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 reals such that 
and 
Find the value of 
be a triangle with circumcentre 
. The points 
lie in the interiors of the sides 
respectively, such that 
is perpendicular to 
and 
is perpendicular to 
. (By interior we mean, for example, that the point 
lies on the line 
and is between 
and 
on that line.)
be the circumcentre of triangle 
. Prove that the lines 
and are perpendicular.
, a line 
to tangent , and another circle 
disjoint from such that and lie on opposite sides of . The tangents to from a variable point 
on meet at 
and 
. Prove that, as varies over , the circumcircle of 
is tangent to two fixed circles.
and we define 
as the set of all products of an element of and an element of . For example, if 
and 
then 
We call a set 
of positive integers good if there do not exist sets 
of positive integers, each with at least two elements and such that the sets 
and are the same. Prove that the set of perfect powers greater than or equal to 
is good.
, , no two elements are equal, but any two or three of these sets may have common elements. A perfect power is an integer of the form 
, where 
and 
are integers.)
be an acute triangle. Let 
be the midpoint of side , and let 
and 
be the feet of the altitudes from and , respectively. Suppose that the common external tangents to the circumcircles of triangles 
and 
intersect at a point , and that lies on the circumcircle of . Prove that line 
is perpendicular to line .
and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
and 
with conditions,
is a prime number.
is a perfect square.
is also perfect square.
be a sequence of positive real numbers such that for every 
, we have:![\[
a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i}
\]](http://latex.artofproblemsolving.com/e/3/0/e30e9e533eb9a41cf847484e676ac4522edda665.png)
Prove that there exists a natural number 
such that for all 
, the following holds:![\[
a_n < \frac{1}{1404}
\]](http://latex.artofproblemsolving.com/0/1/f/01f1a81eed2230332d51a15501ef2a6ad3a7ec82.png)
such that:
for all 
such that:
for all
is a natural number. A function 
is called \textit{
-friendly} if for fewer than 1\% of the integers 
with 
, the equation 
has a solution in natural numbers 
such that ![\( \frac{y_0}{x_0} \in \left[\frac{1}{100}, 100\right] \)](http://latex.artofproblemsolving.com/d/a/6/da67f464a934f890626d188b84940fc82c4f4ff9.png)
, where 
is a solution. Suppose 
, where 
is a polynomial with real coefficients, negative leading coefficients, and total degree greater than 2, and for every real number 
, we have 
as ![\( \frac{y}{x} \in \left[\frac{1}{100}, 100\right] \)](http://latex.artofproblemsolving.com/6/5/d/65d4154dd5fb612f92f24f98406bc9bcfc713806.png)
. Prove that for sufficiently large , the function 
is not -friendly.
such that for all we have:
be defined by![\[
x_1 = 2,\quad x_{n+1} = x_n + \frac{n}{x_n},\quad \text{for all } n \geq 1.
\]](http://latex.artofproblemsolving.com/f/2/8/f28193d8ea3fa6da8bbdb51f9a7e2e9d8a7d3c9f.png)
Prove that the sequences 
and 
have finite limits, and find those limits.
be the number of positive integers that divide the integer 
. For all positive integral divisors 
of 
, what is the sum of numbers 
?
.
.
.