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.
:![\[
2 \left( 5^x + 6^x - 3^x \right) = 7^x + 9^x.
\]](http://latex.artofproblemsolving.com/f/b/6/fb6d5ebb4f448d8eeb73cce3288fea3a56d7f528.png)
, 
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.
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: 
satisfying :![\[n^{2p}=m^2+n^2+p+1\]](http://latex.artofproblemsolving.com/7/4/e/74ee592cb8b3f3a3996ea34f40b78fde9d3672fc.png)
and are integers and 
is a prime number.
be a positive integer and let 
be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of around the circle such that the product of any two neighbors is of the form 
for some positive integer .
be integer numbers 
satisfy that:
. Prove that: are even
is a perfect square number can’t be any power 
of a positive integer number
such that for all 
, we have![\[f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y).\]](http://latex.artofproblemsolving.com/a/a/1/aa1bce053788a3efa59819407b41cfe472f544d0.png)
be a convex quadrilateral and draw regular triangles 
, the first two outward and the other two inward. Prove that 
. What can be said about the quadrilateral 
?
, by spiral symmetry 
and 
, so 
. Thus we have 
. Similarly 
, and 
, so is a parallelogram.
such that:
such that 
