AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
+1 w
such that for all 
the following equality holds ![\[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \]](http://latex.artofproblemsolving.com/f/3/a/f3ac860e90d671f9fb592a6bd55c102a3000daec.png)
where 
is greatest integer not greater than 
that verify the following equation :
be the diameter of semicircle 
. Circle 
is internally tangent to and tangent to at 
. Let 
be a point on and 
a point on such that 
and is tangent to . Prove that 
bisects 
be the midpoint of arc 
of the circumcircle of triangle , let 
be the midpoint of and let 
be the incentres of triangles 
and 
. Prove that points 
lie on a circle. be a triangle with 
. Let 
be the intersection point of the internal bisector of angle 
and the circumcircle of . Let 
be the intersection point of the perpendicular bisector of with the external bisector of angle 
. Prove that the midpoint of the segment 
lies on the circumcircle of triangle 
.
. Prove that
Let 
. Prove that
Let 
. Prove that
Let 
. Prove that
be an acute triangle. Tangents on the circumscribed circle of triangle at points and intersect at point 
. Let and 
be a foot of the altitudes from onto and and let be the midpoint of 
. Prove: is the orthocenter of the triangle 
.
cuts 
in half.
be an integer, and let 
be the set ![\[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\]](http://latex.artofproblemsolving.com/f/9/9/f99d070ae4a6efc4812db4997ec9385542512c2a.png)
Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of .
they state the sloution is 
although I dont think its solavable as it relys on the un-assumable asumption that 
. Thoughts?
is undefined, or it can be equal to 1. is undefined, then the left side of the equation is undefined and therefore so is the whole equation. is 1, then 1-0+4 does NOT equal 1.
they state the sloution is although I dont think its solavable as it relys on the un-assumable asumption that . Thoughts?
unsolvable because it relies on the un-assumable that 
?
then at every 
there is or insnt a sloution say True or False, I would consider an equation solved if for every yoiu have detrimened if is True or False. An equation is unsolavable if you canot detrmine at some value if it is true or false. So that equation I would say is unsolabable because at x=1 we cannot detrimen if x gets a true or false
they state the sloution is although I dont think its solavable as it relys on the un-assumable asumption that . Thoughts?
of occurring, Person A wins.
of occurring, Person A wins.
of occurring, Person A wins.
of occurring. Now, note that there is no possible way for someone to win with less than 
points at this point, so we can ignore that condition.
chance of obtaining a 
-
Solving yields 
.
(as a percent) as a blind guess.
and 
,
.
.
satisfying
Prove
it doesn't work anyways
is solvable but omg we can't assume 
so it's unsolvable