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.
, and 
be positive real numbers, such that 
. Prove the inequality![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
such that ![\[ \left[ \frac{x}{99} \right] = \left[ \frac{x}{101} \right] . \]](http://latex.artofproblemsolving.com/d/b/a/dbaecb29d49c4153c8c03acf03a266eebf3ddb7d.png)
is a polynomial with degree 
. For 
, 
where 
is fixed real numbers such that 
. Find all .
Determine the smallest positive integer 
such that for all positive integers 
to define a sequence of integers 
by letting 
be the smallest integer (strictly) greater than 
making 
into a perfect square. Let 
be the set of positive integers not expressible as the difference of two terms of the sequence 
Prove that 
is finite and determine its size in terms of 
be integers and 
be a prime for which![\[ x^2-3xy+p^2y^2=12p \]](http://latex.artofproblemsolving.com/2/4/c/24cb51eb55144a9a1e0e489d115993496a052e79.png)
.
from the nonnegative integers to the integers satisfying 
and ![\[ f(mn) = f(m) + f(n) + f(m)f(n) \]](http://latex.artofproblemsolving.com/e/f/f/eff10a32cffaec742403ebfc46ff09bc8ef7b7c5.png)
for all nonnegative integers 
and 
.
satisfying the condition: 
with 
, denote its incenter and midpoint of 
by 
, 
, respectively. Let 
be the reflected point of 
wrt 
. Let the lines 
and 
meet at 
. Suppose that 
. Prove that 
.
, 
, with 
, 
. Prove that there exists 
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For and , the notation 
represents 
. ) be the midpoint of side in 
, and 
is such that the quadrilateral 
is cyclic and the circumcircle of 
is tangent to the line 
. If 
is the second common point of the line 
and the circumcircle of , determine the ratio 
.
of positive integers such that 
is divisible by 
.
pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers 
and 
from the board such that:![\[
a^2 + b^2 + 1 = 2ab
\]](http://latex.artofproblemsolving.com/8/f/c/8fc5df8dbc8e1647edfaa96dcfa8a0dfdac0c961.png)
Ways that differ by the order of selection are considered the same. Prove that there exist two numbers 
and 
from the board such that:![\[
c^2 + d^2 + 2025 = 2cd
\]](http://latex.artofproblemsolving.com/2/7/0/2706d72797c5e392b3f068a8174bd257e4d93fa1.png)
pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers and from the board such that:Ways that differ by the order of selection are considered the same. Prove that there exist two numbers and from the board such that:
of nodes, one for each complex number and draw edges if the condition is held.
is form by exactly 
disconnected and simple path.
Condition is equivalent to 
meaning that 
can only be connected to 
and 
, this ensure that is a union of disconnected and simple path. of these path, each of length 
. The we have that 
and 
, then 
.
nodes. So we have a complex number 
and 
and its easy to check that![\[
c^2+(c+45i)^2+2025 = 2c(c+45i)
\]](http://latex.artofproblemsolving.com/a/e/b/aeb34e72265ec8195327125c6953716ab047ab4b.png)
.
