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 an infinite sequence of real numbers satisfying the equation 
for all 
, where and are two different positive reals., , , ... be bounded?
and 
in a non-isosceles triangle 
with incenter 
intersect its circumcircle 
at 
and 
, respectively. The line through , parallel to 
, intersects 
at 
. Prove that 
is tangent to .
be the circumcircle of right-angled triangle (
). The tangent to at point intersects the line 
at point . Suppose that 
is the midpoint of the minor arc 
, and 
intersects for the second time in 
. The tangent to at point intersects at 
. Prove that 
.
such that 
and 
, evaluate 
.
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
. Prove that the points 
and lie on the same line.
be fixed points on this order on a line. Let be a variable circle through and 
and suppose it meets the perpendicular bisector of 
at the points 
and 
. Let 
and 
be the other points of intersection of 
and 
with . Prove that 
passes through a fixed point independent of .
be a positive integer and let 
be distinct integers in the set 
such that divides 
for 
. Prove that does not divide 
:![\[
2 \left( 5^x + 6^x - 3^x \right) = 7^x + 9^x.
\]](http://latex.artofproblemsolving.com/f/b/6/fb6d5ebb4f448d8eeb73cce3288fea3a56d7f528.png)
of non-negative real numbers such that the following three conditions are all satisfied:
;
;
of such that 
[/list]
is a permutation of 
, and they are both permutations of 
. Note that any list is a permutation of itself.
denote the sum of digits function. Does there exist 
such that![\[
n, s(n), \dots, s^{2024}(n)
\]](http://latex.artofproblemsolving.com/c/3/3/c33803a77912ed4dde5a5c0ae05934496f04707f.png)
are all distinct perfect squares?
having integer coefficients. On each move, Vasya pays him a ruble, and calls an integer 
of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equation 
. The game continues until Petya is forced to repeat an answer. What minimal amount of rubles must Vasya pay in order to win?
that satisfies:![\[
\sum_{i=1}^{n} a_{\left\lfloor \frac{n}{i} \right\rfloor} = n^k
\]](http://latex.artofproblemsolving.com/b/b/e/bbe7fe81e366fe390be5d17193d1581cc40dbece.png)
be a positive integer. Prove that for all integers 
, we have:![\[
\frac{c^{a_n} - c^{a_{n-1}}}{n} \in \mathbb{Z}
\]](http://latex.artofproblemsolving.com/0/4/4/04464c1a350e3f9a929ab51534018498680068fc.png)
be real positive numbers such that: 
. Prove that: 
be a triangle. Find the point on its side and the point 
on its side such that the area of triangle 
equals to the area of the quadrilateral 
, and the segment 
has minimum possible length.
be a triangle. Find the point on its side and the point on its side such that the area of triangle equals to the area of the quadrilateral , and the segment has minimum possible length.
and 
.
is the geometric mean of 