[/quote]
,
such that ![\[f(xy-1)+f(x)f(y)=2xy-1,\]](http://latex.artofproblemsolving.com/8/8/8/888ca39f2b7f8cec6d6426bee28d40eade40a66e.png)
for any real numbers 
and 
,
fractions 
The product of these fractions equals 
,
be a function satisfying
for all integers 
and 
.
and 
for all integers 
such that: 
.
of the triangle 
which doesn't coincide with 
or 
intersectes the altitudes from 
and 
at 
and 
respectively. Let 
be the reflection of 
be the reflection of 
Prove that the circles 
and 
are tangent.
be positive real numbers such that 
Prove that we have![\[(a+b)(b+c)(c+a)\geq 8.\]](http://latex.artofproblemsolving.com/2/a/6/2a63f19de634fce70ff4ebc02ad13d442a20c378.png)
such that for any positive integer 
there exists a positive integer 
are equal to 
such that for any positive real numbers 
,![\[
\frac{xy}{x^2 + y^2 + xz} + \frac{yz}{y^2 + z^2 + yx} + \frac{zx}{z^2 + x^2 + zy} \geq C.
\]](http://latex.artofproblemsolving.com/9/1/c/91c8a4af2bd2f3107c8921d2e06f8a09dedd0164.png)
![\[
f : \mathbb{R}^+ \to \mathbb{R}
\]](http://latex.artofproblemsolving.com/e/7/b/e7bfc5b236fb6f00ef328f850b1b14632cbf8416.png)
satisfying the condition that for all 
,![\[
f(x + y^2) \geq f(x) + y.
\]](http://latex.artofproblemsolving.com/3/a/a/3aa20083835682dbd81be692eba65cab19e923e5.png)
be defined as follows:
,
,![\[
a_n = a_{\left\lfloor \frac{n}{3} \right\rfloor} + a_{\left\lfloor \frac{n}{2} \right\rfloor}.
\]](http://latex.artofproblemsolving.com/8/0/0/8005effb2b72a59ce55f49a5204169ec8ffa3b60.png)
Find the sum of all values 
for which there exists a unique positive integer 
.
and 
.
,
are constructed as follows: Points 
and 
lie on the extension of side 
such that 
.
lie on the extension of side 
such that 
.
and 
lie on the extension of side 
such that 
.![\[
\angle ACE + \angle FBD + \angle EAC = 180^\circ.
\]](http://latex.artofproblemsolving.com/c/b/8/cb88e7d6ec129d50bde13e6a08ee16b275f42042.png)
be positive real numbers satisfying![\[
a^2 + c^2 = b(a + c).
\]](http://latex.artofproblemsolving.com/9/c/0/9c0730d15b1177ee23d3f65bc2fe8b2cc85d0c62.png)
Let![\[
m = \min \left( \frac{a^2 + ab + b^2}{ab + bc + ca} \right).
\]](http://latex.artofproblemsolving.com/e/b/6/eb6f340defa4b38e35292062b202d138b9de843b.png)
Find the value of 
.
Y by laf1234
Suppose an infinite non-constant arithmetic progression of integers contains 
in it. Prove that there are an infinite number of perfect cubes in this progression. (A perfect cube is an integer of the form 
, where 
is an integer. For example, 
, 
and are perfect cubes.)
in it. Prove that there are an infinite number of perfect cubes in this progression. (A perfect cube is an integer of the form 
, where 
is an integer. For example, 
, 
and are perfect cubes.)
for some constant 
.
.
for every 
did I fakesolved? I dont know but this is so easy?
be an arithmetic progression with 
is an integer. Take an integer 
,
,
