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.
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 
, consider the 
fractions 
The product of these fractions equals 
, but if you reciprocate (i.e. turn upside down) some of the fractions, the product will change. Can you make the product equal 1? Find all values of for which this is possible and prove that you have found them all.
be a function satisfying
for all integers 
and 
. Show that there exist positive integers 
and such that 
for all integers .
of the triangle 
which doesn't coincide with 
or 
intersectes the altitudes from 
and 
at 
and 
respectively. Let 
be the reflection of in and 
be the reflection of in 
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 such that the last digits of 
are equal to 
such that for any positive real numbers 
, which are the sides of a triangle, the following inequality holds:![\[
\frac{xy}{x^2 + y^2 + xz} + \frac{yz}{y^2 + z^2 + yx} + \frac{zx}{z^2 + x^2 + zy} \geq C.
\]](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:
, and for all positive integers 
,![\[
a_n = a_{\left\lfloor \frac{n}{3} \right\rfloor} + a_{\left\lfloor \frac{n}{2} \right\rfloor}.
\]](http://latex.artofproblemsolving.com/8/0/0/8005effb2b72a59ce55f49a5204169ec8ffa3b60.png)
Find the sum of all values 
for which there exists a unique positive integer such that 
.
, points 
are constructed as follows: Points 
and 
lie on the extension of side 
such that 
. Points and 
lie on the extension of side 
such that 
. Points 
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 
.
there are elements in an arithmetic progression in the order 
such that 
.
(for 
) form an arithmetic progression. If 
is an injective function, then the following properties hold:
, we have 
(i.e. one-to-one).
is an increasing function such that 
.
holds for any set of distinct positive integers that forms an arithmetic progression.
is an injective function, then the following properties hold: is an increasing function such that .
and we focus on only 
, which 
could be decreasing?
. Then, there exists an 
such that 
due to injectivity. WLOG suppose 
. Now, I claim that there exists an 
such that 
for all 
. To show this, assume the contrary, Then, for any 
, there exists an 
such that 
. Then there exists an 
such that 
by the same logic, etc., so we have 
, impossible. So, there exists some 
such that 
for all 
. In particular, 
. Furthermore, since 
, 
and 
. Thus, 
form an arithmetic sequence and 
, as desired.
there are elements in an arithmetic progression in the order such that .
is finite, there are finitely many 
so that 
. Therefore, for sufficiently large we have that 
, so by our assumption we require 
. But then pick such a , and note the chain![\[ f(k) > f(2k - 1) > f(4k - 3) > f(8k - 7) > \cdots \]](http://latex.artofproblemsolving.com/1/d/2/1d22ac0007dbd2448c76bdc835f0560cc46d5f12.png)
contradicting the well ordering principle.
such that: 
.
and 
.
,