[/quote]
,
be a positive integer and let 
be distinct integers in the set 
such that 
for 
.
and 
.
:![\[
2 \left( 5^x + 6^x - 3^x \right) = 7^x + 9^x.
\]](http://latex.artofproblemsolving.com/f/b/6/fb6d5ebb4f448d8eeb73cce3288fea3a56d7f528.png)
,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
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 
.
and 
of non-negative real numbers such that the following three conditions are all satisfied:
;
;
of 
[/list]
is a permutation of 
,
.
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 
.
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 
,![\[
\frac{c^{a_n} - c^{a_{n-1}}}{n} \in \mathbb{Z}
\]](http://latex.artofproblemsolving.com/0/4/4/04464c1a350e3f9a929ab51534018498680068fc.png)
be real positive numbers such that: 
.
satisfying :![\[n^{2p}=m^2+n^2+p+1\]](http://latex.artofproblemsolving.com/7/4/e/74ee592cb8b3f3a3996ea34f40b78fde9d3672fc.png)
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 
for some positive integer 
be integer numbers 
satisfy that
.
is a perfect square number
of a positive integer number
such that:
Y by Adventure10, Mango247
Let 
be an equilateral triangle and let 
be a point outside this triangle, such that 
is an isoscele triangle with a right angle at . A grasshopper starts from and turns around the triangle as follows. From the grasshopper jumps to 
, which is the symmetric point of with respect to 
. From , the grasshopper jumps to 
, which is the symmetric point of with respect to 
. Then the grasshopper jumps to 
which is the symmetric point of with respect to 
, and so on. Compare the distance 
and 
. 
.
be an equilateral triangle and let 
be a point outside this triangle, such that 
is an isoscele triangle with a right angle at . A grasshopper starts from and turns around the triangle as follows. From the grasshopper jumps to 
, which is the symmetric point of with respect to 
. From , the grasshopper jumps to 
, which is the symmetric point of with respect to 
. Then the grasshopper jumps to 
which is the symmetric point of with respect to 
, and so on. Compare the distance 
and 
. 
.
.
,
.
by the distance 2a. Similarly, 
is obtained from 
by the distance 2a and 
from 
by the distance 2a. It follows that the points 
form an equilateral triangle with the side lengths 2a and the sides 
and that the point 
are identical. For similar reasons, 
is obtained from 
is obtained from 
also form an equilateral triangle with the side lengths 2a and the sides 
and that the point 
are identical. Thus the equilateral triangles 
are congruent and have parallel sides and the second is obtained from the first by a translation in the direction 
by the distance 
.
is isosceles, only the fact that 
,
,
is also isosceles. This means that the point 
from the vertex 
and 
.
