AoPS Academy
be a triangle with incenter 
, and let 
be a point on side 
. Points 
and 
are chosen on lines 
and 
respectively such that 
is a parallelogram. Points 
and 
are chosen on side such that 
and 
are the angle bisectors of angles 
and 
respectively. Let 
be the circle tangent to segment 
, the extension of 
past , and the extension of 
past . Prove that is tangent to the circumcircle of triangle .
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 
.
plane, let:
that is divided into trapezoids 
and 
by segment (for 
).![$[ABD] = \frac{kh}{2}=\sqrt{3}$](http://latex.artofproblemsolving.com/e/5/3/e534f058ee89bf76d59e56476106d187b9734c01.png)
. If ![$[AEFD] = \left(\sqrt{3}+\frac{1}{3-\sqrt{3}}\right)[EBCF]$](http://latex.artofproblemsolving.com/3/b/9/3b987310cc14e2eed73eebbde00f56e92a6d2221.png)
, then we obtain:![$\frac{h/2}{2}\left[k + \left(k-\frac{h\cot(A)}{2}-\frac{h\cot(D)}{2}\right)\right] = \left(\sqrt{3}+\frac{1}{3-\sqrt{3}}\right) \cdot \frac{h/2}{2}\left[\left(k-\frac{h\cot(A)}{2}-\frac{h\cot(D)}{2}\right) + (k-h\cot(A)-h\cot(D))\right]$](http://latex.artofproblemsolving.com/a/9/a/a9aa4074cdce94d5eb5934fd96ba609d953454f0.png)
;![$2k + -\frac{h\cot(A)}{2}-\frac{h\cot(D)}{2} = \left(\sqrt{3}+\frac{1}{3-\sqrt{3}}\right) \cdot \left[2k-\frac{3h\cot(A)}{2}-\frac{3h\cot(D)}{2}\right]$](http://latex.artofproblemsolving.com/e/9/c/e9c0fe881acfe91ce47d48f8f4fb6295de62aa47.png)
;![$\frac{4\sqrt{3}}{h} -\frac{h\cot(A)}{2}-\frac{h\cot(D)}{2} = \left(\sqrt{3}+\frac{1}{3-\sqrt{3}}\right) \cdot \left[\frac{4\sqrt{3}}{h}-\frac{3h\cot(A)}{2}-\frac{3h\cot(D)}{2}\right]$](http://latex.artofproblemsolving.com/f/7/c/f7c4c2b49d30fd3f0c91aa98182aaead97e901b4.png)
;![$8\sqrt{3} - h^{2}[\cot(A)+\cot(D)] = \left(\sqrt{3}+\frac{1}{3-\sqrt{3}}\right) \cdot [8\sqrt{3} - 3h^{2}[\cot(A)+\cot(D)]$](http://latex.artofproblemsolving.com/2/d/2/2d2dca5fb41f5d88f5039ab52ba03bf7697d855d.png)
;![$[\cot(A)+\cot(D]h^2 = \frac{56-8\sqrt{3}}{1+7\sqrt{3}}$](http://latex.artofproblemsolving.com/4/a/f/4af6b85c410f195503493def5e8ab84f5d0c5bf3.png)
(i).![$[ABCD] = \frac{h}{2}[2k-h\cot(A)-h\cot(D)] = hk-\frac{[\cot(A)+\cot(D]h^2}{2}$](http://latex.artofproblemsolving.com/4/e/b/4ebecf688f97ea5d4ff48dc0f184535312a154a7.png)
;
.
such that: 
.
