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.
with 
such that ![\[ (a-1)(b-1)(c-1) \]](http://latex.artofproblemsolving.com/f/8/c/f8c33b59bdba1fdd12dec51f16a0b1b93e60754f.png)
is a divisor of 
is called good if there exists a permutation 
of the numbers 
, such that:
and 
have different parities for every 
is a quadratic residue modulo 
for every 
is considered a quadratic residue modulo if there exists an integer 
such that 
.
+2 w
| 
with 
are positive integers and 
is prime (
), we get: 
be a real numbers such that 
Prove that
Where 
be non-negative real numbers. Prove that
Looking for an algebraic solution!
and 
have common external tangents 
and 
meeting at 
. Suppose touches at 
and touches at 
. A circle 
through and intersects again at 
and again at 
, such that quadrilateral 
is convex.
and 
meet at point 
, while lines 
and 
meet at point 
. Show that , , are collinear.
be an acute, scalene triangle with circumcenter 
, circumcircle 
, orthocenter 
. Line 
meets again at 
. Let 
be the midpoint of segments 
respectively. The line through and perpendicular to 
meets line at 
.
meets again at 
. Prove that the intersection of line 
and the perpendicular bisector of 
lies on the circumcircle of triangle 
. and perpendicular to 
meets line at 
. Line 
meets again at 
. Let 
be the intersection of lines 
; 
be the intersection of lines 
; 
be the intersection of lines 
. Let be the intersection of lines 
and 
be the line through and perpendicular to 
. Prove that bisects 
.
such that 
. Show that the equation 
has no integer solution.
, 
, 
, be positive numbers. Prove that
a prime number with 
and 
when 
. then 
else 
.
such that 
. Find the minimum value of:
be a triangle such that 
and 
is obtuse. Point is the circumcenter of triangle , and is the reflection of in . Let be an arbitrary point on line , such that is in between and . Line 
cuts the circumcircle of in 
. Circumcircles of triangles 
and 
cut in 
respectively. Prove that 
is tangent to the circumcircle of triangle 
.
be a prime and 
a nonempty set. Suppose that for each element 
there exist elements , 
such that 
is divisible by . Prove that 
, where 
denotes the cardinality of .
.
![\[
r_1 \ge r_a,\quad r_2 \ge r_b,\quad r_3 \ge r_c
\]](http://latex.artofproblemsolving.com/8/2/1/8214c052ab9ad2c979414f92b6707fa6de33664f.png)
because if any of the radii were smaller than these, the circles would not be able to reach and touch the incircle.![\[
\frac{r_1 + r_2 + r_3}{r} \ge \frac{r_a + r_b + r_c}{r}
\]](http://latex.artofproblemsolving.com/b/b/a/bbadb40e6aa197207abfa907981e0a4a88926969.png)
![\[
\frac{r_a + r_b + r_c}{r} = \frac{h_a - 2r}{h_a} + \frac{h_b - 2r}{h_b} + \frac{h_c - 2r}{h_c}
\]](http://latex.artofproblemsolving.com/5/3/2/5329fec1f02a728b5f34817e0ed43ded663516ae.png)
![\[
= \left(1 - \frac{2r}{h_a}\right) + \left(1 - \frac{2r}{h_b}\right) + \left(1 - \frac{2r}{h_c}\right)
= 3 - 2r \left( \frac{1}{h_a} + \frac{1}{h_b} + \frac{1}{h_c} \right)
\]](http://latex.artofproblemsolving.com/1/e/0/1e087ea87566fb4cf8cd2ae0cda3389247095eea.png)
![\[
\frac{1}{h_a} + \frac{1}{h_b} + \frac{1}{h_c} = \frac{a + b + c}{2S}
\]](http://latex.artofproblemsolving.com/a/4/a/a4ab49131584cf2028270e7248fcbb5d2c6a7adb.png)
![\[
3 - 2r \cdot \frac{a + b + c}{2S} = 3 - \frac{4S}{a + b + c} \cdot \frac{a + b + c}{2S} = 3 - 2 = 1
\]](http://latex.artofproblemsolving.com/0/f/3/0f3ada5ca22deeecc1644d79138942caf04d5134.png)
![\[
\frac{r_1 + r_2 + r_3}{r} \ge 1
\]](http://latex.artofproblemsolving.com/7/1/e/71ea14c8fb8e3860a1db427a0afcdb3b2fd6254b.png)
