/13)^2} = 12/13\]](http://latex.artofproblemsolving.com/c/9/f/c9fa5dfcd05736b8d89e40f2fa30bdba8b9bf898.png)
and ![\[\cos y =\sqrt{1-\sin^2 y}= \sqrt{1-(5/17)^2}= 8/17\]](http://latex.artofproblemsolving.com/8/c/0/8c01f9c21b6126105a491931a68c7e1b12df1276.png)
So:
and 
Hence:
be a quadrilateral with 
parallel to 
and 
.
and 
intersect at a point 
.
distinct from 
such that 
.
distinct from 
lies on the circumcircle of triangle 
such that 
.
and 
intersect at 
.
is parallel to 
.![\[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \]](http://latex.artofproblemsolving.com/a/1/4/a14adf0f1e35df57c734c7df701d5a67da22dc3c.png)
,
,
satisfies 
.![\[
\frac{a+b}{\sqrt{ab(1-ab)}} + \frac{b+c}{\sqrt{bc(1-bc)}} + \frac{c+a}{\sqrt{ca(1-ca)}} \le \frac{\sqrt{2}}{abc}
\]](http://latex.artofproblemsolving.com/4/0/f/40f8383eb6f59646ef6c41b662e9b35831664d5a.png)
such that for all 
the following holds :![\[ f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y) \]](http://latex.artofproblemsolving.com/a/9/9/a99ec11be5ab98c79efcf7a420e46e0584c8dc01.png)
,
,
.
,
bisects 
.
lies on 
,
bisects 
.
.
: 
?
Y by
Charlyn walks completely around the boundary of a square whose sides are each 
km long. From any point on her path she can see exactly 
km horizontally in all directions. What is the area of the region consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the nearest whole number?
Most people jsut subtract the inner square 3*3 but why are there no semicircle areas in the empty spot in the middle?
km long. From any point on her path she can see exactly 
km horizontally in all directions. What is the area of the region consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the nearest whole number?
Most people jsut subtract the inner square 3*3 but why are there no semicircle areas in the empty spot in the middle?
This post has been edited 1 time. Last edited by Spacepandamath13, Jun 6, 2025, 2:08 AM
