[/quote]
,
:o
so 
and since 
is just 
therefore 
so 
(you must use all numbers) and the operations 
ONLY to create all positive integers in order. I will repeat 1 more time: ONLY these 4 operations (no concatenation or anything like totient etc.)
for which there exists a degree 
.
be a positive integer larger than 
,
be integers. It is known that the equation 
has 
and 
are relatively prime.
such that the following set has exactly 2024 elements:![\[
\left\{ \frac{x}{a} + \frac{y}{b} : x, y \text{ are integers with } 0 \leq \frac{x}{a} + \frac{y}{b} \leq 1 \right\}.
\]](http://latex.artofproblemsolving.com/f/6/9/f694675b283928c19a1de6ae886df098eb2cc12d.png)
be an acute triangle with circumcircle 
and 
a point on the minor arc 
.
and 
be the midpoints of 
and 
,
and 
intersect 
and 
,
be the second intersection point of the circumcircles of 
and 
)
,
such that if 
is a integer number with 
and coprime to 
,
such that:![\[ f(x + f(x + y)) = x + f(x) - f(y), \]](http://latex.artofproblemsolving.com/7/0/4/70460ee37a69df33554d994a18ac622cd7bf269f.png)
for all real numbers 
.
of real numbers.
grid 
.
.
in the 
.
.
for all 
,
for all 
.
.
is the circumcenter, 
is the orthocenter. Construct the circumcircles of triangles 
and 
,
,
and 
are congruent, and their nine-point circles coincide.
such that ![\[ f( xf(x) + f(y) ) = f^2(x) + y \]](http://latex.artofproblemsolving.com/9/7/5/975510041aa2af0cfbfefac1ad76f0a4cf12b0ee.png)
for all 
.
Y by
During my math final today at school, the question said stated "When every integer is raised to the power of zero, it is equal to 1". The answers were multiple choice and were : Always, sometimes, never, and I don't know.
I ended up putting the first one, and was informed that it was incorrect. My teacher told me that
is not equal to one. I looked it up, and it said 
. Can someone confirm and prove this. Thanks!
I ended up putting the first one, and was informed that it was incorrect. My teacher told me that
is not equal to one. I looked it up, and it said 
. Can someone confirm and prove this. Thanks!
,
