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.
be a right angled triangle with 
.Let 
be the midpoint of 
, and 
be an arbitrary point on 
. The reflection of 
over 
intersects lines 
and at 
and 
, respectively. The circumcircles of 
and intersect again at 
. Prove that the center of the circumcircle of 
lies on 
.
define 
if 
or 
, and 
where 
are all prime factors of . Find all polynomials 
with nonnegative integer coefficients such that 
divides 
for every nonnegative integer .
satisfy the equation 
. Prove that there exist positive integers 
such that
be a given integer. Prove that infinitely many terms of the sequence 
, defined by ![\[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\]](http://latex.artofproblemsolving.com/8/3/2/832113b1ef1fe211e1eaead25129873aead1dab4.png)
are odd. (For a real number 
, 
denotes the largest integer not exceeding .) there exists an -digit number divisible by 
all of whose digits are odd.
, 
, 
, 
be distinct positive integers, 
. Prove that there exist distinct indices 
and 
such that 
does not divide any of the numbers 
, 
, , 
.
, 
, 
be fixed positive integers. There are 
ducks sitting in a ducks ducks picking paper, and ducks picking scissors., , and , the maximum number of moves which could take
be the circumcenter of triangle . Line 
intersects the altitude from 
at point 
. Let 
be the midpoints of 
, respectively. If 
intersects at 
, and the circumcircle of triangle 
meets at 
, prove that 
is cyclic. be a triangle with incentre 
and circumcentre . Let 
be the touchpoints of the incircle with , 
, respectively. Prove that 
is the Euler line of 
.
. E,F are located on AB such that 
. Prove that the C-excircle of triangle ABC touches the circumcircle (EFC).
is tangent to 
through 
then it is known that 
Inversion with center 
and power 
swaps circle 
and the sideline and takes C-excircle into itself. Since 
and are tangent, then by conformity 
and are tangent as well.
Prove that
Let 
Prove that![$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$](http://latex.artofproblemsolving.com/f/5/e/f5e68204507aa9c3e71c28844bbee3a804546b94.png)
![\[
\text{Find all functions } f : \mathbb{R} \to \mathbb{R} \text{ such that:} \\
f(a^4 + a^2b^2 + b^4) = f\left((a^2 - f(ab) + b^2)(a^2 + f(ab) + b^2)\right)
\]](http://latex.artofproblemsolving.com/2/a/7/2a73a5e58eab4a994fbd83160fb79a0b00152951.png)
.
and 
at 
and 
respectively. It is well-known that 
.
,
and 
map to themselves and the circumcircle of 
maps to 
.
is tangent to the 
and the 
meets the 
is the tangency point between the