such that 
such that![\[ f(x) + f(y) + 2xy = f(x + y) \]](http://latex.artofproblemsolving.com/e/3/d/e3dfbb279ff146cc0923115fe9d3f8d9e3488e17.png)
for all nonnegative real numbers 
and 
.
be a convex quadrilateral such that diagonals 
and 
intersect at right angles, and let 
be their intersection. Prove that the reflections of 
are concyclic.
are either prime numbers or the negatives of prime numbers?
,
,
for 
.![\[A_n=\sum_{i=0}^n s_{i}\]](http://latex.artofproblemsolving.com/3/d/4/3d416f32205acdec15a910d056d4a6e9ad4f85d8.png)
Find 
.
,
,
,
,
are positive integers.
,![\[ \frac{x_{n+1}}{x_n} = \frac{(x_{n+1})^2 + 27}{x_n^2 + 27} \]](http://latex.artofproblemsolving.com/3/0/4/304d56991f9955dc93c16413478b604dd7042ebf.png)
for all positive integers 
.
.
be the maximum value of![\[ \sum_{n=1}^{2025} x_n. \]](http://latex.artofproblemsolving.com/c/4/d/c4da21c0341fdffb56b9a110dd6df874b6e0e8b2.png)
What is 
?
has the property that the sum of its first 
.
be the number of such sequences satisfying 
.
.
has side length 
.
lies on segment 
such that line 
is perpendicular to line 
.
,
,
and 
.
,
,
.
is equal to 
,
be a triangle. Point 
lies on side 
,
lies on side 
,
lies on side 
such that 
,
,
.
be the foot of the altitude from 
to 
,
,![$[AQPR] = \tfrac{3}{7} \cdot [ABC]$](http://latex.artofproblemsolving.com/e/f/1/ef1f89d7ee2b45faac7265d8e2777ba3918ae16e.png)
,
can be expressed as 
and tells Bob that the three resulting remainders are 
,
,
The value of 
can be expressed as 
,
and 
are positive integers such that 
.
and 
inclusive. We say a sequence 
in 
,
that are at least 
is 
.
,
denote the value of the binary number obtained by reading the binary representation of 
such that the equation 
has at least ten positive integer solutions 
such that 
and 
for 
.
.
such that 
and 
is a multiple of 
.
are such that 
and ![\[\frac{x}{26-x^2}=\frac{y}{26-y^2}=\frac{xy}{26-(xy)^2}.\]](http://latex.artofproblemsolving.com/d/7/4/d744d40b3d9e719d30777cc47035fe9e9f6fc44d.png)
What is 
?
.
,
,
.
be the set of permutations of n-degree and 
.
,
(where 
is the identical permutation).
be subgroups of the group 
and 
.
such as 
.
,
for any integer 
and 
is coprime with 
.
for 
.
for some 
where 
.
Y by
Can someone help me with this problem? Let 
. We call a distribution the act of distributing the integers from 
to
represented by tokens to players 
to 
so that they all have the same number of tokens in their urns.
We say that
beats 
when, when and each draw a token from their urn, has a strictly greater chance of drawing a larger number than . We then denote 
. A distribution is said to be chicken-fox-viper when 
What is 
, the number of chicken-fox-viper distributions?
. We call a distribution the act of distributing the integers from 
to
represented by tokens to players 
to 
so that they all have the same number of tokens in their urns.We say that
beats 
when, when and each draw a token from their urn, has a strictly greater chance of drawing a larger number than . We then denote 
. A distribution is said to be chicken-fox-viper when 
What is 
, the number of chicken-fox-viper distributions?
,
represents expected value and 
are the draws of players 
,
.
would satisfy 
with equal probability 
,
.
,
and player 
has 
"
